Interplay between phonons and anisotropic elasticity drives negative thermal expansion in PbTiO$_3$

TL;DR

This study reveals that negative thermal expansion in PbTiO$_3$ results from a complex interaction between phonon behavior and anisotropic elasticity, challenging previous assumptions about the necessary conditions for NTE.

Contribution

It demonstrates that phonon properties alone are not sufficient or necessary for NTE, highlighting the role of anisotropic elasticity in this phenomenon.

Findings

NTE in PbTiO$_3$ involves interplay between phonon properties and elasticity.

Rigid unit phonon modes are not essential for NTE.

New insights into the thermal behavior of materials and NTE mechanisms.

Abstract

We use first-principles theory to show that the ingredients assumed to be essential to the occurrence of negative thermal expansion (NTE) -- rigid unit phonon modes with negative Gr\"{u}neisen parameters -- are neither sufficient nor necessary for a material to undergo NTE. Instead, we find that NTE in PbTiO$_3$ involves a delicate interplay between the phonon properties of a material (Gr\"{u}neisen parameters) and its anisotropic elasticity. These unique insights open new avenues in our fundamental understanding of the thermal properties of materials, and in the search for NTE in new materials classes.

Tables

Table 1. Table 1: Selected elements of the compliance tensor for PbTiO 3 and SnTiO 3 from first-principles calculations in units of 10 − 3 superscript 10 3 10^{-3} GPa -1 .

	$S_{11}$	$S_{12}$	$S_{13}$	$S_{33}$
PbTiO3	7.44	0.49	-11.94	55.69
SnTiO3	7.79	-1.45	-6.83	31.36

Equations

$${\gamma }_{s,\mathbf{k}}^{ij}\equiv -\frac{1}{{\omega }_{s,\mathbf{k}}}\frac{\partial {\omega }_{s,\mathbf{k}}}{\partial {\epsilon }_{ij}},$$

$${\gamma }_{\mathrm{bulk}}^{ij}=\frac{{\sum }_{s,\mathbf{k}}{\gamma }_{s,\mathbf{k}}^{ij}{c}_{s,\mathbf{k}}}{{\sum }_{s,\mathbf{k}}{c}_{s,\mathbf{k}}},$$

$${\alpha }_v=\frac{{\gamma }_{\mathrm{bulk}}{C}_{\eta }}{3B},$$

$${\gamma }_{s,\mathbf{k}}\equiv -\frac{V}{{\omega }_{s,\mathbf{k}}}\frac{d{\omega }_{s,\mathbf{k}}}{dV}.$$

$$\alpha_{a}=(C_{\eta}/V)\bigg{[}(S_{11}+S_{12})\gamma^{a}_{\mathrm{bulk}}+S_{13}\gamma^{c}_{\mathrm{bulk}}\bigg{]},$$

$$\alpha_{c}=(C_{\eta}/V)\bigg{[}2S_{13}\gamma^{a}_{\mathrm{bulk}}+S_{33}\gamma^{c}_{\mathrm{bulk}}\bigg{]},\newline$$

$$2(S_{11}+S_{12}+S_{13}){\gamma }_{\mathrm{bulk}}^a+(S_{33}+2S_{13}){\gamma }_{\mathrm{bulk}}^c<0.$$

$$\left[\begin{array}{c}{\alpha }_a\\ {\alpha }_a\\ {\alpha }_c\end{array}\right]=\left[\begin{array}{ccc}{S}_{11}& S_{12}& S_{13}\\ S_{12}& S_{11}& S_{13}\\ S_{13}& S_{13}& S_{33}\end{array}\right]\left[\begin{array}{c}\frac{C_{\eta }{\gamma }_{\mathrm{bulk}}^a}{V}\\ \frac{C_{\eta }{\gamma }_{\mathrm{bulk}}^a}{V}\\ \frac{C_{\eta }{\gamma }_{\mathrm{bulk}}^c}{V}\end{array}\right].$$

Full text

The first report of negative thermal expansion (NTE) in solids appeared in 1907, with Scheel’s observations that quartz and vitreous silica shrink upon heating Scheel (1907a, b). NTE has since been observed in essentially every materials class, including metals Guillaume (1897), polymers Bruno et al. (1998), metal-organic frameworks Goodwin et al. (2008); Dubbeldam et al. (2007); Han and Goddard (2007), and semiconductors Jiang et al. (2004); Biernacki and Scheffler (1989); Dolling and Cowley (1966). However, despite over a century of study Barrera et al. (2005); Lind (2012); Barron et al. (1980), the microscopic mechanisms of NTE remain poorly understood in all but a handful of cases Tucker et al. (2005); van Schilfgaarde et al. (1999).

In inorganic framework materials, so-called rigid unit phonon modes (RUMs) Pryde et al. (1997); Miller et al. (2009) are widely recognized as being important drivers of NTE. Rather than longitudinal stretching of bonds, these modes typically shorten metal-metal bond distances as temperature increases through relative rotations of rigid (or almost rigid) polyhedral groups, or kinking of oxygen-metal-oxygen bond networks. The RUM model of NTE successfully accounts for the thermal behavior of the canonical NTE material ZrW2O8 Tucker et al. (2005); Alexandra et al. (1996); Mary et al. (1996), as well as a number of zeolites Hammonds et al. (1998); Lightfoot et al. (2001) and Prussian blue materials Goodwin et al. (2005). A second essential ingredient for the occurrence of NTE appears to be the existence of low-frequency RUMs with large, negative Grüneisen parameters, that is, modes with frequencies that decrease with decreasing volume. In fact, negative Grüneisen parameters are sometimes claimed to be prerequisites for NTE behavior Yamamura et al. (2009); Mittal et al. (2017). Indeed, as far as we know, the only exceptions are the elemental metals Zn McCammon and White (1965), Cd McCammon and White (1965), As White (1972), and Sb White (1972), all of which exhibit only uniaxial, rather than volumetric, NTE.

Given the apparent importance of RUMs and phonons with negative Grüneisen parameters, it is curiously rare for ABO3 perovskites to undergo NTE. Most perovskites undergo one or more structural phase transitions involving phonons with strong RUM-like character (typically rotations of the BO6 octahedra), and these types of modes are associated with negative Grüneisen parameters in, for example, $\beta$-cristobalite, $\beta$-quartz, and ZrW2O8 Dove and Fang (2016). In contrast, recent work Abilitt et al. (2017); Senn et al. (2015) has shown that (uniaxial) NTE is common in layered perovskites, such as Ruddlesden-Popper phases, because the combined effects of layering and rotations of the BO6 octahedra enhance their elastic anisotropy compared to bulk perovskites; this enhanced elastic anisotropy appears to be the origin of uniaxial NTE in layered perovskites. Of course, bulk perovskites may exhibit elastic anisotropy, as Refs. Abilitt et al., 2017 and Senn et al., 2015 acknowledge, however the magnitude of the anisotropy is generally not large enough to induce NTE.

In this Letter, we use the ferroelectric perovskite PbTiO3 to demonstrate that neither RUMs nor phonons with negative Grüneisen parameters are necessary (or sufficient) in order for a material to undergo NTE. A key discovery we make is that in non-cubic materials, NTE cannot be predicted based on either the signs or magnitudes of an individual Grüneisen parameter or elastic constant by itself, as is commonly assumed. In these systems, the thermal expansion along a given axis is coupled to multiple Grüneisen parameters through multiple independent elastic constants. We use theory and first-principles calculations to elucidate the microscopic mechanism of NTE in PbTiO3 and show that few of the modes critical to driving NTE have negative Grüneisen parameters and that they are not RUMs, even though PbTiO3 contains many RUM-like modes Hammonds et al. (1998). We then connect the physical mechanism of NTE in PbTiO3 to its electronic structure, and show that its elastic properties are dominated by the stereochemical activity of the Pb2+ 6$s^{2}$6$p^{0}$ lone electron pair. Our results are striking because they suggest that PbTiO3 is unique among well-studied NTE materials, and appears to be the only material that exhibits volumetric NTE well above room temperature, and with positive Grüneisen parameters along all unique crystallographic axes. These results bring clarity to the driving mechanism of NTE in PbTiO3, and broaden the search for new NTE systems to include materials overlooked in the past due to a lack of large, negative Grüneisen parameters.

PbTiO3 is cubic at high temperatures but undergoes a phase transition at 760 K to a tetragonal (P4mm) ferroelectric phase; this phase exhibits volumetric NTE down to approximately room temperature (though only the $c$ axis decreases with temperature while the $a$ axes increase, the net effect is a decrease in volume). We used density functional theory, as implemented in Quantum Espresso with GBRV pseudopotentials Giannozzi et al. (2009); Garrity et al. (2014) (see Supplementary Information (SI) for methods and convergence critera) to calculate the evolution of lattice parameters as a function of temperature in the quasiharmonic approximation (QHA). Figure 1 compares our results obtained from three functionals: LDA Kohn and Sham (1965), PBEsol Ernzerhof and Scuseria (1999) and Wu-Cohen (WC) Wu and Cohen (2006). As expected, all three functionals slightly underestimate the lattice parameters, however each qualitatively reproduces the experimental trend of a shrinking $c$ axis and lengthening $a$ axis as temperature increases. Our calculations reproduce the experimentally observed volumetric NTE, and additional comparisons between data from our first-principles calculations and available experimental data for finite-temperature phonon dynamics and elastic constants indicate that the QHA is a justified approximation for this system (see SI). We report all remaining results for the WC functional only, since it best captures the structural properties of PbTiO3. With this functional, we calculate a volumetric expansion coefficient $\alpha_{v}$ of $-2.29\times 10^{-5}$ K*-1* between 500 K and 700 K, which compares favorably with $\alpha_{v}=-1.8\times 10^{-5}$ K*-1* and $-1.99\times 10^{-5}$ K*-1* from Shirane and Hoshino (1951) and Chen et al. (2005), respectively.

We now turn to elucidating the microscopic mechanism of NTE in PbTiO3. We first calculated the Grüneisen parameters for the equilibrium structure of PbTiO3 at 300 K. Since PbTiO3 is not cubic, the Grüneisen parameter $\gamma$ has a tensor form,

[TABLE]

where $\omega_{s,\mathbf{k}}$ is the frequency of mode $s$ at wavevector $\mathbf{k}$, $\varepsilon_{ij}$ is a strain consistent with crystal symmetry, and $i$ and $j$ are Cartesian directions. We calculated $\gamma^{ij}_{s,\mathbf{k}}$ using a central difference, which required three separate calculations of the full dispersion curve. The quantity usually referred to as the ‘bulk’ Grüneisen parameter is then defined as Barron and Munn (1967),

[TABLE]

where $c_{s,\mathbf{k}}$ is the mode specific heat at constant configuration. We note that the uniaxial stress perturbation method Romao (2017) could be suited to high-throughput studies to screen for NTE along particular axes, and to provide a quick alternative method to compare with results obtained using the strain perturbative definitions as in Eq. (1).

Figure 2 shows phonon dispersion curves for PbTiO3 at 300 K with the magnitude and sign of $\gamma^{a}_{s,\mathbf{k}}\equiv\gamma^{11}_{s,\mathbf{k}}$ and $\gamma^{c}_{s,\mathbf{k}}\equiv\gamma^{33}_{s,\mathbf{k}}$ represented by the thickness and color of the band, respectively. Despite the presence of NTE, both bulk Grüneisen parameters are positive, $\gamma^{a}_{\mathrm{bulk}}$ = 1.42 and $\gamma^{c}_{\mathrm{bulk}}$ = 0.40. The density of states in Figure 2 indicates that the individual phonon modes contributing most strongly to $\gamma^{a}_{\mathrm{bulk}}$ and $\gamma^{c}_{\mathrm{bulk}}$ are low frequency modes with positive Grüneisen parameters, in contrast with the usual expectation that modes with large, negative mode Grüneisen parameters drive NTE. As shown in the Supplementary Information, if we force all modes below 100 cm*-1* to be perfectly harmonic by keeping their frequencies constant with temperature, while allowing the frequencies of all other modes to change, then NTE behavior is completely suppressed. Hence, these low-frequency modes appear to be the primary drivers of NTE and of the positive bulk Grüneisen parameters along both the $a$ and $c$ axes. Additionally, Figure 3 shows that none of the distortions associated with the critical low-frequency modes are RUMs; they are instead dominated by translational Pb motion. Ghosez et al. (1999) Our work so far raises two questions: how do we account for NTE in PbTiO3, given the positive Grüneisen parameters, and do electronic effects play a role in NTE in this material?

Although large, negative mode Grüneisen parameters have been associated with NTE in the literature, this relationship need not hold for anisotropic systems. In cubic materials, the bulk Grüneisen parameter is scalar and the coefficient of volumetric thermal expansion can be expressed as Ashcroft and Mermin (1976)

[TABLE]

where $B$ is the bulk modulus and $C_{\eta}$ is the specific heat at constant configuration. The bulk Grüneisen parameter $\gamma_{\mathrm{bulk}}$ is still defined as in Eq. (2), however the individual mode Grüneisen parameters are reduced to simple volume derivatives,

[TABLE]

Looking at Eq. (3), $\gamma_{\mathrm{bulk}}$ is the only term on the right-hand side that can be negative. Hence, a negative $\alpha_{v}$ in cubic systems is impossible without a negative $\gamma_{\mathrm{bulk}}$, which must itself arise from large, negative mode Grüneisen parameters.

Anisotropic materials have additional structural degrees of freedom that complicate the relationship between the coefficient of thermal expansion, Grüneisen parameters, and elastic constants. For example, in tetragonal systems like PbTiO3, the coefficients of thermal expansion along $a$ ($\alpha_{a}$) and $c$ ($\alpha_{c}$) are defined as Wallace (1972); Munn (1972):

[TABLE]

and

[TABLE]

where $S_{ij}$ is the $ij^{th}$ component of the elastic compliance tensor $\mathbf{S}$ in Voigt notation. Since $\alpha_{v}\equiv 2\alpha_{a}+\alpha_{c}$ Wallace (1972), in order for $\alpha_{v}$ to be negative, we require

[TABLE]

Eq. (7) shows that negative bulk Grüneisen parameters are not a prerequisite for NTE in anisotropic materials. The Sij can be positive or negative, and the relative sign and magnitude of these terms constrain the signs of $\gamma^{a}_{\mathrm{bulk}}$ and $\gamma^{c}_{\mathrm{bulk}}$ required for NTE. Without knowledge of the full compliance tensor, no definitive statements about how the signs or magnitudes of $\gamma^{a}_{\mathrm{bulk}}$ and $\gamma^{c}_{\mathrm{bulk}}$ relate to $\alpha_{v}$ can be made. However, the dependence of $\alpha_{v}$ on $\gamma^{c}_{\mathrm{bulk}}$ and $\gamma^{a}_{\mathrm{bulk}}$ can be determined directly if Sij is known.

Table 1 shows selected elements of the compliance tensor for the tetragonal phase of PbTiO3 from first principles calculations, which compare well with experiment Kalinichev et al. (1997) (note the large magnitude of $S_{13}$, previously correlated Goodwin et al. (2008); Abilitt et al. (2017) with NTE behavior). The $a$-axis expands as temperature rises, and so $\alpha_{a}$ is positive. From Eq. (5), since $S_{11}+S_{12}>0$ for PbTiO3, $\gamma^{a}_{\mathrm{bulk}}$ must be large and positive in order for $\alpha_{a}>0$. Since positive Grüneisen parameters are known to be associated with positive thermal expansion, our results so far are unsurprising. However, we will now argue that it is the tendency of PbTiO3 to expand along $a$ with increasing temperature that drives the system to shrink along $c$, giving rise to volumetric NTE.

The key to understanding NTE in PbTiO3 is to examine the relationship between the Grüneisen parameters, and applied stress along the $a$ axis and induced strain along the $c$ axis. Since the $c$ axis contracts as temperature increases, $\alpha_{c}<0$. The conventional wisdom suggests that this requires a negative $\gamma^{c}_{\mathrm{bulk}}$, while we know that $\gamma^{c}_{\mathrm{bulk}}$ is actually positive in this system. From Eq. (6), since $S_{33}$ is also positive, the only way for $\alpha_{c}$ to be negative is for the product $S_{13}\gamma^{a}_{\mathrm{bulk}}$ to be negative – indeed the case for PbTiO3, as shown in Table 1. Physically, a large, negative $S_{13}$ in tetragonal systems functions in a similar way to a large, positive Poisson’s ratio in cubic systems. It is this temperature-induced negative strain that drives $\alpha_{c}$ to be negative, and gives rise to NTE in PbTiO3.

If $S_{13}\gamma^{a}_{\mathrm{bulk}}$ was positive, or even slightly less negative, then it would be energetically favorable for the $c$ axis to expand with increasing temperature. Hence, the NTE is not due to a $c$ axis driven to contract by a negative $\gamma^{c}_{\mathrm{bulk}}$, but in spite of a positive $\gamma^{c}_{\mathrm{bulk}}$. We illustrate this by rewriting equations 5 and 6 in matrix form:

[TABLE]

The column vector on the right has units of pressure/Kelvin – a temperature-induced stress – linked through the compliance tensor to the column vector on the left, which has units of strain/Kelvin. In this arrangement, $\gamma^{a}_{\mathrm{bulk}}$ and $\gamma^{c}_{\mathrm{bulk}}$ define the magnitude of a thermal stress normal to the $a$ and $c$ axes, respectively, and are linked to thermal strain through $\mathbf{S}$. Just as a mechanical stress state involving positive stress along both axes could result in negative strain along $c$ due to coupling through $S_{13}$, thermal stress driven by positive $\gamma^{a}_{\mathrm{bulk}}$ and $\gamma^{c}_{\mathrm{bulk}}$ could result in $\alpha_{c}<0$, where the sign of $\alpha_{c}$ depends on a careful balance between all of $S_{ij}$, $\gamma^{a}_{\mathrm{bulk}}$ and $\gamma^{c}_{\mathrm{bulk}}$.

Understanding that NTE in PbTiO3 arises from a careful balance between $S_{ij}$, $\gamma^{a}_{\mathrm{bulk}}$, and $\gamma^{c}_{\mathrm{bulk}}$ terms, we can explore the origin of the sign and magnitude of these parameters through investigating a similar ABO3 perovskite, SnTiO3. Previous ab initio studies predict that SnTiO3 also adopts a $P4mm$ ground state structure at 0 K Parker et al. (2011). Though synthesis of bulk SnTiO3 has proven difficult Matar et al. (2009); Konishi et al. (2002) and few reliable experimental measurements of physical properties exist, it still provides a useful comparison to illuminate the origin of NTE in PbTiO3. We find that within the QHA, SnTiO3 exhibits repressed NTE behavior compared to PbTiO3, and positive or near-zero volumetric NTE at every temperature (see SI).

The behavior of SnTiO3 can be linked to terms in the compliance matrix (Table 1) and bulk Grüneisen tensor ($\gamma^{c}_{\mathrm{bulk}}=7.72$, $\gamma^{a}_{\mathrm{bulk}}=3.42$). SnTiO3 has similar compliance as PbTiO3 along the $a$ axis, but is much stiffer (lower compliance) along $c$ and exhibits much less coupling between stress along $a$ and strain along $c$ (a lower $S_{13}$). SnTiO3 exhibits a slightly more positive $\gamma^{a}_{\mathrm{bulk}}$ at 300 K than PbTiO3 ($3.42$ to $1.42$), but a $\gamma^{c}_{\mathrm{bulk}}$ over an order of magnitude larger ($7.72$ to $0.40$). Referring to Eqs. (7) and (6), we can confirm that these differences should result in a system where NTE is suppressed compared to PbTiO3.

As $\gamma^{a}_{\mathrm{bulk}}$ and $\gamma^{c}_{\mathrm{bulk}}$ are vibrational, dynamic properties, the difference between these values in each material could be due to either the mass difference between Sn and Pb (118.71 a.u. and 207.2 a.u., respectively) or from differences in electronic structure, primarily the tendency of the A-site cation to form stereochemically active lone pairs. We rule out mass difference as the driving effect by performing a computational “experiment” in which the QHA is performed for PbTiO3, but with the mass of Pb changed to that of Sn (see SI). PbTiO3 then exhibits lattice parameters as a function of temperature nearly identical to those for the original system in Figure 1, and a 500-700 K $\alpha_{v}$ of $-2.38\times 10^{-5}K^{-1}$, which is within 5% of $\alpha_{v}$ for the original system. Though this experiment is not physically realizable, the results show that the absence of NTE in SnTiO3 cannot be due to the mass difference between Pb and Sn alone.

Previous studies Walsh et al. (2011); Stoltzfus et al. (2007); Pitike et al. (2015); Waghmare et al. (2003) have shown that the n$s^{2}$n$p^{0}$ lone electron pair on Pb and Sn is stereochemically active and is responsible for driving the symmetry-breaking distortion that lowers the symmetry of each material from cubic to tetragonal. The Pb/Sn-$s$-anion-$p$ interaction at the top of the valence band is actually antibonding and therefore energetically destabilizing. These states could be stabilized by mixing with the unfilled Pb/Sn-$p$ states at the bottom of the conduction band, but such mixing is forbidden by symmetry in the cubic perovskite structure. This electronic instability (known as a pseudo- or second-order Jahn-Teller distortion) manifests in the lattice as a structural phase transition: the ferroelectric transition lowers the atomic site symmetries such that Pb/Sn-$s$-anion-$p$-Pb/Sn-$p$ mixing is allowed. The resulting localized, non-bonding state at the top of the valence band is the lone pair.

The closer the cation-$s$ and anion-$p$ states are in energy, the more cation-$s$ character will be present in the anti-bonding states, and the greater the stabilization gained from hybridization between these anti-bonding states and the unfilled cation-$p$ states. Since the Sn 5$s$ states are closer in energy to the O 2$p$ states than the Pb $6s$ states are Walsh et al. (2011), SnTiO3 exhibits a stronger tendency towards a structural distortion than PbTiO3. In fact, Crystal Orbital Hamiltonian Population (COHP) analysis Dronskowski and Blöchl (1993); Deringer et al. (2011); Maintz et al. (2013, 2016) shows that while the Jahn-Teller distortion makes the Pb-$s$ – O-$p$ interaction slightly less antibonding, in the case of SnTiO3, it brings the Sn-$s$ – O-$p$ interaction from antibonding in the cubic phase to bonding in the tetragonal phase. Since these states are filled, there is a larger energy gain associated with the Jahn-Teller distortion in SnTiO3 compared with PbTiO3, and a larger corresponding structural distortion.

Next, we link the size of the structural distortion in SnTiO3 to $S_{13}$ and its NTE behavior. It is well known in the ferroelectrics literature Ederer and Spaldin (2005); Lee et al. (2007) that $c/a$ in tetragonal BaTiO3 can be enhanced signficantly with epitaxial strain, thereby signficantly enhancing ferroelectric polarization. In contrast, $c/a$ (and therefore polarization) in PbTiO3 is quite insensitive to epitaxial strain. This behavior is attributed to the fact that PbTiO3 in the tetragonal phase is already very structurally distorted, and hence resists further distortions. Since $c/a$ for SnTiO3 (1.18) is larger than PbTiO3 (1.08), we predict that $S_{13}$ for SnTiO3 is less negative than for PbTiO3 because SnTiO3 is even more resistant to further structural distortions than PbTiO3. We tested this hypothesis by calculating selected elements of the compliance tensor for a hypothetical SnTiO3 structure with a $c/a$ ratio set to that of PbTiO3. Compared with SnTiO3 with its equilibrium $c/a$, this hypothetical structure has a more negative $S_{13}$, a larger (more positive) $S_{33}$, and a $\gamma^{c}_{bulk}$ of -1.71. Inserting these values of $S_{ij}$ and Grüneisen parameters into Eq. (5) and (6), we find $\alpha_{a}$ = 7.5 ($\times$10*-4K-1*) and $\alpha_{c}$ = -26.0 ($\times$10*-4K-1*). Since $\alpha_{v}\equiv 2\alpha_{a}+\alpha_{c}$, this hypothetical SnTiO3 structure should exhibit volumetric negative thermal expansion. Hence, the electronic structure of the material is important in determining the size of the structural distortion away from the high-symmetry phase. However, in the distorted structure, it is primarily the shape of the unit cell that determines the elastic properties and Grüneisen parameters (see SI for further details).

Our results suggest that revisiting anisotropic systems with positive bulk Grüneisen parameters may be a promising direction in the search for new NTE materials. Since the number of unique indices of $\mathbf{S}$ and $\gamma^{ij}_{\mathrm{bulk}}$ are larger for Bravais lattices of lower symmetry, the number of degrees of freedom in expressions like Eq. (7) increases, providing more pathways for NTE. We have also shown that, critically, as volumetric NTE arises from the product of multiple $\gamma^{ij}_{\mathrm{bulk}}$ and $S_{ij}$, a search for NTE materials that prioritizes either large magnitudes or particular signs of certain indices of these tensors will miss favorable candidates. Additionally, in light of previous work Nielsen et al. (2013) suggesting that the presence of lone pair electrons reduces thermal conductivity, exploring the link between phonon modes driving NTE in PbTiO3 and those driving thermal expansion could yield new insights with respect to mode-level control of thermal properties.