Electronic transport and the thermoelectric properties of donor-doped SrTiO3

Strontium titanate (SrTiO3) is widely recognised as an environmentally-benign perovskite material with potential for thermoelectric applications. In this work we employ a systematic modelling approach to study the electronic structure and thermoelectric power factor (PF) of pure SrTiO3 and donor-doped Sr(Ti0.875M0.125)O3 (M = Cr, Mo, W, V, Nb, Ta). We find that the carrier concentration required to optimise the PF of SrTiO3 is on the order of 1021 cm−3, in line with experimental studies. Substitution at the Ti (B) site with 12.5 mol% Nb or Ta is predicted to yield the best PF among the six Group V/VI dopants examined, balancing the Seebeck coefficient and electrical conductivity, and doping with the more abundant Nb would likely give the best price/performance ratio. Although W doping can significantly improve the electrical conductivity, this is at the expense of a reduced Seebeck coefficient. The first-row elements V and Cr have a significantly different impact on the electrical properties compared to the other dopants, forming resonant levels or creating hole carriers and leading to poor thermoelectric performance compared to the second- and third-row dopants. However, the reduction in the bandgap due obtained with these dopants may make the materials suitable for other applications such as photovoltaics or photocatalysis. Our modelling reveals the critical carrier concentrations and best B-site dopants for optimising the electrical properties of SrTiO3, and our predictions are supported by good agreement with available experimental data. The work therefore highlights avenues for maximising the thermoelectric properties of this archetypal oxide material.


Introduction
The immense amount of waste heat generated in industrial and domestic environments, and the slow scale up of renewable energy to meet rising demand, makes thermoelectric (TE) technology for the conversion of waste heat to electric power an important part of a coordinated strategy for mitigating climate change [1,2].The conversion efficiency of a TE material is commonly defined by a dimensionless figure of merit zT given by: where S is the Seebeck coefficient, σ is the electrical conductivity, S 2 σ is the power factor (PF), κ l and κ e are the lattice (phonon) and electronic components of the thermal conductivity κ, and T is the absolute temperature [3].The zT values of promising TE materials, in particular chalcogenides such as PbTe and SnSe-based compounds [3][4][5][6], have vastly improved over recent decades.However, many of these high-zT materials are composed of heavy, toxic, and/or rare elements, and some materials also show poor thermal stability at high temperature, both of which restrict their applications [7][8][9][10].As a result, there has been increasing focus on oxide-based materials as non-toxic, cheap, earth-abundant alternatives with superior high-temperature stability [11][12][13].Donor-doped (n-type) strontium titanate (SrTiO 3 ; STO) is regarded as a promising oxide TE, with a high melting point of 2080 • C [14] and an outstanding PF up to 3600 µW m −1 K −2 at room temperature due to a large Seebeck coefficient [15].The latter arises from the large effective mass of the Ti 3d-based carriers [16].STO is a wide-bandgap insulator (E g = 3.2 eV) [17], and electron doping has been universally adopted as a means to improve its TE performance, with typical routes based on La doping at the Sr (A) site [9,[18][19][20] and Nb doping at the Ti (B) site [12,21,22].Other potential dopants for tuning the electrical properties include trivalent rare-earth elements such as Pr [22], Nd [23], and Ce, Sm, Gd, Dy and Y [24] at the A site and other pentavalent or hexavalent elements including Ta [25], Mo [19], and W [26] at the B site.For example, Kovalevsky et al investigated experimentally the effects of rare-earth A-site dopants in STO [24], while the behaviour of a number of B-site dopants have been studied theoretically including V [27], Nb [28,29] and Ta [29].However, a comprehensive survey of potential B-site dopants for STO has yet to be carried out.
Maximising the PF S 2 σ requires a large Seebeck coefficient and high electrical conductivity.The Seebeck coefficients of metals and heavily-doped (degenerate) semiconductors are inversely proportional to the concentration of charge carrier n according to: where k B , h and m * are, respectively, the Boltzmann constant, the Planck constant and the carrier effective mass [30].On the other hand, electrical conductivity is directly proportional to the carrier concentration through: where µ is the carrier mobility and depends inversely on the effective mass.The two electrical properties are therefore interrelated, and the highest PF is obtained as a compromise between a large S and large σ.For STO, this balance is achieved at 10-20 mol% doping levels, which yield typical PFs above 1000 µW m −1 K −2 from 300-1000 K [31,32].Doping has limited impact on phonon scattering, and strategies to suppress the lattice thermal conductivity instead mostly focus on introducing a high density of scattering centres, for example by forming composites [33], nanoprecipitates [34] and defects [35].However, this can also result in electron scattering and negatively impact the electrical properties [35].For STO, the largest zT in the vast majority of studies is in the range of 0.3-0.4 at ∼1000 K, and these are typically underpinned by large PFs [21,23,31,32,36].It is preferable to optimise the electrical properties over the thermal transport properties since the PF also determines the maximum output power of a TE device [23,37].With this in mind, it is essential to establish the optimum carrier concentration to maximise the PF in STO, and to identify suitable doping strategies for achieving it.
In this work we have used first-principles calculations with density-functional theory (DFT) to model the electronic structure of STO and supercell models of Sr(Ti 0.875 M 0.125 )O 3 with 12.5 mol% of the B-site atoms substituted by one of six Group V or VI elements M = Cr, Mo, W, V, Nb, Ta.We determine the dependence of the Seebeck coefficient, electrical conductivity and PF of STO on the carrier concentration, and we evaluate the impact of the different dopants on the electrical properties to provide guidance for selecting dopants for high-performance STO-based thermoelectrics.

Computational methods
The calculations in this work were performed using DFT as implemented in the Quantum ESPRESSO package [38,39].
Pristine STO was modelled in the cubic phase with five atoms in the crystallographic unit cell, and doped models were created by substituting one Ti (B) atom in a 2 × 2 × 2 supercell (40 atoms) with one of the six dopants V, Nb, Ta, Cr, Mo, and W (figure S1).Electron exchange and correlation were modelled with the PBE [40] and PBEsol [41] functionals.The plane-wave cutoff energies for the wavefunctions and charge densities were set to 52 and 576 Ry, respectively, except for V-doped STO, for which higher cutoffs of 70/700 Ry were used as recommended for our chosen pseudopotential.k-point meshes with 4 × 4 × 4 and 2 × 2 × 2 subdivisions were used to model the electronic structures of STO and the doped supercells.The ion cores were modelled using projector augmented-wave [42] pseudopotentials [43,44].Each model was fully relaxed using the Broyden-Fletcher-Goldfarb-Shanno algorithm.Electronic density of states (DoS) curves and band structures were obtained by performing non self-consistent calculations to evaluate the electronic band energies on dense uniform 20 × 20 × 20 and 10 × 10 × 10 k-point meshes for pristine STO and the doped supercells, and at k-points along a high-symmetry path through the cubic perovskite Brillouin zone (figure S1, supporting information).
Electronic transport calculations were performed on the pristine and doped STO models using semi-classical Boltzmann transport theory using the calculated electronic structures and the BoltzTraP code [45].For these calculations, the k-point density was increased by a factor of 5 using interpolation, and the temperature-dependent relaxation times were set as described in the section 3.1 using the data from experimental measurements on pristine STO performed by Kinaci et al [29].For pristine STO we explored temperatures from 300-1200 K in steps of 50 K and extrinsic electron carrier concentrations of 10 19 , 4 × 10 19 , 7 × 10 19 , 10 20 , 4 × 10 20 , 7 × 10 20 , 10 21 , 4 × 10 21 , 7 × 10 21 and 10 22 cm −3 .As described in the text, calculations were also performed using a scissors operator to correct the calculated bandgap to the experimental value.For the doped models we used the same temperatures but used the intrinsic carrier concentrations obtained by BoltzTraP using the calculated Fermi energies E F .
Finally, the formation energy ∆E F of pristine STO and the doping energies ∆E D for each of the six dopants were calculated using the PBEsol + U total energies of the optimised STO and doped supercell models, the oxides SrO, TiO 2 , V 2 O 5 , Nb 2 O 5 , Ta 2 O 5 , CrO 3 , MoO 3 and WO 3 , and the O 2 molecule.).Electron exchange and correlation and the plane-wave cutoff energies for the wavefunctions and charge densities for the oxides and O 2 were set the same as the pristine and doped STO models.The oxides were modelled in the published crystal structures, and a model of O 2 was created by placing the molecule at the centre of a large cubic box with a minimum (initial) distance of 15 Å between periodic images.Details of these calculations are given in section S1 of the supporting information, and the technical parameters are summarised in table S1.

Results and discussion
3.1.Pristine SrTiO 3 STO adopts a cubic perovskite structure with a lattice constant a = 3.905 Å [46].The Ti and O atoms are located at the body and face centres, forming a network of corner-sharing TiO 6 octahedra, and the Sr atoms are located at the corners of the cubic unit cell in the cuboctahedral cavities (figure S2, supporting information).Table 1 shows the optimised lattice constants obtained with the PBE and PBEsol functionals.The PBE functional overestimates the cell volume by 2.6% compared to the experimental value, whereas PBEsol underestimates the volume by 0.9% but is closer to experiments.These results agree with the tendency for PBE to 'underbind' and overestimate the cell volume, whereas PBEsol is optimised for densely-packed solids [41] and as such tends to predict more accurate lattice constants [47,48].
The electronic band structures calculated using the two functionals are presented in figures 1(a) and (b).The two functionals predict very similar electronic structures, with band gaps E g ≈ 1.8 eV and Fermi levels (E F ) located inside the gap.The atom-projected DoS show that the top of the valence band is dominated by O 2p states with a small contribution from Ti states, while the bottom of the conduction band is mostly dominated by Ti 3d states with a small contribution from O. The lowest-energy conduction bands are generated by the Ti 3d t 2g orbitals [49], which form a sharp feature in the DoS.PBE predicts the Fermi level to be closer to the valence-band edge E V (E F − E V = 0.56 eV), whereas PBEsol predicts the E F to be closer to the middle of the gap (E F − E V = 0.89 eV).In addition, while the crystal field splitting of the Ti 3d e g and t 2g bands is comparable in both calculations, PBEsol predicts a larger bandwidth for the t 2g , which is possibly due to the smaller lattice constant increasing the orbital overlap and the interaction strength between atoms.
We further investigated the effect of applying a Hubbard U correction [50] to the Ti 3d states with both functionals to attempt to optimise the lattice constants and the band gap [51].The lattice constant and change in the band gap as a function of U are plotted in figures 2(a) and (b).The lattice constant shows a non-linear increase with U while the E g peaks at ∼2.7 eV, at U = 8 eV.With the PBE functional, this value of U leads to an overestimation of the cell volume by ∼6% compared the experimental value.With PBEsol, the experimental lattice constant and volume are recovered with U = 3.25 eV but at this value of U the band gap is underestimated by ∼33% compared to the experimental E g = 3.2 eV [17].A larger U = 8 eV recovers the experimental gap by 80%, but this results in the cell volume being overestimated by 6%.
We also studied the effect of the DFT + U correction on the positions of the valence and conduction band edges (E V at k = R, E C at k = Γ) relative to the E F (figures S3-S5, supporting information).Starting from U = 0 (i.e.no correction), the differences E F − E V and E C − E F decrease and increase with U, respectively, indicating that the correction widens the gap and moves the E F closer to the valence band.With PBE + U, at U = 5.5 eV the Fermi level is close to the valence band edge, resulting in a very small E F − E V (figures S3 and S4).Increasing the U to 6 eV causes the E F to 'jump' the middle of the gap, with a corresponding sharp increase and decrease in E F − E V and E C − E F , respectively.A further increase in U to 6.5 eV shifts the E F towards the conduction band edge, and at this and larger U the difference  behaviour is not observed with PBEsol functional, for which the E F − E V and E C − E F show a monotonic decrease and increase, respectively, up to the largest U = 7 eV we tested (figures S3 and S5).Comparison of the band structures for selected U also shows that the E F remains close to the valence-band edge in all cases.
Given that STO is a wide-gap insulator, and we would not therefore expect the underestimation of the bandgap to affect the transport properties, we selected PBEsol with U = 3.25 eV to perform the remainder of our transport calculations, as a balance between predicting an accurate structure and a reasonable bandgap.The band structure of STO obtained with this functional is shown in figure 2(c).Compared to the 'bare' PBEsol band structure in figure 1(b), both band structures have near-identical form but the PBEsol + U calculation predicts a wider bandgap of 2.04 eV compared to the bare PBEsol value of 1.83 eV, which is closer to the experimental value.This is primarily achieved through a lowering in energy of the valence band edge and an increase in energy of the conduction band edge.
The correction also leads to a larger DoS at the bottom of the conduction band, consistent with a band flattening.This will increase the band effective masses m B , as these are inversely proportional to the band curvature [52], and may influence the PF by lowering the carrier mobility and electrical conductivity and/or enhancing the Seebeck coefficient (equations ( 2) and ( 3)).
Based on the widely-reported n-type behaviour of STO, the electronic transport properties of STO were calculated based on the PBEsol + U electronic structure as a function of electron carrier concentration n from 10 19 -10 22 cm −3 using the semi-classical Boltzmann transport theory implemented in the BoltzTraP code (figure 3).By default, BoltzTraP uses the constant relaxation-time approximation (CRTA) in which the electron scattering times (inverse scattering rates) are treated as an unknown constant τ .In this model, the S are independent of the τ , while the σ is obtained as the 'reduced' quantity σ/τ .Kinaci et al estimated a set of temperature-dependent relaxation times for single-crystal STO and found that the τ were comparable over a  wide range of dopants and doping levels [29].We therefore used these in conjunction with our calculations to obtain 'absolute' values of σ and S 2 σ.
We note that we could in principle adjust the τ to explore the impact of changes to carrier mobility on the doping.Dopants can introduce electron scattering and reduce the carrier mobility by modulating the crystal structure through fluctuations in atomic masses and ionic radii, which would be reflected in a reduced τ .However, the impact of a given dopant is in general difficult to predict.On the other hand, the carrier concentration is less sensitive to the nature of the dopants and is proportional to the doping level (i.e. the number of carriers that can be provided to the system by a given concentration of Nb 5+ and Ta 5+ is the same).If we make the reasonable assumption that a dopant will typically degrade rather than enhance the mobility, quantifying a target optimum carrier concentration is a more useful starting point for designing experiments.We note that dopants can also modulate the band structure and therefore the effective mass of the carriers, which will impact both the carrier mobility and the Seebeck coefficient [53].This is not accounted for in the rigid band approximation used in the present calculations on STO, but we address this in the calculations on doped supercell models presented below.
The calculated Seebeck coefficient S of STO is negative, as shown in figure 3(a), indicating the expected n-type semiconducting behaviour.The absolute S decreases with carrier concentration but increases with the temperature, mirroring the trends seen in experiments [30].With a fixed n = 10 19 cm −3 , the calculated S are predicted to increase from −305 µV K −1 at 400 K to −450 µV K −1 at 1,200 K, whereas with a larger n = 10 22 cm −3 these values are reduced to −15 and −56 µV K −1 .At an intermediate, but still high, n = 10 20 cm −3 we predict a reasonable S = −220 µV K −1 at 400 K.The strong dependence of the Seebeck coefficient on carrier concentration may be associated with the flat conduction-band minima and resulting large band effective masses [52].
We find that the measured Seebeck coefficients tend to be larger than the calculated ones, e.g. for n = 5 × 10 19 cm −3 , the estimated S = −269 µV K −1 at 400 K is 37% smaller than the reported S = −430 µV K −1 [54].The discrepancy is smaller at larger n, with the S underestimated by ∼30% at n = 10 20 cm −3 .These differences can be ascribed to the presence of defects (e.g.ionised impurities) in the experimental samples and/or to the bandgap underestimation with PBEsol + U.In the former case, defects in doped single crystals can trap and scatter carriers, thus increasing the effective scattering rates, and/or enhance the energy filtering effect, and both of these can lead to an enhanced Seebeck coefficient [55].On the other hand, for a low (extrinsic) n an underestimation of the bandgap could result in larger intrinsic carrier concentrations and a reduced Seebeck coefficient (or vice versa if the bandgap is overestimated).
The calculated σ and S 2 σ are plotted and compared to experimental measurements in figures 3(b) and (c) [15,54].The calculated conductivities show a large deviation from the experimental measurements, with σ consistently around an order of magnitude larger than the measured values and with the difference falling at larger n.Considering the discrepancies between the calculated and measured S, this may again be associated with enhanced carrier scattering or lower effective carrier concentrations in the experimental samples.The calculations predict an increase in σ of over two orders of magnitude at 400 K as n is increased from 10 19 to 10 22 cm −3 , from 1.5 × 10 4 -1.7 × 10 6 S m −1 , which again mirrors the trends in the experimental measurements.
Most importantly, the underestimation of the Seebeck coefficients and overestimation of the conductivity balance to yield a satisfactory reproduction of the experimental PF (figure 3(c)).The predicted PF of ∼2600 µW m −1 K −2 at 400 K with n between 5 × 10 20 and 1 × 10 21 cm −3 is higher than reported by Ohta et al [54] but comparable to the largest values of 2800-3600 µW m −1 K −2 obtained in the experiments on single-crystal strontium titanate performed by Okuda et al (T = 300 K, n between 2 × 10 20 and 2 × 10 21 cm −3 ) [15].The PF decreases sharply at carrier concentrations larger than 10 21 cm −3 , indicating that heavy doping has a significant detrimental impact on the thermoelectric performance.The calculations predict an optimal doping level on the order of 10 21 cm −3 , which is a level that has been employed in a number of experimental investigations [15,54,56].However, it is unlikely that our predicted PFs can be reproduced exactly in real experiments, for the reasons discussed above.Furthermore, the predicted optimum carrier concentrations may need to be increased for polycrystalline samples, perhaps significantly so, to mitigate the impact of defects and grain boundaries on the carrier concentration and mobility and, consequently, the electrical conductivity.
Finally, to test the impact of the bandgap underestimation with PBEsol + U on the calculated transport properties, we repeated the BoltzTraP calculations with a scissors operator applied to correct the bandgap to the experimental value (figure S6 in the supporting information).The wider bandgap leads to an increased S and lower σ at small carrier concentrations, where the extrinsic n is smaller than the thermally-generated intrinsic carrier populations, but the results become comparable at higher n and are identical for n > 4 × 10 20 cm −3 .This confirms that the bandgap underestimation does not significantly affect the accuracy of our calculations at the typical doping levels employed in experimental studies, and the optimum doping level identified in this study.

Group V and VI B-site doped SrTiO 3
Having demonstrated that our calculations on pristine STO compare well to experiments, we performed similar calculations on explicitly doped systems by considering a range of potential atomic substitutions at the Ti (B) site.Substitution of a single Ti atom in a 2 × 2 × 2 supercell results in a doping level of 12.5 mol%, which is within the range employed in experiments [31,32].We considered six elements from Groups V and Table 2. Doped systems examined in this work together with the Hubbard U values applied to the transition-metal d orbitals, the ionic radii of the dopant atoms in their possible oxidation states (with a coordination number of 6), the optimised lattice constants, and the total magnetisation of the simulation cell.
Hubbard U (eV) Ionic Radius (pm) [62]  a The lattice constants of the supercells were halved to compare with pristine SrTiO3.b BM = Bohr Magneton.c The U = 3.25 eV determined for the Ti atoms was applied to all of the doped systems.
VI of the periodic table, viz.V, Nb, Ta, Cr, Mo, and W. While we were able to determine a suitable Hubbard U value for Ti by comparing to experiments, it is not possible to do so for the dopants due to the scarcity of experimental data, and we therefore used U values from the literature (table 2) [57][58][59][60].Due to the unpaired electrons introduced by the dopants, spin polarisation was taken into account in these calculations.
To assess the stability of the six dopants, we calculated the doping energies ∆E D with respect to the Group V sesquioxides (V) 2 O 5 and Group VI trioxides (VI)O 3 , and molecular O 2 (table S2, supporting information).The calculated ∆E D range from 0.13-2.06eV.The first-row elements V and Cr have the lowest ∆E D , of 0.13 and 0.76 eV, respectively, Mo has the largest E D = 2.06 eV, and the other three dopants Nb, Ta and W have similar ∆E D of 1.23-1.26eV.These are all relatively small, and would easily be offset by the calculated formation energy ∆E F of STO of −1.17 eV per formula unit (−9.36 eV for the pristine STO supercell the doped models are based on).This suggests that incorporating all six dopants into STO should be possible and, indeed, relatively facile.
Under the assumption that unpaired spin density predominantly resides on the dopant ions, the oxidation states can be loosely inferred from the total magnetisation M of the doped supercells models (table 2).For the Group V dopants, M = 0 and 1 correspond to the +5 and +4 oxidation states, the former of which is indicative of complete ionisation.For the Group VI dopants, M= 0, 1 and 2 correspond to the +6, +5 and +4 oxidation states, of which +6 and +5 correspond to complete and partial ionisation.The data suggests that none of the dopants are fully ionised.The calculations predict that the additional valence electrons of V and Cr will remain localised on the dopant atoms (M = 1 and 2 respectively).Nb and Ta are predicted to be partially ionised, corresponding to (athermal) carrier concentrations of approx.3.3 × 10 20 and 6.23 × 10 19 cm −3 respectively.On the other hand, Mo adopts an oxidation state close to +5, while W is almost fully ionised, corresponding to n = 1.75 × 10 21 and 3.15 × 10 21 cm −3 .For comparison, full ionisation of the Group V and VI dopants would produce n of around 2.1 × 10 21 and 4.2 × 10 21 cm −3 , respectively.
The lattice constants of the doped models obtained after geometry optimisation are compared to that of pristine STO in table 2. For completeness, we also performed optimisations with 'bare' PBEsol, and the  S3.All the doped systems retained the cubic perovskite structure, as expected from the Goldschmidt tolerance factor (table S3) [61].
With both PBEsol and PBEsol + U the changes in the lattice constant are generally consistent with differences in the ionic radii of the doping elements in the oxidation states inferred from the magnetism (tables 2 and S3).The ionic radii of V 4+ /Cr 4+ and Nb 4+ /Ta 4+ are smaller and larger than Ti 4+ , respectively, resulting in a reduction and expansion of the lattice constants.On the other hand, the radii of Mo 4+ / 5+ and W 5+ / 6+ are comparable to that of Ti 4+ , resulting in a comparable lattice constant to pristine STO.
It is also noteworthy that the lattice constants of V-and Cr-doped STO are comparable, despite the ionic radius of V 5+ (54 pm) being much larger than Cr 6+ (44 pm), which lends further support to these elements being in the +4 oxidation states with comparable radii (55/58 pm, table 2).For V-and Cr-doped STO, the total magnetisation values are 1 and 2, respectively, suggesting both V and Cr ions exhibit a valence state of 4+.The ionic radius of V 4+ and Cr 4+ are comparable (58 pm and 55 pm [62], respectively).
The calculated electronic band structure and the DoS curves for each of the doped models in the vicinity of the Fermi level are presented in figures 4 and 5.The relative position for the Fermi level and width of the band gap are listed in table 3.
We begin by discussing the Group V-doped models (figure 4).Doping with V introduces an isolated three-fold flat band into the host STO bandgap in one of the two spin channels (figure 4(a)).This can be viewed as a defect [63] or resonant level [27], although we note that the previous report of the formation of a resonant level with V doping was based on doping at the Sr (A) rather than at the Ti (B) site as in the present study [27].The participation of Ti and O in the DoS associated with the defect band suggests the V states are to some extent incorporated into the Ti-O bonding network.The E F is located within the band, suggesting metallic behaviour, and the steep DoS at the top of the band potentially suggests an enhanced Seebeck coefficient [63,64].In the other spin channel, the V states form part of the host STO conduction band and the E F remains within the gap, indicating semiconducting behaviour.
For Nb-doped STO, the Nb states are spread throughout the conduction band.The E F in both spin channels is located at the bottom of the conduction band and is deeper into the band in one spin channel than the other (figure 4(b), table 3).The DoS at the Fermi level is small, suggesting that, in contrast to V, Nb generates shallow defect levels rather than resonant levels.Ta-doped STO shows similar behaviour to Nb-doped STO, with the E F located in and close to the bottom of the conduction band in the two spin channels (table 3), but with steeper band curvature indicating a lower band effective mass (figure 4(c)).
Turning to the Group VI-doped models (figure 5), Cr-doped STO shows qualitatively similar behaviour to the V-doped model (figure 5(a)).Cr contributes a three-fold band near the top of the valence band in one of the spin channels, and the Fermi level is located within this band in one of the spin channels and close to the valence-band edge in the other (table 3).The DoS again suggests participation of Cr states in the bonding, and the strong intensity of the Cr feature in the DoS of one of the spin channels is again indicative of the formation of a resonant level.On the other hand, the predicted electronic structures of Mo-and W-doped STO (figures 5(b) and (c)) are similar to Nb-and Ta-doped STO.Defect levels from Mo and W contribute to the conduction band and the E F are located within the conduction bands (table 3).The DoS curves indicate higher participation from Mo and W and are more segregated and sharper than the DoS of the Nb-and Ta-doped models.The E F are also deeper within the conduction bands in Mo-and W-doped STO, due to the larger number of carriers from these dopants (tables 2 and 3).The abnormal behaviour in Cr-doped STO may indicate that the +4 oxidation state inferred from the total magnetisation of the doped model, which is the difference in the electron density in the two spin channels, results from a lower oxidation state in one spin channel and a higher oxidation state in the other.
Across the six doped models, the general trend is that the lightest two elements result in the formation of midgap levels in one of the two spin channels, and Cr is unusual in that the Fermi energy is predicted to lie close to the valence band.These phenomena could have a significant impact on the thermoelectric performance, and the narrowing of the host bandgap due to the resonant levels could also impact the performance in photovoltaic or photocatalytic applications.
On the other hand, the four heavier elements all contribute shallow defect levels to the host conduction band shift the E F to the bottom of the conduction band, which is expected to yield the more usual n-type metallic behaviour.
Having analysed the electronic structures, the electrical-transport properties of doped models were then estimated using BoltzTraP as for pristine SrTiO 3 (figure 6), with the carrier concentrations determined automatically from the DoS and the position of the Fermi energy (figure S7 in the supporting information).
The calculations predict a positive Seebeck coefficient for Cr-doped STO in both spin channels, a positive S for Mo-doped STO in one of the spin channels, a transition from n-to p-type behaviour at around 650 K in one spin channel for V-doped STO, and negative S for Nb-, Ta-and W-doped SrTiO 3 in both spin channels (figure 6(a)).The behaviour of Cr-, Nb-, Ta-and W-doped STO, and V-doped STO at low temperature, is consistent with the positions of the E F in the calculated electronic structures (cf figures 4 and 5).For V and Cr-doped STO, the calculations predict large (absolute) S up to −2400 and +1000 µV K −1 at 300 K in one of the two spin channels and a fall in the absolute values with temperature.For W-, Nb-and Ta-doped STO, the calculations predict modest S values of up to −300 µV K −1 and an increase in the absolute values with temperature, as expected [30].The predicted hole-dominated transport in one of the two spin channels of Mo-doped STO is at odds with the location of the E F within the conduction band (cf figure 5(b)).In this spin channel, the E F is located at the top of the three-fold band introduced by Mo, and this band does not fully overlap with the host conduction band, resulting in dominant p-type behaviour.In the other spin channel, where the defect and host bands do overlap, the calculations predict the expected n-type behaviour.This unusual behaviour results in a predicted decrease in the absolute S with temperature in one spin channel and a more complex temperature dependence in the other.We attribute the latter to the balance between the increase in intrinsic carrier concentration with temperature and the change in the DoS and carrier effective mass with the corresponding position of the Fermi energy [30].
As noted above, it has been found that the carrier concentration and the nature of the dopants in STO have a small impact on the electronic relaxation times [29], and we therefore used the same relaxation times as for pristine STO to estimate the absolute electrical conductivities and PFs of the doped systems (figures 6(b) and (c)).We note that the relaxation times are only available from 400-1000 K and not the wider range of 300-1200 K over which our transport calculations were performed.
With the exception of one of the spin channels of V-and Cr-doped STO, for which the E F are located in the band gap (cf figures 4 and 5), all the calculations predict metallic behaviour from 400-1000 K. Overall, W-doped STO has the highest predicted σ, up to 10 6 S m −1 (over the temperature range examined), which we attribute to the larger carrier concentration from its estimated oxidation state (3.15 × 10 21 cm −3 ).Ta and Nb-doped STO are predicted to exhibit lower, but still relatively large σ on the order of 10 5 -10 6 S m −1 .While we would have expected doping with Mo to yield higher electrical conductivity than Nb, due the higher potential carrier concentrations, our calculations suggest this is not the case, and the marked differences in the electronic structures of the two spin channels suggests this might be attributed to (partial) electron localisation and consequent semi-metallic behaviour.Cr-and V-doped STO are predicted to have σ as low as 10 −4 S m −1 but with a strong temperature dependence, particularly in the spin channels with the largest associated Seebeck coefficients, which we attribute to the localisation of the additional electrons and consequently more significant increase in the number of thermally-activated carriers at elevated temperature.For most of the models the calculations predict conductivities of the order of 10 5 -10 6 S m −1 at 400 K, which decrease by an order of magnitude at 1000 K.These values are in good agreement with the measured σ on the order of 10 5 S m −1 for doped single-crystal SrTiO 3 at 300 K with similar doping levels [15].
The predicted PFs are compared in figure 6(c).Nb-and Ta-doped STO are predicted to have the highest PFs from 400-1000 K in one of the two spin channels.The highest predicted PF for Nb-doped STO of ∼2250 µW m −1 K −2 at 450 K is close to the largest PF of 2800 µW m −1 K −2 at 300 K reported for doped STO with a carrier concentration of 2 × 10 21 cm −3 [15].Mo-and W-doped STO show moderate PFs of 800-1700 µW m −1 K −2 in one of the two spin channels due to the balance of a large S and modest σ in Mo-doped STO, and a high σ but modest S in W-doped STO.Finally, Cr-and V-doped STO are predicted have the lowest PFs due to their relatively low σ.We therefore conclude that Cr and V doping do not result in competitive thermoelectric performance compared to doping with heavier second-and third-row Group V/VI transition metals.With the aim of avoiding heavy elements and using more abundant and lower-cost elements, Nb is much more attractive as a dopant due to its higher abundance and lower cost [65,66] together with the relatively high predicted PFs of Nb-doped STO (figure 6).Finally, we also compared our predictions for Nb-doped STO with experimental measurements on STO with Nb doping levels of 10-15 mol% [56], bracketing the 12.5% in our model (figure 7).In view of the polycrystalline nature of the experimental samples, and the fact that carriers are believed to be fully activated above 450 K based on the σ (T) being similar to single crystals, we selected data points from the experimental work at T > 450 K.The predicted electrical-transport properties for one of the two spin channels are in good agreement with the experimental work, which further validates our modelling approach.We also find that our predicted S and σ are closer to the experimental data at higher doping levels and temperatures, which suggests a more prominent impact of defects and grain boundaries at lower n and T, in keeping with our earlier observation that larger carrier concentrations may be needed in polycrystalline samples to mitigate the impacts of defects and grain boundaries.

Conclusions
In this work, we have modelled the electronic structure and electrical transport properties of pristine SrTiO 3 and developed a series of models for Sr(Ti 0.875 M 0.125 )O 3 doped with Group V and VI transition metal elements.
Careful testing of the calculation parameters against experimental lattice constants and bandgaps indicated that PBEsol + U with a Hubbard U correction of 3.25 eV applied to the Ti 3d orbitals accurately predicts the experimental lattice constant while also giving an improved prediction of the bandgap compared to 'bare' PBEsol.Transport calculations using electron relaxation times from the literature reproduce measurements on single-crystal STO very well and in particular provide a good estimate of the PF.Based on these calculations, we predict an optimum carrier concentration of the order of 10 21 cm −3 for STO, and our highest predicted PF of 2600 µW m −1 K −2 at 400 K is comparable to the PFs measured in experiments on single crystals.Although the calculations predict that the PF will be reduced significantly for n > 10 21 cm −3 , we suggest that larger carrier concentrations may well be needed in polycrystalline STO to mitigate the impact of defects, including grain boundaries, on the transport properties.
Of the six dopants considered, the lighter first-row elements V and Cr lead to the formation of intermediate bands/resonant levels, resulting in low electrical conductivity and poor thermoelectric performance.The reduction in the bandgap due to these levels may however make these materials suitable for other applications such as photovoltaics or photocatalysis.Doping with Mo is predicted to yield a large Seebeck coefficient and moderate conductivity, resulting in a reasonable PF.On the other hand, doping with W produces a high conductivity but a low Seebeck coefficient, resulting in a more modest PF.The result for W may be indicative of a higher than optimum carrier concentration, and doping with W at a lower level may yield better performance.Our calculations predict that Nb-and Ta-doped STO will have the highest overall PFs, but Nb is much more attractive as a dopant due to its higher abundance and lower cost.

Figure 1 .
Figure 1.Calculated electronic band structure and atom-projected density of states (DoS) for SrTiO3 in the vicinity of the Fermi level EF obtained using (a) PBE, and (b) PBEsol.

Figure 2 .
Figure 2. (a)/(b) Dependence of (a) the optimised lattice constant a of SrTiO3 and (b) the band gap (Eg) on the Hubbard U correction applied to the Ti 3d states in calculations using the PBE and PBEsol GGA functionals.On both plots the experimentally-reported values are shown for comparison [17, 46].(c) Electronic band structure and DoS for SrTiO3 obtained with PBEsol + U = 3.25 eV in the vicinity of the Fermi level EF.

Figure 3 .
Figure 3.Comparison of the calculated (a) S, (b) σ and (c) power factor (PF, S 2 σ), obtained using semi-classical Boltzmann transport calculations with the PBEsol + U electronic structure and the relaxation times from Kinaci et al [29], with experimental single-crystal data from Ohta et al (circles) [54] and Okuda et al (triangles) [15].

Figure 4 .
Figure 4. Calculated electronic band structures and density of states (DoS) in the vicinity of the Fermi level EF for SrTiO3 doped with Group V elements: (a) V, (b) Nb, (c) Ta.The band dispersions in the two spin channels are shown by amber and cyan lines, respectively, and the corresponding DoS curves are shown to the left and right of the central axis line.

Figure 5 .
Figure 5. Calculated electronic band structures and density of states (DoS) in the vicinity of the Fermi level EF for SrTiO3 doped with Group VI elements: (a) Cr, (b) Mo, (c) W. The band dispersions in the two spin channels are shown in amber and cyan lines, respectively, and the corresponding DoS curves are to the left and right of the central axis line.

Figure 6 .
Figure 6.Calculated electrical-transport properties of donor-doped SrTiO3: (a) Seebeck coefficient S, (b) electrical conductivity σ, and (c) power factor S 2 σ (PF).The transport coefficients associated with the two spin channels are shown in the left-and right-hand columns respectively.The σ and PFs in (b) and (c) were calculated using the reported electron relaxation times τ for SrTiO3 [29].

Table 1 .
[46]mised lattice constant and cell volume of SrTiO3 obtained with the PBE and PBEsol functionals compared to the experimental values[46].

Table 3 .
Estimated bandgaps Eg and position of the Fermi level (EF) relative to the valence-and conduction-band edges Ev/Ec in pristine and donor-doped SrTiO3.Italicised Eg indicate indirect gaps.Note that V generates midgap states in one of the two spin channels, resulting in a significantly reduced Eg compared to the other systems.Negative values of Ec-EF or EF-Ev indicate that the EF lies within the host valence/conduction band.
lattice constants are compared to the PBEsol + U values in table