Impact of crystal structure on the thermoelectric properties of n-type SrTiO₃

Abstract

We present a detailed first-principles study of the electrical and thermal transport, and the thermoelectric figure of merit zT, of the oxide perovskite SrTiO₃ in the orthorhombic Pnma, tetragonal I4/mcm and cubic Pm3̄m phases. Analysis of the lattice thermal conductivity shows that the “particle-like” contribution, κ_p, is highest in the Pm3̄m phase due to larger phonon group velocities. We also find that all three phases show significant heat transport through glass-like interband tunnelling. On the other hand, we predict the cubic and orthorhombic phases to show superior n-type conductivity, due to significantly stronger polar-optic phonon scattering and shorter electron lifetimes in the tetragonal phase. Due to its superior electrical properties, we predict that the Pm3̄m phase will attain a 25% larger high-temperature zT than the I4/mcm phase, while we predict the best zT can be obtained for the Pnma phase due to its favourable electrical properties and low κ_latt. This work provides new insight into the impact of structure type on the thermoelectric performance of oxide perovskites, and indicates targeting particular structure types, e.g. through chemical doping, could provide a facile route to optimising the zT of SrTiO₃ and related systems.

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1 Introduction

Climate change due to anthropogenic greenhouse gas (GHG) emissions is the foremost scientific and technological challenge of the 21^st century. In 2024, global temperatures exceeded 1.5 °C relative to the pre-industrial 1990 baseline for the first time, with clear – and devastating – consequences from an increased prevalence of extreme weather events.¹ The majority of GHG emissions come from burning fossil fuels for energy, and achieving net zero emissions is key to mitigating climate change. This requires significant changes to how we produce and consume energy. In 2023, the US produced 27.4 Petawatt hours (PWh) of energy, with 82.4% derived from fossil fuels. Of this, only 34.5% was used in energy services, and the remaining 65.7% was lost as waste heat.^2–4 This scale of energy waste means that technologies to improve energy efficiency are an important enabler of net zero.

Thermoelectric (TE) power harnesses the Seebeck effect in a TE material to converts heat to electricity and thus recover waste heat as electrical energy. A thermoelectric generator (TEG) combines two TE materials with dominant p-type (hole) and n-type (electron) carriers, connected electrically in series and thermally in parallel, to harvest electrical energy from the temperature gradient between a heat source and sink. TEGs are solid state devices with no moving parts, are maintenance free and easily scalable, and produce no noise or emissions during operation, making them suitable for a wide range of applications. These include, but are not limited to: battery-free power sources for wireless sensors, wearables and medical devices;^5–7 heat recovery from automobile powertrains and data centres;^8,9 and as a primary geothermal power source.¹⁰ The conversion efficiency of a TE material is typically measured by the dimensionless figure of merit zT:

A large zT requires a high (absolute) Seebeck coefficient |S| and electrical conductivity σ to optimise the power factor S²σ (PF), and a low electronic and lattice thermal conductivity κ = κ_latt + κ_el. High σ is characteristic of metals, whereas large |S| is typically associated with semiconductors and insulators, and low κ is associated with amorphous materials (glasses). The optimum physical properties for high-performance TEs therefore straddle three traditionally distinct classes of material. The three electrical properties are a product of the electronic structure, and the σ/κ_el and |S| are typically proportional and inversely proportional, respectively, to the charge carrier concentration n.¹¹ The κ_latt depends on the structure and chemical bonding and is typically minimised in materials composed of heavy elements, with weak chemical bonding, and/or with features such as “active” lone pairs that lead to strong phonon anharmonicity.^12,13

Currently, the most widely used thermoelectric (TE) materials are Bi₂Te₃ for near-room temperature applications (zT from ∼1–1.5 at 350–450 K)^4,14–16 and PbTe for high-temperature power generation (zT up to 2.2 at 915 K),¹⁷ due to a combination of a favourable electronic structure and intrinsically low thermal conductivity.^18,19 However, Te is scarce and the environmental toxicity of Pb raises potential concerns over end-of-life disposal, which restrict these materials to niche applications.²⁰ Widespread adoption of TEGs thus requires alternative materials that balance high performance with sustainability and cost efficiency.

Oxides are attractive candidate thermoelectric materials due to their low toxicity, low cost, and exceptional chemical stability, the latter of which makes them particularly well suited to high temperature heat recovery in automotive and industrial settings. Oxides often exhibit intrinsically large Seebeck coefficients, but typically also suffer from low electrical conductivity and high lattice thermal conductivity, resulting in overall modest zT values. However, the discovery of high thermoelectric power factors in cobalt oxides in the 1990s led to the identification of several families of prospective oxide TEs. p-type oxides include the layered cobaltites Na_xCoO₂,^21–24 Ca₃Co₄O₉,^25–27 and Bi₂Sr₂Co₂O_y.^28–30 n-type oxides include ZnO³¹ and the ABO₃ oxide perovskites, in particular SrTiO₃^32–34 (STO) and CaMnO₃.^35,36

SrTiO₃ in particular has emerged as one of the most promising oxide perovskite TEs due to its chemical versatility, high thermal stability, and tunable electronic properties.³⁷ Compared to the related CaTiO₃ and BaTiO₃, STO has an intrinsically large |S|, reasonable σ, superior PF, and impressive thermal stability at elevated temperature.^37–41 Computational modelling has further determined that these advantageous properties are due to STO supporting a significantly higher σ while maintaining comparable |S| to other titanates.^42,43

STO is typically prepared and tested in thin-film form, with high-temperature zT competitive with other benchmark thin-film systems, e.g. zT ≈ 0.29 at T = 1000 K in Sr(Ti_0.8Nb_0.2)O_2.75⁴⁴ vs. zT ≈ 0.3 in Sb₂Te₃,⁴⁵ zT ≈ 0.45 in unoptimised SnSe,⁴⁶ and zT ≈ 0.4 in carbon nanotube films.⁴⁷ Control of nanostructure can potentially push this much higher, with an unprecedented zT ≈ 1.6 at T = 298 K recently reported for Sr(Ti_0.8Nb_0.2)O₃ through precisely controlled strain and interfacial polarisation.⁴⁸ This is significantly higher than the zT = 0.02–0.08 obtained for bulk STO at room temperature, where the large κ = 9–12 W m⁻¹ K⁻¹ limits the thermoelectric performance.⁴⁹

The wide variation in experimental results is in part due to the high structural and chemical flexibility inherent to perovskites, which can adopt multiple crystal phases and can accommodate doping at the A and B cation sites as well as significant levels of oxygen deficiency. Optimising STO and other oxide perovskite thermoelectrics therefore requires a good fundamental understanding of the underlying structure–property relationships. In a previous study, we applied a fully ab initio modelling approach to study the thermoelectric performance of orthorhombic CaTiO₃, tetragonal SrTiO₃ and rhombohedral BaTiO₃.⁴³ Among our key findings was that the crystal structure had a significant impact on the physical properties and in particular the κ_latt. In this work, we build on this with a detailed first-principles study of STO in the orthorhombic, tetragonal and cubic phases. We find that the smaller primitive cell of the Pm3̄m phase results in larger phonon group velocities and high κ_latt, whereas the I4/mcm phase shows significantly stronger polar-optic phonon carrier scattering and lower electrical conductivity and power factors. As a result, the hypothetical Pnma phase, which combines the lower κ_latt of the tetragonal phase and the high PF of the cubic phase, is predicted to show a ∼25% larger zT than the cubic phase and ∼50% larger zT than the tetragonal phase at high temperature. This work therefore demonstrates that materials engineering strategies to stabilise particular structural distortions from the cubic aristotype could be a facile route to optimising the performance of STO and other oxide perovskite thermoelectrics.

2 Computational modelling

2.1 Prediction of thermoelectric properties

2.1.1 Lattice thermal conductivity. The lattice thermal conductivity due to “particle-like” heat transport through phonons, κ_p, was initially calculated by solving the phonon Boltzmann transport equation (BTE) within the single-mode relaxation-time approximation (SM-RTA).⁵⁰

In the SM-RTA, the macroscopic κ_p is obtained as an average over contributions κ_qj from microscopic phonon modes qj, with wavevector q and band index j:⁵⁰

The normalisation factor is the product of the unit-cell volume V and the number of wavevectors N_q included in the summation. The phonon mode heat capacities C_qj and group velocities ν_qj are calculated from the harmonic phonon frequencies ω_qj as:The n_qj in eqn (3) are the Bose–Einstein occupation numbers given by:The phonon lifetimes τ_qj are determined from the inverse of the phonon linewidths (scattering rates) Γ_qj:The linewidths are calculated using third-order perturbation theory as outlined in ref. 50, with the Γ_qj obtained as a sum of contributions from anharmonic energy- and (crystal) momentum-conserving three-phonon scattering processes.

The SM-RTA model in eqn (2) only considers phonon depopulation events, whereas in real materials the flow of thermal energy can result in repopulation.⁵¹ This additional contribution can be taken into account by full solution of the linearised Boltzmann transport equation (LBTE).^51,52

Furthermore, significant overlap between phonon modes, due to small interband frequency spacing and/or to short τ_qj (broad Γ_qj) can lead to heat conduction through “wave-like” interband tunnelling characteristic of amorphous materials (glasses).⁵¹ This additional contribution κ_w can be obtained by solving the Wigner transport equation using the approach outlined in ref. 51. The κ_w can then be combined with either the RTA or LBTE models for the κ_p to compute the total lattice thermal conductivity as:

κ_latt(T) = κ_p(T) + κ_w(T) (7)

The κ_p, κ_w and κ_latt are 3 × 3 tensors, with the transport along the three Cartesian directions α = x, y and z given by the diagonal elements κ_αα. As thermoelectric materials typically take the form of consolidated powders or thin films with randomly-oriented crystal grains, we compute a (scalar) average κ_latt from:

where Tr denotes the trace of the κ tensor. We distinguish the scalar and tensor quantities respectively using Roman and bold type.

2.1.2 Electrical conductivity, Seebeck coefficient and electronic thermal conductivity. The electrical conductivity σ, Seebeck coefficient S and electronic thermal conductivity κ_el were determined by solving the electron BTE within the momentum relaxation-time approximation (MRTA).^53,54

The spectral conductivity Σ is calculated according to:⁵⁴

The summation and integral run over electronic states kj, with wavevector k and band index j, band energy ε_kj, group velocity ν_kj and lifetime τ_kj. The ν_kj are given by the gradient of the electronic band structure as:The τ_kj are calculated by combining approximate models of the electron scattering rates from acoustic deformation potential (ADP), polar-optic phonon (POP), ionised impurity (IMP) and piezoelectric (PIE) scattering using Matthiessen's rule:⁵⁴

τ_kj⁻¹(T) = Γ^ADP_kj(T) + Γ^POP_kj(T) + Γ^IMP_kj(T) + Γ^PIE_kj(T) (11)

Expressions for the scattering rates can be found in ref. 54.

The Σ is then used to obtain the generalised moments of the transport distribution function, [script L]ⁿ, from:

where ε_F is the Fermi energy and f⁰ is the Fermi–Dirac distribution function:

Finally, the σ, S and κ_el are calculated from the [script L]ⁿ(T) according to:^53,54

σ(ε_F, T) = [script L]⁰(ε_F, T) (14)

The [script L]ⁿ, and hence the σ, S and κ_el, depend on the Fermi energy. Under the assumption that dopants do not affect the host band structure, the ε_F can be adjusted to specify an extrinsic carrier concentration (“doping level”) n to predict the effect of chemical doping. (This is termed the “rigid band approximation” (RBA).)

As for the κ_latt, the σ, S and κ_el are 3 × 3 tensors, and scalar averages σ, S and κ_el can be obtained in an analogous manner to eqn (8). The power factor S²σ in eqn (1) is also orientation dependent, and the PF along a given Cartesian direction can be calculated from:

(S²σ)_α = S_αα²σ_αα (17)

The scalar average PF is obtained in the same way, but using the averaged S and σ to match how this quantity is evaluated from experimental measurements (i.e. we do not form and take the diagonal average of a “PF tensor”).

2.1.3 Thermoelectric figure of merit. The calculated κ_latt and σ, S and κ_el can be combined to determine the thermoelectric figure of merit zT from first principles using eqn (1). The κ_latt is a function of temperature, while the three electrical properties are a function of both T and the Fermi energy ε_F (extrinsic carrier concentration n; cf. eqn (7) and eqn (14)–(16)). We therefore determine the zT as a (two-dimensional) function of both T and n.

The zT is orientation dependent and the values along the three Cartesian directions can be calculated as:

As for the power factor, the scalar average zT is calculated using the scalar averages of the four constituent properties, as opposed to the diagonal average of a “zT tensor”).

As noted in Section 2.1.1, heat transport through three different mechanisms can be taken into account when computing the κ_latt (particle-like transport with the SM-RTA or LBTE models, and interband tunnelling). We discuss the contributions of these different mechanisms in detail in Section 3.1 below.

2.2 Density-functional theory calculations

Calculations were performed using pseudopotential plane wave density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) code.⁵⁵

Initial structures of SrTiO₃ in the tetragonal I4/mcm and cubic Pm3̄m phases were taken from the materials project (MP) database⁵⁶ (mp-5229, mp-4651), and an orthorhombic Pnma structure was built by substituting the A-site cation in CaTiO₃ (mp-1185232). All three structures were then fully optimised to tight tolerances of 10⁻⁸ eV on the total energy and 10⁻³ eV Å⁻¹ on the forces.

Electron exchange and correlation were modelled using the PBEsol generalised gradient approximation (GGA) functional.⁵⁷ The ion cores were modelled using projector-augmented (PAW) pseudopotentials^58,59 with the following valence configurations: Ti – 3d² 4s²; O – 2s² 2p⁴; Sr – 4s² 4p⁶ 5s². The valence wavefunctions were described using a plane-wave basis with an 800 eV kinetic-energy cutoff, and the electronic Brillouin zones were sampled using Γ-centred Monkhorst–Pack k-point meshes⁶⁰ with 6 × 6 × 6, 5 × 5 × 3 and 3 × 2 × 4 subdivisions for the Pm3̄m, I4/mcm and Pnma phases respectively. These parameters were chosen based on explicit testing to converge the absolute total energies to <1 meV atom⁻¹ and the external pressures to <1 kbar (0.1 GPa).

2.2.1 Lattice dynamics and thermal conductivity. Lattice-dynamics and thermal-conductivity calculations were performed using the supercell finite-differences approach implemented in the Phonopy and Phono3py codes.^50,61 The 2^nd-order (harmonic) force constants were computed in 4 × 4 × 4 expansions of the Pm3̄m unit cell and I4/mcm primitive cell (320/640 atoms), and a 4 × 2 × 4 expansion of the Pnma unit cell (640 atoms). Atom-projected phonon density of states (PDoS) curves were computed by interpolating the phonon frequencies onto regular Γ-centred q-point grids with 12 × 12 × 12 subdivisions using the linear tetrahedron method for Brillouin-zone integration. Phonon dispersions were obtained by evaluating the frequencies at strings of q points passing through the high-symmetry points in the respective Brillouin zones. Non-analytical corrections to the dynamical matrices at q → Γ were included with the approach in ref. 62 using Born effective-charge tensors Z* and high-frequency dielectric constants ε_∞ computed from density-functional perturbation theory (DFPT).⁶³

The 3^rd-order (anharmonic) force constants were computed in a 3 × 3 × 3 expansion of the Pm3̄m unit cell (135 atoms), a 2 × 2 × 2 expansion of the I4/mcm primitive cell (80 atoms), and a 2 × 1 × 2 expansion of the Pnma unit cell (80 atoms). The “particle-like” lattice thermal conductivities for band transport were calculated using the SM-RTA model by combining the 2^nd- and 3^rd-order force constants to evaluate the heat capacities C_qj, group velocities ν_qj and lifetimes τ_qj on uniform Γ-centred 22 × 22 × 22, 8 × 8 × 8 and 12 × 12 × 12 q-point sampling meshes for the Pm3̄m, I4/mcm and Pnma phases respectively. These meshes were chosen based on explicit testing to converge the scalar average of the κ_p tensors (eqn (2) and (8)) to within 5% of the values obtained with larger sampling meshes. We also solved the full linearised phonon BTE to obtain the κ_p including repopulation effects, and the Wigner transport equation to determine the κ_w contribution from interband tunnelling.⁵¹

2.2.2 Electronic transport. Electronic transport calculations were performed using the AMSET code.⁵⁴ Accurate band gaps E_g were obtained from non-self-consistent calculations with the HSE06 hybrid functional.^64,65 These calculations were also used to obtain electronic PDoS curves using a Gaussian smearing with a width σ = 0.01 eV. Uniform band-structure calculations were then performed with PBEsol and denser 12 × 12 × 12, 10 × 10 × 6, and 6 × 4 × 8 k-point meshes for the Pm3̄m, I4/mcm and Pnma phases, respectively, and the bandgaps increased to the HSE06 values using scissors operators. These meshes were further interpolated to approx. 20×, 30× and 10× higher densities when calculating the electrical transport properties. The charge densities from the calculations were also used to calculate the band energies at strings of k-points along the high-symmetry paths in the Brillouin zones to obtain electronic band structures, with the band gaps again corrected using scissors operators. The combined band structure and PDoS curves were plotted using sumo.⁶⁶

AMSET estimates the electronic relaxation times by summing scattering rates from four different processes, viz. acoustic deformation potential (ADP), polar optic phonon (POP), piezoelectric (PIE), and ionised impurity (IMP) scattering, requiring a range of material properties to be calculated. (We note that PIE scattering is negligible in the three systems examined in this work, as they have centrosymmetric space groups and the piezoelectric moduli vanish by symmetry.)

Deformation potentials were computed by performing a series of single-point energy calculations on deformed structures, generated using AMSET, with PBEsol. The high-frequency, ionic and static dielectric constants ε_∞, ε_ionic and ε_s = ε_∞ + ε_ionic, together with the Born effective charges Z* and piezoelectric moduli, were determined using DFPT.⁶³ Explicit tests found that 2× (Pm3̄m/Pnma phases) and 3× denser k-point sampling (I4/mcm phase) compared to the “base” meshes used for geometry optimisation were required to converge the ε_∞. The Z* were used to determine the infrared (IR) activities of the phonon modes at q = Γ,^67,68 evaluated with Phonopy, and to calculate the polar optic phonon (POP) frequency [script L]_po. Finally, the elastic constant matrix elements C_ij were computed using PBEsol and the finite-differences routines in VASP⁶⁹ (denser k-point sampling was not required for these calculations).

3 Results and discussion

3.1 Structure, phonon spectra and lattice thermal conductivity

The optimised structures of the three SrTiO₃ phases are shown in Fig. 1. For all three systems the lattice parameters are in excellent agreement with experimental measurements, with maximum differences of 0.55% in the lattice parameters and 0.42% in the unit cell volumes of the Pm3̄m and I4/mcm phases (Table S1). We note that the hypothetical Pnma phase of STO has not been observed experimentally, but is adopted by heavily-doped derivatives, and our predicted lattice parameters and cell volumes are within 0.56 and 1.58%, respectively, of the measurements on SrTi_0.9Fe_0.1Ta_0.1O₃ reported in ref. 70.

Fig. 1 Optimised structures of SrTiO₃ in the cubic Pm3̄m (a), tetragonal I4/mcm (b) and orthorhombic Pnma phase (c). The I4/mcm and Pnma structures are oriented to show the equivalence to the cubic structure.⁷¹ The Sr, Ti and O atoms are shown in yellow, light blue and green respectively. These images were prepared using the VESTA software.⁷²

Both the lower-symmetry tetragonal I4/mcm and orthorhombic Pnma phases can be generated from the “parent” Pm3̄m structure by rotations of the TiO₆ octahedra.⁷¹ The I4/mcm phase is obtained by an anti-phase tilt of successive “layers” of TiO₆ octahedra around the z axis, such that the a, b and c lattice vectors of the I4/mcm structure correspond to the (110), (1̄10) and (002) directions in the cubic phase. The Pnma phase is obtained by a more complex combination of an in-phase tilt around x and anti-phase tilts of the same angle around y and z, and the a, b and c lattice vectors correspond to the (011), (020) and (011̄) directions in the Pm3̄m parent.

The calculated phonon spectra of the three SrTiO₃ phases are shown in Fig. 2. The cubic Pm3̄m structure has n_a = 5 atoms in the primitive cell, resulting in 3n_a = 15 branches at each phonon wavevector q. In contrast, the I4/mcm and Pnma structures have n_a = 10 and 20, respectively, and thus considerably more complex phonon spectra with 30 and 60 bands per q. The increase in complexity due to the larger primitive unit cells is further compounded by the lower band degeneracy from the reduced symmetry.

Fig. 2 Calculated phonon dispersion and density of states (DoS) of Pm3̄m (a), I4/mcm (b) and Pnma SrTiO₃ (c). The DoS curves show the total DoS in black/grey with the projections onto the Sr, Ti and O atoms in green, blue and purple respectively.

In all three systems, the A-site cation makes the largest contribution to the low-frequency modes, with sharp features in the DoS consistent with some degree of “rattling” behaviour. The mid-frequency optic modes are predominantly due to the motion of the Ti and O atoms, and can be attributed to deformations of the TiO₆ octahedra, while all three systems show prominent high-frequency features associated with motion of the O atoms. The calculated phonon spectra of the Pm3̄m and I4/mcm phases agree well with previous calculations.^73,74

With the exception of some small interpolation artefacts around q = Γ in the Pnma dispersion, the dispersions of the orthorhombic and tetragonal phases show no imaginary harmonic modes, indicating that both structures are dynamically stable. On the other hand, the dispersion of the Pm3̄m phase shows prominent dynamical instabilities at q = R and M, in agreement with previous calculations.^75,76 These off-Γ instabilities are related to the octahedral tilts that generate the lower-symmetry structures.⁷⁷ We were not able to find reference phonon spectra for the Pnma phase, but the good agreement for the Pm3̄m and I4/mcm phases compared to previous studies gives us confidence in our calculations on the orthorhombic phase.

The scalar-average “particle-like” conductivity, κ_p, of the three SrTiO₃ phases obtained using the single-mode relaxation-time approximation (SM-RTA) are compared in Fig. 3(a), and values at T = 1000 K are compared in Table 1. At 1000 K, we predict κ_p = 1.2–1.9 W m⁻¹ K⁻¹, with an ordering of Pm3̄m > I4/mcm > Pnma. Compared to the intermediate I4/mcm phase, which we examined in our previous study,⁴³ the lower-symmetry orthorhombic phase has a 13% lower κ_p, but the κ_p is still 26% larger than that of the isostructural CaTiO₃ (0.96 W m⁻¹ K⁻¹). On the other hand, the high-symmetry Pm3̄m phase has a 34% higher κ_p than the I4/mcm phase. Taken together, this suggests the effects of crystal symmetry and A-site cation mass on the κ_p are both strong and of similar magnitude. On the other hand, the three phases of STO have very similar density ρ = 5.1–5.12 g cm⁻³ (Table S1), which in our view cannot explain the significant variation in the κ_p. This provides further support for crystal symmetry being the key factor in the reduction in κ_p from the cubic to the orthorhombic phase.

Fig. 3 “Particle-like” (band transport) contributions κ_p to the averaged lattice thermal conductivity κ_latt of the Pm3̄m, I4/mcm and Pnma phases of SrTiO₃ (eqn (2)).⁵⁰ Plot (a) shows the scalar-average κ_p, calculated using eqn (8), while plots (b) and (c) show the decomposition into the harmonic summation κ_p/τ^ph and weighted-average lifetime τ^ph defined in eqn (19).¹³

Table 1 Analysis of the “particle-like” (band transport) contributions κ_p to the scalar-average lattice thermal conductivity κ_latt of the three phases of SrTiO₃ examined in this work at T = 1000 K. For each κ_p we show the harmonic function κ_p/τ^ph and weighted-average lifetime τ^ph, defined in eqn (19), together with the integrals of the “scattering phase space” functions N̄₂ from 0 to the maximum frequencies ω_max in the phonon spectra, the weighted-average three-phonon interaction strengths P̃, and the average number of scattering channels Ñ₂, defined in eqn (26), (21) and (27). The latter three quantities are all scaled to allow for comparison between the three phases

	κp [W m^−1 K^−1]	κp/τ^ph [W m^−1 K^−1 ps^−1]	τ^ph [ps]		P̃ × (3na)^2 [eV^2]	Ñ2/(3na)^2 [THz^−1]
Pm3̄m	1.88	21.29	0.09	2.75 × 10^−2	5.36 × 10^−7	7.95 × 10^−2
I4/mcm	1.4	10.65	0.13	0.56 × 10^−2	5.08 × 10^−7	5.76 × 10^−2
Pnma	1.21	11.01	0.11	0.16 × 10^−2	6.09 × 10^−7	5.72 × 10^−2

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To explore the origin of the differences in the SM-RTA κ_p, we employ the analysis from our previous studies¹³ and define a weighted-average phonon lifetime τ^ph such that the κ_p can be written:

where we have taken the scalar average of the group velocity outer product (cf. eqn (8)). The summation κ_p/τ^ph is temperature dependent due to the heat capacities C_qj and the high-temperature saturation value provides a metric for comparing differences in group velocities between materials, while the τ^ph can be used to compare phonon lifetimes.¹³

The κ_p/τ^ph and τ^ph as a function of temperature for the three phases of STO are compared in Fig. 3(b) and (c), respectively, and values at T = 1000 K are listed in Table 1. The calculated κ_p/τ^ph of the Pnma and I4/mcm phases are similar, differing by only 3.8% at 1000 K, whereas the group velocities of the Pm3̄m phase are around a factor of two larger. Given that the group velocities ν_qj are the gradient of the phonon dispersion, this difference can be attributed to the impact of the smaller primitive cell of the cubic phase on the phonon dispersion (cf. eqn (4), Fig. 2). On the other hand, the τ^ph fall in the order Pm3̄m < Pnma < I4/mcm, i.e. the cubic phase has the shortest averaged phonon lifetimes. The τ^ph of the Pnma phase is 17% shorter than the I4/mcm phase at 1000 K, resulting in a lower overall κ_p, whereas the much larger κ_p/τ^ph of the Pm3̄m phase offsets the short τ^ph and results in the largest predicted κ_latt among all three phases.

In the SM-RTA, the phonon lifetimes τ_qj are calculated as the inverse of the linewidths Γ_qj, which are obtained perturbatively from the imaginary parts of the phonon self-energies.⁵⁰ The Γ_qj can be written approximately as:

The P_qj are the averaged three-phonon interaction strengths Φ_{qj,q′j′,q′′j′′} given by:The function N₂(q,ω,T) defines the “scattering phase space”, i.e. the number of energy- and momentum-conserving scattering pathways available to a phonon with wavevector q and frequency ω at temperature T:

N₂(q, ω, T) = N⁽¹⁾₂(q, ω, T) + N⁽²⁾₂(q, ω, T) (22)

The functions N⁽¹⁾₂ and N⁽²⁾₂ define separately the phase spaces for collision/coalescence (Type 1, two phonons → one phonon) and decay/emission events (Type 2, 1 → 2) and are given by:where the functions Δ and δ enforce the conservation of (crystal) momentum and energy, respectively. The three-phonon interaction strengths are determined from the harmonic frequencies and displacement vectors (eigenvectors) and the 3^rd-order force constants Φ⁽³⁾, while the N₂ depend on the harmonic frequencies only. With reference to eqn (20), the average phonon lifetimes can therefore be interpreted in terms of a parameter P̃, describing the average strength of the three-phonon interactions and the level of phonon anharmonicity, and the size of the scattering phase space.¹³

To simplify the comparison between systems, it is useful to average the N₂ over q to obtain functions of frequency only, i.e.:

Also, whereas the P̃ is a single parameter, and can in principle be compared quantitatively between systems, the N̄₂ can only be compared qualitatively. We therefore compute and use one of two metrics to compare the scattering phase spaces.¹³ First, we compute the integral of the N̄₂, which provides a crude measure of the size of the scattering phase space:where ω_max is the highest frequency in the phonon spectrum.

Secondly, we compute weighted-average Ñ₂ calculated from the τ^ph and P̃ using eqn (20):

Finally, we note that for comparison between systems with different n_a the P̃ and N₂/N̄₂ need to be multiplied and divided, respectively, by (3n_a)².

The calculated N̄₂ for the Pm3̄m phase of STO at T = 1000 K is shown in Fig. 4, and equivalent plots for the I4/mcm and Pnma phases are shown in Fig. S1. We find that the scattering phase space of the low-frequency phonon modes is dominated by collision processes, while decay pathways become available from ∼3 THz, are competitive with collisions up to 13 THz, and are the dominant scattering mechanism for the high-frequency modes up to the f_max ≈ 16 THz (cf. Fig. 2).

Fig. 4 Average scattering phase space function N̄₂ defined in eqn (25) for the Pm3̄m phase of SrTiO₃ at T = 1000 K. The separate phase spaces for collision and decay processes (N̄⁽¹⁾₂/N̄⁽²⁾₂) are shown in blue and purple, respectively, and the total phase space is shown in grey.

The calculated P̃ and the phase space integral and Ñ₂ defined in eqn (26) and (27) for all three phases are listed in Table 1. The scaled P̃ and Ñ₂ indicate that the three-phonon scattering strengths and phase spaces are similar across all three phases, with P̃ between 5.08–6.09 × 10⁻⁷ eV² and Ñ₂ between 5.72–7.95 × 10⁻² THz⁻¹. The Ñ₂ fall in the order of Pnma ≈ I4/mcm < Pm3̄m, i.e. the cubic phase has a notably larger scattering phase space. On the other hand, the P̃ fall in the order I4/mcm < Pm3̄m < Pnma. The short τ^ph of the cubic phase compared to the other two phases is therefore a product of moderate three-phonon interaction strengths and a large scattering phase space, whereas the longer τ^ph of the tetragonal phase is due to comparatively weak three-phonon interactions and a small phase space.

The trend in the κ_p/τ^ph with crystal symmetry qualitatively mirrors our previous calculations on the Group IV–VI chalcogenides, including the “flagship” thermoelectric materials SnS and SnSe, where low group velocities were found to be characteristic of low-symmetry structures with large primitive cells.¹³ In absolute terms, however, the κ_p/τ^ph of all three STO phases are 10–40× larger than chalcogenide phases with comparable symmetry and smaller primitive cells. On the other hand, the τ^ph at T = 300 K range from 0.3–0.5 ps (Table S2), placing them on the same order of magnitude as the shortest averaged lifetimes calculated in ref. 13. The Ñ₂ at room temperature are on the same order as the largest values obtained for the chalcogenides, while the averaged three-phonon interaction strengths fall within the mid-to-upper range of P̃. Compared to these other flagship thermoelectric materials, the oxides thus have relatively large three-phonon scattering phase spaces, moderate-to-strong phonon anharmonicity, and, consequently, short τ^ph.

The trends in the group velocity and lifetimes compared to other classes of materials can be partially interpreted in terms of the chemical bonding. The harmonic frequencies and group velocities are both derived from the 2^nd-order force constants (FCs), which describe the changes in atomic forces in response to small displacements of the atoms. Strong chemical bonding favours large FCs, high phonon frequencies, and, consequently, high group velocities (cf. eqn (4)). The three-phonon interaction strengths additionally depend on the 3^rd-order FCs, which describe the changes in forces in response to pairwise atomic displacements. While we are not aware of any systematic studies to this effect, it is not unreasonable to expect the 2^nd- and 3^rd-order FCs to be at least loosely correlated, such that stronger bonding results in larger 2^nd-order and 3^rd-order FCs. From this point of view, the stronger bonding in the oxides results in larger group velocities, but also stronger three-phonon interactions and shorter phonon lifetimes. In these systems, the former dominates the κ_p.

The generality of our finding that the κ_latt depends sensitively on crystal structure is also supported by a recent theoretical study of Zintl compounds.⁷⁸ In particular, this study found that low group velocities and, in some cases, short lifetimes are favoured by inhomogeneous chemical bonding, for which low crystal symmetry is an indicator, which mirrors the trends for the three phases of SrTiO₃ in the present study.

We also compared the κ_p obtained from the SM-RTA in eqn (2) with that obtained by full solution of the linearised phonon Boltzmann transport equation (LBTE) (Table 2).^50,52 The conceptual difference between the two models is that the SM-RTA only considers depopulation events, whereas the LBTE approach also accounts for repopulation. In practice, the full LBTE solution gives similar results to the SM-RTA, with differences between −2.5 and 6.9% in the κ_p at T = 1000 K.

Table 2 Contributions to the scalar-average lattice thermal conductivity, κ_latt = κ_p + κ_w of the three phases of SrTiO₃ examined in this work from “particle-like” conductivity κ_p and “wave-like” (interband tunnelling) conductivity κ_w at T = 1000 K. Values of the κ_p obtained with the single-mode relaxation-time approximation (SM-RTA) and full solution of the linearised Boltzmann transport equation (LBTE) are given, together with the equivalent κ_latt

	[W m^−1 K^−1]
	κp (SM-RTA)	κp (LBTE)	κw	κlatt (SM-RTA)	κlatt (LBTE)
Pm3̄m	1.88	2.01	1.0	2.88	3.01
I4/mcm	1.4	1.44	1.32	2.72	2.77
Pnma	1.21	1.18	1.36	2.57	2.54

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Fig. 5 Contribution to the scalar-average lattice thermal conductivity κ_latt of the Pm3̄m, I4/mcm and Pnma phases of SrTiO₃ from “particle-like” (band) and “wave-like” (interband tunnelling) transport, κ_p/κ_c, as a function of temperature. Plot (a) compares the κ_p obtained with the single-mode relaxation-time approximation (eqn (2)) and by full solution of the linearised Boltzmann transport equation (LBTE).^50,52 Plot (b) compares the κ_p obtained by solving the LBTE and the total κ_latt = κ_p + κ_w obtained by solving the Wigner transport equation to include interband tunnelling.⁵¹

In our previous work on the lanthanide cobalates LnCoO₃⁷⁹ we found that “wave-like” tunnelling made a significant contribution to the overall κ_latt. We find the same for all three phases of SrTiO₃, with the κ_w accounting for 33–35% of the κ_latt of the Pm3̄m phase at 1000 K, depending on whether the κ_p is calculated using the SM-RTA or LBTE model, and a larger 47–49 and 52–54% of the κ_latt of the lower-symmetry I4/mcm and Pnma phases (Table 2). The larger κ_w of the tetragonal and orthorhombic phases can be explained by the more complex phonon dispersions leading to a smaller interband spacing, which more than compensates for the narrower linewidths (longer lifetimes) compared to the cubic phase (cf. Fig. 2 and Table 1). While we only considered the Pnma phases of the LnCoO₃, we predicted a similar ∼50% contribution from the κ_w to the κ_latt.⁷⁹ We have also found that the κ_w is much more significant in the more structurally-complex “π” phases of the chalcogenides SnS and SnSe than the simpler orthorhombic phases.⁸⁰ In conjunction with the present results, this suggests that there may be a universal trade-off between high κ_p in higher-symmetry phases and high κ_w in lower-symmetry phases.

Taken together, our analysis shows that the κ_latt of SrTiO₃ is a balance of several competing factors. The “particle-like” conductivity, κ_p, is largest in the high-symmetry Pm3̄m phase, where the large phonon group velocities compensate for shorter lifetimes. On the other hand, the interband tunnelling contribution, κ_w, is larger in the lower-symmetry I4/mcm and Pnma phases, where the smaller interband frequency spacing outweighs the narrower linewidths. Overall, however, we predict the Pm3̄m phase to have a 5–10% larger κ_latt than the I4/mcm phase, and a 10–20% larger κ_latt than the Pnma phase, at 1000 K, and we therefore conclude that the phonon group velocities are the biggest factor in determining the higher thermal conductivity of the cubic phase. The difference between the tetragonal and orthorhombic phases is more subtle, but the shorter phonon lifetimes of the Pnma phase outweigh the slightly larger group velocities and κ_w to yield a 5–10% lower κ_latt.

More generally, the short phonon lifetimes in comparison to the chalcogenides suggests that the most fruitful approach to optimising the κ_latt of the oxide perovskites is likely to be targeting lower group velocities, e.g. through weaker or less homogeneous chemical bonding, the latter through structure types with larger/lower-symmetry unit cells. Our recent set of similar calculations on the LnCoO₃ identified lower group velocities, attributed to the weaker ionic bonding, as a key factor in the lower κ_latt, which lends support to this hypothesis.⁷⁹

Finally, we also consider the directional anisotropy in the κ_latt. Due to the symmetry of the Pm3̄m, I4/mcm and Pnma structures, the equivalences of the diagonal elements of the κ_latt are κ_xx = κ_yy = κ_zz = κ_latt, κ_xx = κ_yy ≠ κ_zz and κ_xx ≠ κ_yy ≠ κ_zz. The independent components of the κ_p, obtained using the SM-RTA and LBTE, the κ_w and the κ_latt at T = 1000 K are compared in Tables S3 and S4. The κ_p of the I4/mcm phase show modest anisotropy, with the thermal conductivity along the a = b and c directions differing from the scalar average κ_p in Table 2 by 7.6 and 15% respectively, and with similar % differences obtained for both the SM-RTA and LBTE κ_p. The anisotropy in the Pnma phase is much smaller, at 2.1–4.4% of the κ_p. The anisotropy in the κ_w is also relatively small, with maximum differences of 2.8 and 4.1% in the orthorhombic and tetragonal phases. Notably, differences in the κ_p and κ_w combine to minimise the anisotropy in the κ_latt, with differences of 2.9–6% and 1.4–4.1%, respectively, for the I4/mcm and Pnma phases. Our calculations therefore predict minimal anisotropy in the κ_latt in both non-cubic phases of STO.

As noted in the previous section, the c = z and b = y directions in the I4/mcm and Pnma phases are equivalent to the a = b = c directions in the Pm3̄m phase. For the tetragonal phase, we predict the largest κ_p along the c direction, and the κ_latt along this direction obtained using the SM-RTA and LBTE κ_p are equal to and 2.7% smaller than, respectively, the κ_latt along the three axes in the cubic phase. For the orthorhombic phase, on the other hand, we predict the lowest κ_p and κ_w along the b direction, with the result that the κ_latt is 14–19% smaller than in the cubic phase. This suggests that the phonon transport along the c direction in the I4/mcm phase remains similar to the parent cubic phase, but is significantly disrupted in the Pnma phase, which again highlights the large impact of the lower crystal symmetry on the κ_latt.

3.2 Electronic structure and transport properties

The electronic band structure and density of states (DoS) of the three SrTiO₃ phases are compared in Fig. 6. In all three systems, the valence- and conduction-band edges are formed respectively from O 2p and Ti 3d states. The impact of crystal symmetry on the electronic band structure is similar to that on the phonon dispersion, with the increase in the complexity of the unit cells from the Pm3̄m to the Pnma phase resulting in more complex band structures. In the Pm3̄m phase, the valence-band maximum (VBM) is located at k = R, whereas in the I4/mcm and Pnma phases the VBM is located along the Γ → X and Γ → W paths, respectively. The conduction-band minimum (CBM) of all three phases is located at k = Γ.

Fig. 6 Electronic band structure and density of states (DoS) of Pm3̄m (a), I4/mcm (b) and Pnma SrTiO₃ (c). On each band structure the valence and conduction bands are coloured blue and yellow, respectively, and the valence- and conduction-band minima are marked by green and red circles. On each DoS plot, the total DoS is shown in grey and the projections onto the O 2p and Ti 3d states are shown in blue and pink, respectively.

Using the HSE06 hybrid functional, we predict bandgaps E_g between 4–4.25 eV for the three phases (Table 3), which are considerably larger than both the direct E_g = 3–3.25 eV and the indirect E_g = 3.75 eV reported in experiments.^81,82 However, given the large E_g we would not expect the overestimation of the bandgaps to significantly impact the calculated electronic properties. A measured bandgap for Pnma STO is not available, but our predicted band gap value is consistent with the insulating nature predicted in ref. 83.

Table 3 Calculated electronic band gaps E_g of the three phases of SrTiO₃ examined in this work compared to experimental studies

	Eg,dir [eV]	Eg,indir [eV]
Pm3̄m	4.23	4.0
I4/mcm	4.15	4.14
Pnma	4.13	—
Expt^81,82	3.75	3–3.25

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Fig. 7 shows the predicted scalar-average electrical conductivity σ, absolute Seebeck coefficient |S|, power factor S²σ (PF) and electronic thermal conductivity κ_el of the three STO phases with n-type doping as a function of doping level n at a fixed T = 1000 K. A similar plot showing the temperature dependence of the properties at a fixed n = 2 × 10²¹ cm⁻³ is provided in Fig. S2.

Fig. 7 Predicted scalar-average electrical properties of n-type Pm3̄m, I4/mcm and Pnma SrTiO₃ as a function of extrinsic carrier concentration n (“doping level”) at a fixed temperature T = 1000 K: (a) electrical conductivity σ, (b) absolute Seebeck coefficient |S|, (c) power factor S²σ (PF), and (d) electronic thermal conductivity κ_el.

The relationship between n and σ is given by:

τ^el is a weighted-average electron lifetime, comparable to the τ^ph obtained from analysing the particle-like lattice thermal conductivity (cf. eqn (19)). Similarly, is a weighted-average carrier effective mass for electronic conductivity, averaged over the bands that contribute to the σ. The approximately linear rise in σ with n results in a steep rise in σ when the carrier concentration is plotted on a logarithmic scale as in Fig. 7(a). We predict that n > 10²⁰ cm⁻³ are required to obtain a reasonable σ ≈ 10s of S cm⁻¹.

On the other hand, the S is inversely proportional to the n according to:

where q = ±e for hole and electron carriers and is the Seebeck effective mass, which is related to through:¹¹where N_V is the valley degeneracy. This dependence results in a monotonic decrease in |S| with n when the carrier concentration is plotted on a logarithmic scale (Fig. 7(b)).

The opposing behaviour of σ and |S| result in a peak in the S²σ, with the PFs optimised at n ≈ 2 × 10²¹ cm⁻³. This is in line with previous studies.^43,84 While pristine stoichiometric SrTiO₃ can have relatively low n on the order of 10¹⁵ cm⁻³,⁸⁵ exposure of the material to reducing conditions can produce high concentrations of oxygen vacancies, which act as electron donors, and yield carrier concentrations above 10²⁰ cm⁻³ without explicit chemical doping.⁸⁶ It is possible to obtain n > 10²¹ cm⁻³ with explicit doping, for example by replacing Sr with La and/or Ti with Nb.^86–88 The carrier concentration we predict is required to optimise the PFs should therefore be readily accessible in experiments.

Finally, large n results in “degenerate” semiconducting behaviour with a metallic-like dependence of the conductivity on temperature (Fig. S2). Under these circumstances, the κ_el is related to the σ through the Wiedemann–Franz law:

κ_el = LσT (31)

where L is the Lorentz number. As a result, the κ_el rises steeply above n ≈ 10²⁰ cm⁻³, reaching values of ∼1 W m⁻¹ K⁻¹ at the n ≈ 2 × 10²¹ cm⁻³ that optimises the PF, and becomes comparable in magnitude to the κ_latt at n ≈ 5 × 10²¹ cm⁻³ (Fig. 7(d), cf. Table 2).

We note that we have excluded the temperature dependence of the properties in eqn (28)–(31) for brevity, but the effective masses , τ^el and L are in general temperature dependent. All four parameters also in general depend on the n.

Table 4 lists the values of the four electrical properties at n = 2 × 10²¹ cm⁻³ and T = 1000 K. We predict the Pm3̄m and Pnma phases to show similar electrical properties, with the former having a 10% higher σ and the latter an 11% larger |S|. The net result is that the Pnma phase is predicted to have a 10% larger PF and, due to its lower σ, a 10% lower κ_el. On the other hand, while all three phases have similar |S|, the I4/mcm phase has notably lower σ, which results in a 32/62% smaller peak PF compared to the Pm3̄m and Pnma phases, albeit with the benefit of a 40–60% lower κ_el.

Table 4 Analysis of the scalar-average n-type electronic properties of the three phases of SrTiO₃ examined in this work at T = 1000 K and n = 2 × 10²¹ cm⁻³. For each phase we show the electronic conductivity σ, Seebeck coefficient S, power factor S²σ and electronic thermal conductivity κ_el. We also show the conductivity and Seebeck effective masses and weighted-average electronic lifetime τ^el calculated as described in the text

	σ [S cm^−1]	S [μV K^−1]	S^2σ [mW m^−1 K^−2]	κel [W m^−1 K^−1]			τel [fs]
Pm3̄m	88.1	−130	1.5	1.61	0.78	2.21	1.22
I4/mcm	51.9	−140	1.02	0.93	0.82	2.34	0.76
Pnma	78.9	−145	1.65	1.45	0.8	2.4	1.12

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Given the similar band structure and E_g, the lower electrical conductivity of the I4/mcm phase is somewhat surprising. To investigate this further, we followed the procedure in ref. 89 to obtain an estimate of the and . This entails performing calculations using the constant relaxation-time approximation (CRTA), in which the τ_kj in eqn (9) are replaced by a constant relaxation time τ^el. In these calculations, the τ^el act as a scaling factor for the σ (cf. eqn (28)), and the CRTA conductivity and the σ calculated with per-state scattering rates can thus be used to determine weighted-average τ^el for comparing between systems in the same way as the τ^ph in eqn (19).

The calculated , and τ^el at n = 2 × 10²¹ cm⁻³ and T = 1000 K are included in Table 4. We calculate very similar and for the three systems. The 5.9 and 8.6% larger of the I4/mcm and Pnma phases compared to the cubic phase account for the majority of the 7.7/11.5% larger |S|. On the other hand, the 2.6–5.1% difference in the of the Pnma and I4/mcm phases compared to the Pm3̄m phase indicate that the much lower σ of the tetragonal phase is not due to differences in the carrier effective masses. Instead, the calculated τ^el indicate that the I4/mcm phase has significantly stronger electron scattering compared to the other two phases.

Analysis of the mobility μ as a function of n at T = 1000 K shows that the μ is dominated by polar-optic phonon (POP) scattering (Fig. S3). The associated μ_POP are between 1.5–2× larger in the Pnma and Pm3̄m phases, indicating that the shorter τ^el in the I4/mcm phase is due to stronger POP scattering. The treatment of POP scattering in this work is based on the Frölich model for an electron in a dielectric medium interacting with a dispersionless optic phonon mode with frequency ω_po.⁹⁰ The scattering matrix elements g are of the form:⁹¹

where ε_∞ and ε_S are the scalar averages of the high-frequency and static dielectric constants. The three systems have similar ω_po = 5.2–6.5 THz and similar ε_∞ = 6.75–7 ε₀. The ε_S span a much larger range of 220–350 ε₀ but are larger than the ε_∞, so the latter dominate the right-hand expression in eqn (32). AMSET uses a modified version of the Frölich model that accounts for anisotropy in the dielectric properties, but this is unlikely to explain the stronger POP scattering in the I4/mcm phase because the ε_∞ tensors of all three phases are almost isotropic, with the directional components ε_∞,αα differing by at most 2% from the scalar averages. The matrix elements in AMSET also incorporate a term for the overlap of the electronic states. Moreover, to calculate the scatteing rates (inverse lifetimes), the g are combined with an expression accounting for thermal population and the conservation of energy similar to those in eqn (23) and (24).⁵⁴ It is currently not possible to separate these two types of contribution from the “nature” of the electronic states, but we can nonetheless infer that the stronger POP scattering in the I4/mcm phase is due to either or both of these and not to differences in the model optic phonon and dielectric properties.

We recently performed a similar analysis for the chalcogenides SnS and SnSe,⁸⁰ which yielded and and 0.7–0.81 respectively. The oxides have a larger , resulting in lower σ that requires roughly an order of magnitude larger n to offset. While the oxides have a 3–4× larger m_S, potentially resulting in a larger |S|, this is counterbalanced by the large n required to optimise the PF. Also, while ref. 80 did not calculate τ^el, similar results to those obtained with per-state τ_kj were obtained using a typical τ^el = 10 fs (10⁻¹⁴ s), which suggests a weighted-average relaxation time around an order of magnitude longer than we calculate for STO. The analysis of the κ_latt in the previous section suggested that the stronger bonding in the oxides may lead to stronger phonon anharmonicity and shorter phonon lifetimes. Drawing a parallel, analysis of the τ^el suggests that this also results in stronger electron scattering and shorter electron lifetimes.

Our previous study of the LnCoO₃ yielded and at the n that maximise the PF.⁷⁹ In terms of optimising the properties, this suggests that, as for the κ_latt, exploring alternative perovskite chemistries may be fruitful, although weaker chemical bonding may have the undesirable side effect of increasing the conductivity effective masses. It would be of particular interest to determine whether the much lower σ of the I4/mcm phase is replicated over other perovskite materials, as, if so, it would imply that this phase should ideally be avoided. The results in the present study are however insufficient to confirm, or otherwise, the generality of this finding.

Finally, we again consider the directional anisotropy in the electrical properties. The diagonal σ_αα, S_αα and κ_el,αα components of the σ, S and κ_el tensors, and the corresponding power factors S_αα²σ_αα, calculated at n = 2 × 10²¹ cm⁻³ and T = 1000 K are shown in Table S5. The tetragonal phase shows fairly minimal anisotropy, with up to 5% differences in the directional σ, S and S²σ compared to the scalar averages, and a slightly larger 7.5% variation in the κ_el. The electrical properties of the orthorhombic phase show more anisotropy, with up to 6–9% variation in the σ, S and κ_el, and 12.7/10.9% higher/lower S²σ along the crystallographic b and c directions compared to the scalar average, which, for this system, is close to the PF along the a direction. While larger than the anisotropy in the κ_latt, we still consider this to be relatively modest.

The I4/mcm structure shows the largest σ and κ_el but the smallest |S| along the c direction. The |S| is similar to the cubic phase, but the σ and κ_el are much smaller, as is the S²σ. The equivalent comparison for the Pnma phase shows the smallest σ, largest |S| and intermediate κ_el along the b direction. The σ and κ_el are lower than those in the Pm3̄m phase, but more comparable than those in the tetragonal phase, whereas the |S| and, consequently, the PF, are significantly larger. We therefore infer that the distortions in the two lower-symmetry phases have a significant impact on the electrical properties relative to the cubic phase, even when compared along equivalent directions.

3.3 Thermoelectric figure of merit

Combining the lattice thermal conductivity and electrical-transport properties discussed in the two preceding sections allows us to predict the figure of merit zT. Our analysis of the κ_latt found that the additional contribution to the “particle-like” conductivity, κ_p, obtained by solving the linearised Boltzmann transport equation over the less computationally-demanding single-mode relaxation-time approximation was relatively small, but that the wave-like contribution from interband tunnelling, κ_w, was significant in all three phases. On the basis that combining the LBTE κ_p with the κ_w should give the most accurate description of the κ_latt, we used these values for calculating the zT.

Fig. 8 shows the predicted scalar-average zT of the three phases of SrTiO₃ as a function of doping level n and temperature T. Given the fall in the κ_latt with temperature and the behaviour of the power factor S²σ with n (cf. Fig. 5 and 7), the best performance is obtained at high T and relatively large n ≈ 10²¹ cm⁻³.

Fig. 8 Predicted scalar-average n-type thermoelectric figure of merit zT of Pm3̄m (a), I4/mcm (b) and Pnma SrTiO₃ (c) as a function of extrinsic carrier concentration (“doping level”) n and temperature T.

Table 5 lists the maximum predicted zT, zT_max, at T = 400, 600 and 1000 K, roughly matching the low-, mid- and high-temperature heat-recovery scenarios outlined in ref. 92. We predict zT_max of <0.1, 0.13–0.21 and 0.31–0.47 at the three temperatures, all of which are lower than the typical benchmark zT = 1 but which are consistent at the higher temperature with experimental measurements. With the exception of the cubic and tetragonal phases having similar predicted zT_max at 400 K, we predict that the zT fall in the order of Pnma > Pm3̄m > I4/mcm at all three temperatures. The relatively poor performance of the tetragonal phase is primarily due to its low conductivity, which does not compensate for its larger absolute Seebeck coefficient and results in a low power factor. On the other hand, the superior performance of the Pnma phase compared to the Pm3̄m phase is primarily due to its 15–30% lower κ_latt and correspondingly lower total thermal conductivity, as well as to the combination of slightly lower σ and slightly larger |S| producing a 2–4% larger S²σ.

Table 5 Predicted maximum scalar-average n-type zT, zT_max, of the three phases of SrTiO₃ examined in this work at T = 400, 600 and 1000 K, roughly corresponding to the low-, medium- and high-temperature heat-recovery scenarios outlined in ref. 92. For each predicted zT we also show the associated extrinsic carrier concentrations (“doping levels”) n, Seebeck coefficients S, electrical conductivities σ, power factors S²σ (PFs), and lattice, electronic and total thermal conductivities κ_latt, κ_el and κ_tot

	T [K]	n [cm^−3]	ZT	S [μV K^−1]	σ [S cm^−1]	S^2σ (PF) [mW m^−1 K^−2]	κ [W m^−1 K^−1]
							κel	κlatt	κtot
Pm3̄m	400	7.5 × 10^20	0.06	−109	900	1.06	0.76	5.79	6.55
	600	7.5 × 10^20	0.15	−150	581	1.31	0.7	4.29	4.99
	1000	1 × 10^21	0.39	−182	448	1.48	0.79	3.01	3.8
I4/mcm	400	7.5 × 10^20	0.06	−128	484	0.79	0.4	4.9	5.3
	600	1 × 10^21	0.13	−148	432	0.94	0.49	3.77	4.26
	1000	1 × 10^21	0.31	−195	261	0.99	0.45	2.77	3.22
Pnma	400	5 × 10^20	0.09	−148	500	1.08	0.4	4.25	4.65
	600	7.5 × 10^20	0.21	−166	487	1.34	0.58	3.33	3.91
	1000	1 × 10^21	0.47	−198	392	1.54	0.7	2.54	3.24

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Table S6 compares the directional values of the zT, S²σ and κ_tot under the same conditions as the zT_max in Table 5. The zT of the I4/mcm phase show modest anisotropy, with a 3.2% larger zT_max along the a = b directions and a 6.5% smaller zT_max along the c direction compared to the scalar average at T = 1000 K. This is due primarily to differences in the thermal conductivity. The Pnma phase shows more significant anisotropy, with a 3.9% larger PF and 4.3% smaller κ_tot along the b direction resulting in an 8.4% higher zT_max compared to the scalar average, and a 7.8% smaller PF and 0.93% higher κ_tot along the c direction producing a 10% lower zT_max. As for the electrical properties, we find little correspondence between the zT of the cubic phase and the zT along the c direction in the I4/mcm phase and the b direction in the Pnma phase, which indicates that the large impact of the structural distortions on the electrical properties also carries through to the figure of merit.

Experimental values of zT span a range of 0.27–0.37, with σ = 200–303 S cm⁻¹, S = −168 to −233 μV K⁻¹, and κ = 2.6–3.1 W m⁻¹ K⁻¹ at T = 1000 K.^93–95 Assuming the Pm3̄m phase, we predict values of between zT = 0.2 (n = 10²⁰ cm⁻³, σ = 46.1 S cm⁻¹, S = −367 μV K⁻¹, κ = 3.09 W m⁻¹ K⁻¹) and zT = 0.37 (n = 5 × 10²⁰ cm⁻³, σ = 227 S cm⁻¹, S = −152 μV K⁻¹, κ = 3.41 W m⁻¹ K⁻¹). These predictions, in particular those at the larger n, are reasonably consistent with experimental measurements, confirming the accuracy of our modelling approach. In particular, with reference to Table 2, and as found in our previous work,⁷⁹ accounting for wave-like tunnelling contributions to the κ_latt results in a notably better match to experiments. With this in mind, our predictions suggest that experiments have likely achieved close to the maximum possible for pristine bulk-like cubic STO, but that significant performance enhancements could be possible by targeting chemical modifications (e.g. doping) that stabilise the orthorhombic phase – according to our analysis, doing so would largely retain the favourable electrical properties of the Pm3̄m phase while significantly reducing the κ_latt.

4 Conclusions

In summary, we have carried out a detailed ab initio modelling study of the impact of crystal structure on the thermoelectric performance of the archetypal oxide perovskite SrTiO₃ in the orthorhombic Pnma, tetragonal I4/mcm and cubic Pm3̄m phases.

Analysis of the structural dynamics and lattice thermal conductivity shows that the primary impact of structure type is that the smaller primitive cell of the Pm3̄m phase results in larger phonon group velocities and a larger contribution from “particle-like” (band) transport to the κ_latt, which more than offsets its shorter phonon lifetimes. More generally, compared to the flagship Group IV–VI chalcogenide thermoelectrics the oxides show relatively strong phonon scattering and short phonon lifetimes, but this is more than compensated by larger group velocities. Based on this, efforts to optimise (i.e. reduce) the κ_latt of STO and other oxide perovskites should focus on chemical modifications or alternative perovskite chemistries that favour lower-symmetry crystal structures and/or which have weaker/less homogeneous chemical bonding. A second notable finding from our analysis of the κ_latt is the contribution from wave-like transport, which has a large impact on all three phases but is most significant for the lower-symmetry structures. Taken together with our previous work on the lanthanide cobalates, we suggest that accounting for transport through this mechanism is important for an accurate description of the κ_latt in oxide perovskites generally, and that the increase in this term in lower-symmetry structures may be universal across different oxide perovskite chemistries.

Analysis of the electrical properties indicates that the I4/mcm phase has significantly lower electrical conductivity due to strong polar-optic phonon scattering, whereas the Pnma phase shows comparable or higher power factors to the cubic Pm3̄m phase due to the combination of a lower σ and larger absolute Seebeck coefficient. Comparing again to the Group IV–VI chalcogenides, we find that the power factors of the oxides are limited by high conductivity effective masses and short electron lifetimes, and that the high carrier concentrations required to offset this effectively nullify the higher |S|. From a fundamental perspective, it is interesting to note that the oxides show both short phonon and electron lifetimes, which could indicate that the electrical properties and κ_latt may be coupled in a way which is yet to be explored in detail. We also believe the low predicted σ of the I4/mcm phase warrants further investigation, particularly to determine whether similar behaviour is observed across different perovskite chemistries.

Finally, our predicted zT show that, while STO cannot achieve the benchmark zT ≥ 1 even at a high T = 1000 K, it may be possible to significantly improve upon existing experimental studies by targeting the stabilisation of lower-symmetry structures and in particular the orthorhombic Pnma phase.

We hope that the results presented here will ultimately provide some important insight to support ongoing efforts to obtain high-performance oxide thermoelectrics for sustainable energy recovery.

Author contributions

Conceptualization, methodology, software and writing – all authors; data curation, formal analysis and investigation – AZK and JMF; funding acquisition – JMS; project administration and supervision – JMF and JMS; resources – JMS.

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information: Optimised lattice parameters and comparison to experimental measurements; analysis of the “particle-like” contributions to the lattice thermal conductivity at 300 K; anisotropy in the κ_latt at 1000 K; “scattering phase space”; functions for the I4/mcm and Pnma phases of SrTiO₃; anisotropy in the electrical transport at 1000 K; analysis of the electron mobility as a function of carrier concentration at 1000 K; and anisotropy in the thermoelectric figure of merit at 400, 600 and 1000 K. See DOI: https://doi.org/10.1039/d5ya00105f.

Raw data from this study will be made available to download free of charge after publication from an online repository at https://doi.org/10.17632/9h4b89b79r. Our analysis code is available on GitHub at https://github.com/skelton-group/ZT-Calc-Workflow.

Acknowledgements

AZK is supported by a University of Manchester PhD studentship, and JMF and JMS are supported by a UK Research and Innovation (UKRI) Future Leaders Fellowship (MR/T043121/1, MR/Y033973/1). The calculations used the ARCHER 2 HPC facility via the UK Materials Chemistry Consortium, which is funded by the UK Engineering and Physical Sciences Research Council (EP/R029431, EP/X035859).

Notes and references

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