A Unified View of the Substitution-Dependent Antiferrodistortive Phase

Oct 21, 2016 - ... and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, .... Ma, Alvarado, Xu, Clément, Kodur, Tong, Grey, a...
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A Unified View of the Substitution-Dependent Antiferrodistortive Phase Transition in SrTiO3 Eric McCalla, Jeff Walter, and Chris Leighton* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States S Supporting Information *

ABSTRACT: The cubic-to-tetragonal antiferrodistortive transition at 105 K in the most widely studied perovskite, SrTiO3, is perhaps the preeminent example of a second-order structural phase transition. Extensive investigations since the 1960s have tracked the softening of the phonon mode associated with this transition, the development of octahedral rotations, and the interplay with incipient ferroelectricity and superconductivity. It is less well known that modest ionic substitutions alter the transition temperature (Ta) over a remarkable range in SrTiO3, from 0 K to above ambient. Here, we first study the thermodynamics of the transition via specific heat and determine a Ta shift of +10.9 K/ atomic % in Nb-substituted crystals, the most heavily studied in terms of electronic transport. We then present the first quantitative rationalization of the trends in Ta with substitution in SrTiO3. We demonstrate that the apparently complex behavior versus tolerance factor occurs simply due to a nonlinear dependence of the rate of change of Ta with substituent concentration. We then connect this to ionic valence mismatch, using bond valence concepts to establish a new parameter, ⟨ε4⟩, which exhibits a universal linear dependence with Ta for all known substitutions. This provides the first unified view of the substituent-dependent Ta in SrTiO3, deepens our understanding of the phase transition (including a theoretical maximum in the rate of Ta suppression), and demonstrates predictive power via a simply computed parameter.

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ferroelectricity.4,5,7 SrTiO3 is thus a quantum paraelectric, a low-T ferroelectric instability being prevented only by quantum fluctuations. Finally, in doped SrTiO3 (with O vacancies, Nb, or La), these mode softenings either are known to have or have been thought to have dramatic impacts on electronic transport.8−15 Electron scattering from soft phonons around Ta has been shown to be essential to understanding mobility in SrTiO3,8 while recent work has revealed transport anisotropy due to the onset of tetragonality.9 Moreover, the very large dielectric constant resulting from the T → 0 mode softening leads to large donor Bohr radii and weak impurity scattering in SrTiO3, resulting in a remarkable, low-density, high-mobility metallic state.10−12 This state supports superconductivity at the lowest electron densities in any solid.13 Superconductivity in SrTiO3 thus challenges conventional explanation, and the zone boundary soft phonons have been suggested to play a role in an unusual pairing mechanism.9,14,15 Perhaps less well known in SrTiO3 is the fact that the cubicto-tetragonal antiferrodistortive phase transition temperature, Ta, varies over a wide range with even modest ionic substitution. The literature on this topic can be mined for Ta data as a function of A site substitution (i.e., Sr1−xAxTiO3), with A = Ba2+, Pb2+, Ca2+, Y3+, Gd3+, La3+, or Bi3+, and as a function

INTRODUCTION The fact that the most heavily studied perovskite oxide, SrTiO3, undergoes a cubic-to-tetragonal structural phase transition at 105 K has been known for more than 50 years.1−3 The essential character of this transition has been understood since the 1960s, the TiO6 octahedral units rotating around the c-axis in an antiferrodistortive pattern upon cooling (i.e., with alternating senses of rotation in adjacent octahedra), displacing the O ions, and lowering the space group symmetry from 3−5 The structures above and below the Pm3m ̅ to I4/mcm. transition temperature, Ta, are illustrated in panels a and b of Figure 1, the octahedral tilt angle reaching ∼2° at low temperatures, and the c/a ratio deviating from unity by ∼0.001.2,6 For a number of reasons, this structural transition has taken on unusual importance in solid-state chemistry and condensed matter physics. First, polymorphic phase transitions are abundant in perovskites, and the status of SrTiO3 as the most intensively studied example has afforded this transition particular importance. Second, this transition was one of the first structural phase transformations understood to be secondorder, driven by softening of a Brillouin zone boundary phonon as Ta is approached from above.3−5 SrTiO3 was in fact one of the first systems studied via soft-mode spectroscopy, using both Raman3,5 and inelastic neutron scattering.4 Third, this mode softening is accompanied by a second softening as T → 0, this time of a zero wavevector phonon, leading to incipient © 2016 American Chemical Society

Received: August 31, 2016 Revised: October 7, 2016 Published: October 21, 2016 7973

DOI: 10.1021/acs.chemmater.6b03667 Chem. Mater. 2016, 28, 7973−7981

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made to rationalize this behavior, with various degrees of success.15,17−19 Several authors, for example, have advanced strain-based arguments, pointing out (consider Figure 1b) the fact that the octahedral tilting in the tetragonal phase should be favored by a smaller rA, which liberates more space for BO6 octahedral rotations.15,17−19 Ta would thus be expected to increase for low-rA substitutions, as is indeed the case for Y3+, Gd3+, La3+, Ca2+, etc. Conversely, large rA substitutions such as Pb2+ and Ba2+ would be expected to decrease Ta, which is also the case. Similar qualitative thinking can be applied to the B site, a smaller rB being expected to decrease the stability of octahedral rotations, thus suppressing tetragonality to lower Ta. This is evidenced by the case of Mn4+, although data for this system are sparse. It is noteworthy in this context that Ta data have also only very recently become available for SrTi1−xNbxO3,9 despite this system’s importance in the literature on doped SrTiO3.12 These data derive from the recently discovered resistance anisotropy and are included in Figure 1c, showing an increase in Ta with Nb content, as expected on the basis of strain. It must be noted that the arguments mentioned above are qualitative and that, despite attempts, quantification has not been satisfactorily demonstrated. For example, Tkach et al.19 used strain-based arguments to attempt to collapse the individual branches in the Ta versus t plot for a partial set of the substitutions mentioned above (Y3+, Gd3+, and La3+ on the A site) but to do so were forced to include an adjustable value for the radius of Sr vacancies, which were simultaneously introduced to balance charge. In cases such as Sr1−xLaxTiO3, however, it has been established, in highly perfect samples,23 that trivalent La substitution is actually charge balanced by electron doping, bringing into question even the existence of charge balancing Sr vacancies. In addition, while data are available in some cases for the behavior of the octahedral tilt angle with substitution, this has also evaded quantitative understanding. As an example, the tilt angle increases to 3.5° with Nb doping24 and 3.2° with Ca doping,15 as Ta increases and the tetragonal phase is stabilized, but quantitative understanding of the Ta−octahedral tilt angle relation remains lacking. One further potential limitation of the Ta(t) approach shown in Figure 1c is that the average tolerance factor does not capture the ionic radius variance, σr2. Well-known examples of the importance of this ionic radius variance exist in the literature on magnetic and electronic ordering temperatures in perovskite manganites25 and Sr2FeMoO6-based double perovskites,26 as well as the superconducting transition in La2CuO4based cuprates.27 In these systems, the ordering temperatures exhibit clear correlations with the variance in radii (i.e., quenched disorder), and it is possible that similar effects could occur in the data shown in Figure 1c. Looking at Ta as a function of average ionic radius and radius variance, however (see Figure S1), we find that Ta versus σr2 also exhibits no universal behavior among the various substituents that have been studied. The slopes of plots of Ta versus σr2 for Pb and Ba substitution, for example (which decrease Ta), differ by a factor of 4, while the equivalent slopes for Nb and Y substitution (which increase Ta) differ by a factor of 2. In short, there is currently no approach from the literature that provides any collapse of the available Ta data for all substituents to a single universal relation or provides any convincing quantitative or semiquantitative rationalization of the observed trends.

Figure 1. SrTiO3 perovskite structure (a) above and (b) below the cubic-to-tetragonal phase transition at Ta. Note the antiferrodistortive rotation of the BO6 (i.e., TiO6) octahedra in the tetragonal phase. The axes refer to the pseudocubic unit cell (dashed outline in panel a). (c) Published Ta values for various cation substitutions in SrTiO3 (on sites A and B) vs the resulting tolerance factor, t, calculated from the average value of the ionic radii. Data are from refs 9 and 15−21. The black dashed line marks the t of unsubstituted SrTiO3, while the colored lines are straight line fits for individual substituents; the green dotted line is an extrapolation to 0 K for Ba2+ substitution.

of B site substitution (i.e., SrTi1−xBxO3), with B = Mg2+, Mn4+, or Mg2+1/3Nb5+2/3.15−21 The results can be plotted as in Figure 1c, where the reported Ta’s are shown versus the Goldschmidt tolerance factor, t = (rA + rO)/[√2(rB + rO)], where rA, rB, and rO are the average ionic radii of the ions at sites A, B, and O, respectively. Substitutions such as Ba2+ at the A site drive Ta rapidly toward 0 K (green line in Figure 1c),17 while just a few percent Ca2+ on the A site stabilizes the tetragonal phase up to room temperature (black line in Figure 1c).21 These literature results are shown here versus t in an attempt to expose any connections with average ionic size, an approach that has yielded insight in many cases, e.g., magnetic ordering temperatures in manganites.22 The t value for unsubstituted SrTiO3 (1.0016) is marked by the vertical dashed line in Figure 1c, deviating only slightly from the ideal cubic perovskite value of 1. As shown in the plot, intriguing correlations between Ta and t indeed exist, but with significant limitations. Most notably, while substitutions that drive t upward tend to lower Ta, and the opposite tend to increase Ta, the Ta(t) relationships are typically consistent for only a given substituent. Explicitly, a linear Ta(t) occurs in a given system (e.g., Sr1−xCaxTiO3 or Sr1−xBaxTiO3), but with clear differences between substituents. In addition, certain unusual substitutions, such as Mg2+ for Ti4+, yield Ta’s that appear to be anomalous (Figure 1c). While we are unaware of any prior work that has compiled a picture as complete as Figure 1c, attempts have certainly been 7974

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Chemistry of Materials In this work, we address these shortcomings in the understanding of Ta in SrTiO3, both by filling gaps in the literature for substitution of Nb5+ for Ti4+ and by presenting a simple theoretical analysis of the stability of the low-T tetragonal phase. The former is derived from specific heat measurements that directly confirm Ta versus concentration for Nb substitution, as well as providing insight into the thermodynamics of the transition. We then proceed to demonstrate that the apparently complex behavior of Ta versus t shown in Figure 1c, which has frustrated prior attempts at rationalization, is simply due to a nonlinear t dependence of dTa/dx, the rate of change of Ta with substituent concentration. We also find that this quantity is best correlated not with the ionic size mismatch (i.e., strain), but rather with the ionic valence mismatch. Using analysis based on bond valence concepts applied to the low-T tetragonal phase, we show that this naturally leads to the definition of a new parameter, ⟨ε4⟩, essentially an averaged value of the fourth power of a relative valence mismatch. This is demonstrated to exhibit a universal linear relationship with Ta for all known substitutions in the literature on SrTiO3. The individual branches on the Ta versus t plot shown in Figure 1c thus collapse to a single universal curve, providing a considerably deepened, and predictive, understanding of the nature of this transition. As examples, an unanticipated theoretical maximum in the rate of T a suppression of −30 K/atom % is elucidated, from which we deduce an elastic interaction length scale in SrTiO3 of ∼2 unit cells, of potential use in understanding other properties of this fascinating material.

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EXPERIMENTAL METHODS

Commercial single crystals of SrTi1−xNbxO3 with x = 0, 0.2, 1.0, 1.4, or 2.0% from three suppliers (Crystec, Crystal GmbH, and MTI Corp.) were used for specific heat (Cp) measurements in a Quantum Design Physical Property Measurement System (PPMS). These crystals were previously characterized by X-ray diffraction,12 trace element analysis,28 and electronic transport.12 The PPMS employs relaxation calorimetry, during which a heat pulse is applied and the ensuing temperature increase is monitored. When a short pulse is used (“pseudostatic relaxation mode”29), the heat capacity is determined at a single temperature with high precision. This method was used for all samples here, with pulses corresponding to 2% of the measurement temperature, between 1.8 and 270 K. Complementary to this, a longpulse method was also applied. In this “scanning mode”,29 the heat capacity is determined at every temperature during the pulse, which can span >10 K. Although this method is less precise, it is of known utility at phase transitions, particularly first-order ones, where the short-pulse method can result in broadening.29 Here, the long-pulse method was used for unsubstituted SrTiO3 only, simply to confirm that the transition at Ta is not artificially broadened by the short-pulse method. The addenda heat capacity was first determined, measuring only the sample stage and Apiezon N grease used to affix the sample. Crystals were then added and measured at the same temperatures. The sample/calorimeter thermal coupling was checked at all temperatures, verifying coupling values of >95%. This criterion was violated only for unsubstituted SrTiO3, and only below 10 K, which is irrelevant for this study. The ratio of sample to addenda heat capacity was also maintained above 0.5 for all data presented here.30

Figure 2. (a) Temperature (T) dependence of the specific heat (Cp) of a 0.2 atomic % Nb-substituted SrTiO3 single crystal. The black solid line shows the background used to extract the excess specific heat due to the phase transition, ΔCp, as discussed in the text. (b) Cp vs T − Tmax for multiple Nb contents, where Tmax is the onset temperature of the transition upon cooling. The legend in panel a applies to all panels. (c) ΔCp vs T for all five SrTi1−xNbxO3 crystals. In panel c, the solid lines are guides to the eye. The temperatures parametrizing the phase transition (Tmax and Ta) are shown.

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shown more clearly in Figure 2b, for all five compositions studied, by making a close-up plot of Cp versus T − Tmax, where Tmax is the onset temperature of the Cp anomaly. The anomaly takes the form of a relatively sharp onset of excess specific heat upon cooling, with a low-T tail.31,32 To quantitatively estimate the excess specific heat, ΔCp, a smooth background was subtracted from the data. The method developed by Salje et al.

RESULTS A representative Cp(T) for a SrTi1−xNbxO3 crystal is shown in Figure 2a, in this case for x = 0.2%. The overall behavior is typical (detailed analysis of Debye temperatures and low-T electronic contributions vs x will be published elsewhere), but with a small but easily resolved anomaly at Ta. This anomaly is 7975

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Chemistry of Materials for pure SrTiO331 was employed to do this, essentially combining a quadratic fit to Cp(T) above the transition (e.g., the solid black line in Figure 2a) with a thermodynamically consistent treatment of the low-T tail (see the Supporting Information for further details). The resulting ΔCp(T) is shown for all crystals in Figure 2c, the most striking observations being the monotonic increase in onset temperature with Nb substitution and the shape of ΔCp(T), which is qualitatively independent of x. A number of details in Figure 2c require further comment. First, and as illustrated in the figure, two temperatures are needed to parametrize ΔCp(T): the onset temperature of the transition (Tmax) and the peak temperature (Ta). To reconcile the transition temperature from these ΔCp(T) data with earlier X-ray diffraction data on the onset of tetragonality, we note that we have to associate the peak temperature (not Tmax) with Ta. Second, comparisons to prior work, particularly for unsubstituted SrTiO3, require that attention be paid to the precise method of background subtraction. Reference 15, for example, shows ΔCp(T) values quite different from those depicted in Figure 2c, but this is solely due to a different choice of background (see the Supporting Information). Reassuringly, given that we used their method of subtraction, our own data on ΔCp(T) of unsubstituted SrTiO3 are similar to those of Salje et al.;31,32 we favor this method because of its improved accuracy in the integration of ΔCp(T) to determine entropy, as discussed below. Third, it must be stressed that ΔCp(T) exhibits no evidence of thermal hysteresis within our resolution (see Figure S2). Long-pulse and short-pulse calorimetry methods also provide essentially identical ΔCp(T) values (also shown in Figure S2). Taken together, these observations [specifically the “lambda anomaly” shape of ΔCp(T), the absence of hysteresis, and the equivalence of long- and shortpulse methods29] are consistent with a transition that is, at least within the limitations of our methods, second-order. This is consistent with the literature consensus about the antiferrodistortive transition in SrTiO3, and with the gradual mode softening picture, although we note that Ginzburg−Landau analysis33 indicates that cubic-to-tetragonal phase transformations are constrained to be first-order. Reconciliation of these two pictures may involve recognition that the symmetry breaking internal to the unit cell due to the TiO6 rotations can allow for a second-order cubic-to-tetragonal transition as part of Ginzburg−Landau analysis.33 Interesting in the context of this discussion is the fact that Salje et al. reported evidence that unsubstituted SrTiO3 is actually close to a tricritical point.31,32 On the basis of the data depicted in Figure 2c, Figure 3 summarizes the Nb content dependence of the key thermodynamic parameters. Figure 3a first shows Ta(x) extracted from the peak in ΔCp(T). Ta is found to shift monotonically upward with x, reaching 127 K at 2 atomic % Nb, a straight line fit (blue solid line) yielding a slope of 10.9 K/atomic %. This is in good agreement with the recent electrical resistance anisotropy data of Tao et al.9 A simple measure of the “width” of the transition is shown in Figure 3b, which plots the x dependence of the full width at half-maximum (FWHM) of the ΔCp(T) peak. Despite some potential evidence of a more rounded peak at x = 2.0 atomic % (Figure 2c), the FWHM is rather independent of substituent concentration. This latter point is better quantified in Figure 3c, which plots the x dependence of the entropy associated with T the transition, calculated from ΔS = ∫ Tmax (ΔCp/T) dT, where min

Figure 3. Summary of the phase transition thermodynamic parameters vs Nb content, as extracted from specific heat: (a) transition temperature, Ta, (b) FWHM (full width at half-maximum) of the excess heat capacity peak, and (c) entropy associated with the transition, ΔS. Dashed lines simply connect points, while the solid line in panel a is a straight line fit.

Tmin is a minimal temperature below which the excess specific heat can no longer be reliably measured and is thus not plotted in Figure 2c. This leads to unavoidable but very minor underestimation of ΔS. Regardless, ΔS is seen from Figure 3c to be essentially x-independent, over a range in which Ta shifts by 21%. To a good approximation, at least in this composition range, Ta is thus the only parameter that varies with Nb substitution in SrTiO3, the entropy removed across the phase transition being approximately constant. This is a point to which we will return in the theoretical analysis below. We also note that these entropy values, ∼0.1 J mol−1 K−1, are rather small, presumably related to the small size of the structural perturbation. As a point of comparison, equivalent values across second-order ferromagnetic ordering transitions in perovskite ferromagnets lie at ∼0.5−1 J mol−1 K−1, values that are themselves considerably suppressed compared to the full spin entropy.34 A full summary of all thermodynamic parameters associated with the transition in SrTi1−xNbxO3 is provided in Table S1. Theoretical Analysis and Discussion. Building on the results presented above on Nb substitution, we now present an analysis that provides a unified view of the substituentdependent Ta in SrTiO3, providing predictive power and a deepened understanding. The first step is the realization that the problematic multiple Ta(t) branches in Figure 1c occur simply due to a nonlinear but monotonic t dependence of dTa/ dx, the rate of change of Ta with substitution. This is illustrated in Figure 4a, which plots dTa/dx versus t for a 1 atomic % substitution of each of the elements in Figure 1c. With the exception of Mg2+, which we disregard for now but explicitly return to below, reasonable adherence to a single curve is found, revealing a strong and systematic dependence of dTa/dx on t. This holds for both positive and negative dTa/dx (i.e., t values above and below that of SrTiO3), for both A and B site 7976

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Figure 4. Differential rate of change of the transition temperature with substitution, dTa/dx, for multiple substituents vs (a) the tolerance factor at a substitution of 1 atomic %, t1%, (b) the ionic radius mismatch, Δr, and (c) the valence mismatch, ΔVX. The black dashed lines mark unsubstituted SrTiO3, and the solid lines are quadratic fits (ignoring Mg-based substitutions). In panel c, the green arrow for SrTi1−x(Mg1/3Nb2/3)xO3 points from Mg coordination of 6 to 5, while for SrTi1−xMgxO3, it points from 6 to 4. In each panel, the R2 value is shown for the quadratic fits.

ΔVX = VX − Videal

substitutions, and for both isovalent and aliovalent cases. The latter point underscores the fact that Ta is not strongly affected by electronic doping, or the formation of any charge-balancing point defect. To provide a simple quantification of the extent of collapse to a single curve, the data were fit to a quadratic form, dTa/dx = α(t1%)2 + βt1% + χ, where α, β, and χ are free parameters. As shown in Figure 4a, this fit (solid blue line) results in a correlation coefficient (R2 = 0.972) indicating a reasonable, but certainly not perfect, description of the data. Motivated by the statements in the Introduction, in Figure 4b we explore a slightly modified approach, plotting dTa/dx directly as a function of ionic radius mismatch. To do so, we define Δr = rX − rA for substitution on the A (i.e., Sr) site and Δr = −(rX − rB) for substitution on the B (i.e., Ti) site, where X is the substituent element. The change of sign here ensures that Δr is negative for strains that stabilize the tetragonal phase (small rA/large rB) and positive for the opposite case (large rA/ small rB). The result is again a strong systematic correlation with dTa/dx, with a marginal improvement in collapse to a single curve. This is evidenced by an R2 value of 0.977 from a fit to a similar quadratic form, dTa/dx = α(Δr)2 + βΔr + χ. Figure 4c presents an alternative approach to the correlation of dTa/dx with substituent element in SrTiO3, which we argue through the remainder of this paper to be more insightful, useful, and predictive. This is based not on ionic radius mismatch but rather ionic valence mismatch, specifically in the low-T tetragonal phase. A standard bond valence sum approach is utilized to do this, defining the bond valence for cation X with oxygen (O) as Sxo = e(R c − RXO)/ bc

and on the B site ΔVX = −(VX − Videal)

(1)

nX

∑ SXO i=1

(4)

This convention simply associates ΔVX values of 0 with destabilization. Our approach to implement this for various substitutions in SrTiO3 is to compute ΔVX in a single, model tetragonal structure, specifically the relatively highly tetragonally distorted one reported at low T in SrTi0.875Nb0.125O3, with average bond lengths as summarized in Table S2.24 While the exact choice of this structure is not critical because of the modest magnitude of the structural distortions across Ta, we choose here a relatively heavily distorted reference point. The |ΔVX|’s thus obtained provide a measure, in this relatively heavily distorted tetragonal phase, of the extent to which the substituent valence would deviate from the formal valence. The result for dTa/dx versus ΔVX is shown in Figure 4c, the first striking feature being the absolute magnitudes of the valence mismatches. While the ΔV’s for substituents such as Nb are modest, cases such as Y and Ba, which induce the fastest rates of Ta enhancement and/or suppression, are accompanied by |ΔV| ≈ 1.5. Explicitly, the Y ions in Sr1−xYxTiO3 in this model tetragonal structure would effectively be in a Y1.5+ state, very different from the formal Y3+ state. Such large |ΔV|’s derive from the large local strains induced by substituting cations as small as Y, qualitatively explaining how modest concentrations can have such an influence on Ta. We emphasize that the real structure in Sr1−xYxTiO3 may well distort to relieve some of this valence mismatch (continued Y substitution can result in second phases beyond 4%,38 and eventually Y-rich compositions enter the Pnma YTiO3 structure39) but that the |ΔV|’s calculated this way nevertheless provide a useful measure of the stability of the tetragonal phase. This will be further borne out by the correlations presented below. Figure 4c further shows that the collapse of dTa/dx for each substituent to a single universal curve is also improved when considering valence mismatch, as opposed to ionic size mismatch (Figure 4b) or tolerance factor (Figure 4a). This is clear from close inspection of the data and is confirmed by a fit to a quadratic form similar to that used above, i.e., dTa/dx = α(ΔVX)2 + βΔVX + χ. The resulting R2 value is 0.994, significantly improved over those for panels a and b of Figure 4.

where Rc and bc are known constants for a given X−O bond35−37 and RXO is the X−O bond distance. The valence of cation X, VX, is then obtained by summing the SXO’s over cation X’s nX nearest neighbors:

VX =

(3)

(2)

We then define the valence mismatch for a given cation, |ΔVX| = |VX − Videal|, where Videal is the ideal formal valence of the cation, e.g., 3.0 for Y3+, 2.0 for Ba2+, etc. The sign of the mismatch needs to be properly handled, and we define on the A site 7977

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any substituent in SrTiO3, and we thus define a new parameter, ε = ΔVmax − ΔVX, a relative ionic valence mismatch. Figure 5a

It must be emphasized here that the choice to correlate dTa/dx with ionic valence mismatch improves the collapse to a single universal curve regardless of the choice of fit function. This was verified by using numerous functions capable of describing the trend shown in Figure 4. Further evidence that ionic valence mismatch is more powerful than ionic size mismatch for understanding Ta in SrTiO3 is provided by the case of Mg2+, a troubling outlier in Figures 1c and 4a,b. Substitution of Mg2+ for Ti4+ is expected to be accompanied by O vacancy formation;20 this was the motivation for the study of Sr(Mgx/3Nb2x/3)Ti1−xO3−δ, where the Nb5+ is designed to charge balance the Mg2+, resulting in a net isovalent substitution for Ti4+.20 As acknowledged by Lemanov et al., however, O vacancy formation may nevertheless occur.20 An advantage of the valence mismatch approach is that this possibility is easily accounted for simply by performing the summation in eq 2 over differing numbers of O nearest neighbors. As illustrated by the green arrows in Figure 4c, significantly improved collapse to a universal curve indeed occurs for Mg and Mg1/3Nb2/3 substitutions when O vacancy formation is accounted for. This was optimized with an O coordination of 4 for Mg (as expected from charge balance) and 5 for Mg1/3Nb2/3. While the collapse remains imperfect for Mg, the situation is nevertheless distinctly improved compared to that depicted in Figures 1c and 4a,b. The remaining discrepancy could be due to the assumptions made here or structural distortions/accompanying defect formation for which we do not account. The fact that a lower O vacancy density is required for Mg1/3Nb2/3 compared to Mg also fits expectations, confirming that co-doped Nb mitigates O vacancy formation. A further intriguing feature of Figure 4c is that while dTa/dx continues to grow at large negative ΔVX’s (i.e., at Gd, Y, and beyond), at large positive ΔVX’s (near Ba) this is not the case. In fact, one could hypothesize on the basis of these data that an asymptotic negative value of dTa/dx is being approached at large positive ΔVX’s, between approximately −20 and −30 K/ atomic %. This would imply a maximal possible rate of destabilization of the tetragonal phase with substituent addition in SrTiO3, which we now argue has a simple physical origin. As an extension of the arguments in the Introduction, this would require a substituent that maximizes the ideal A−O bond length or minimizes the ideal B−O bond length; these ideal bond lengths refer to the values obtained when VX = Videal. Both of these limits are actually easily understood: The ideal B−O bond length being driven to zero corresponds to complete collapse of the BO6 octahedra, while the ideal A−O bond length obviously has a maximal value, the hard limit being rA → 0.5a, corresponding to touching of hard sphere ions. In hindsight, there are thus simple geometrical reasons to expect such a maximal negative value of dTa/dx. Considering an isovalent (i.e., Videal = 4) B site substituent that achieves this maximal destabilization rate, we set RXO equal to 0 in eq 1 and use the near-universal value of 0.37 for bc, along with eqs 2−4, to calculate the maximal possible ionic valence mismatch for any substituent in our model tetragonal state (ΔVmax = 3.98). We note that this corresponds to VX = 0.02, i.e., close to VX = 0 (essentially the point at which the B site ion is no longer a cation), which is also consistent with the concept of a hard limit on B−O bond length. As would be expected, achieving this level of valence mismatch on the A site requires an ionic radius of 0.45a, rather close to the 0.5a limit for the A site. On the basis of the observations described above, we argue that this ΔVmax of 3.98 is a natural reference point for ΔV for

Figure 5. (a) Differential rate of change of the transition temperature with substitution, dTa/dx, vs relative valence mismatch, ε = ΔVmax − ΔVX, for all data in Figure 4, excluding Mg substitutions. The solid line is the fit discussed in the text to dTa/dx = γεg + η. (b) Transition temperature, Ta, vs ⟨ε4⟩, along with a straight line fit (solid line) ignoring Mg. The green arrow illustrates the effect of changing the oxygen coordination of Mg; the arrow points from coordination of 6 to 4. All symbol colors match the legend in Figure 1c, the relevant references being 9 and 15−21. The inset of panel b shows only the Nb-doped samples, from this work (black filled circles) and ref 9 (blue diamonds), along with the fit from the main panel.

shows dTa/dx versus ε, which indeed shows the expected behavior: collapse to a universal curve, dTa/dx passing through zero near ε = 3.98, and a negative intercept on the dTa/dx axis. The latter observation is supported by the solid blue line, which is a fit to dTa/dx = γεg + η, with γ, η, and g being fitting parameters. The best fit yields the following: γ = 0.138 K/ atomic %, η = −30.2 K/atomic %, and g = 3.97. We will return to this fit below, providing justification both for the choice of this form and for the significance of g ≈ 4, but for now, we use it only to illustrate the existence of the −30.2 K/atomic % maximal rate of destabilization of the tetragonal phase. This −30.2 K/atomic % value itself actually yields new insight into SrTiO3. Specifically, this value indicates that the hypothetical substituent capable of maximally destabilizing the tetragonal phase in SrTiO3 would bring Ta to 0 K with only 3.5% substitution. We first note that Ba is actually quite close to this situation, rBa being 0.41a (compared to the maximum of 0.5a) and dTa/dx reaching −25 K/atomic %.17 Ba would thus 7978

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Chemistry of Materials be expected to suppress Ta to 0 K with only 4.2% substitution, a situation that is prevented in reality only by the onset of ferroelectricity.17 Explicitly, at x > 3% in Sr1−xBaxTiO3, the softening of the zero wavevector phonon occurs faster than the zone boundary phonon softening, leading to a ferroelectric Curie point that cuts off the cubic-to-tetragonal transition. The second interesting point about the special value of −30.2 K/ atom %, or the 3.5% critical concentration, is that it can be used to extract at least an approximate length scale for ionic/elastic interactions in SrTiO3. Specifically, 3.5% is essentially a concentration of 1 part in 27, corresponding, in the oversimplified case of ordered impurities, to 1 substituent per 3 × 3 × 3 unit cells. We propose that this is the concentration at which interactions between impurities become significant in SrTiO3, stabilizing the cubic phase. Recognizing that the impurities are randomly distributed leads to a three-dimensional site percolation problem, analogous to rigidity percolation.40 The percolation threshold in such a problem is 31% on a simple cubic lattice,41 which can be combined with our 3.5 atomic % critical concentration to yield an approximate length scale over which the impurities influence the local structure. Explicitly, this length scale, essentially the diameter of the sphere of influence of a single impurity, can be approximated by (31/3.5)1/3, which yields 2 unit cells. At this mean separation, the substituent ions interact sufficiently to stabilize the cubic phase to T = 0, preventing octahedral rotations and tetragonality. Notably, this length scale compares well to others in related materials. In doped BaTiO3, for instance, Wang et al. deduced that the Coulombic interaction length scale required to maintain ferroelectricity is 1−2 unit cells.42 Similarly, a number of works have determined the length scale over which octahedral tilt patterns imposed by lattice matching relax in perovskite films, again yielding a few unit cells.43 We return now to further understanding of Figure 5a, particularly the fit to dTa/dx = γεg + η. The fact that this is worthwhile is dramatically illustrated by Figure 5b, which, on the basis of the determination of g ≈ 4 in Figure 5a, plots Ta versus ⟨ε4⟩, where N

⟨ε 4⟩ =

∑ X=1

(ΔVmax − ΔVX)4 N

to the various substitutions (a single model tetragonal structure was used in the computation of ε). Importantly, in addition to describing all data on cation substitutions available in the literature, this relation is also predictive. This we demonstrate by applying this analysis in a situation significantly different from those considered above. This arises from the fact that such aggressive reduction has been achieved in SrTiO3−δ that the resulting O vacancy density has been observed to shift Ta. Hastings et al.44 and Bauerle et al.45 found that Ta decreased to ≥70 K after high-temperature annealing in H2, deducing from Hall effect measurements of the free electron density, n, a shift of dTa/dn of 1.1−1.4 × 10−19 K cm3. Within our approach, the relevant defects are fivecoordinated Ti ions, for which ΔV = 0.714, Figure 5a predicting dTa/dx = −15.0 K/atomic %. Assuming two free electrons per O vacancy (i.e., per five-coordinated Ti) then yields a predicted dTa/dn = 0.89 × 10−19 K cm3. While this is already in very reasonable agreement with experiment, it should be emphasized that there are also good reasons to expect fewer than two electrons per O vacancy, including compensation by defects,12,46 and trapping of O vacancy-generated electrons at in-gap states.47,48 Perfect agreement between our analysis and experiment can in fact be achieved with 1.3−1.6 free electrons per O vacancy, in very reasonable agreement with compensation ratios found in Nb-doped SrTiO3 at similar n values.12 With further calibration, we note that measurement of Ta could conceivably be developed as a means to estimate O vacancy density in SrTiO3 samples. Clearly, the final issue to address is the physical basis for the dTa/dx = γεg + η relation deduced from Figure 5a, particularly the meaning of our finding that g ≈ 4, which is central to Figure 5b. We first note that a number of approaches in inorganic solid-state chemistry have linked ionic valence mismatches from bond valence sums to the binding (cohesive) energies of crystals. One popular approach uses a parameter termed the global instability index, G, defined as ⎡ N (ΔV )2 ⎤1/2 X ⎥ G = ⎢∑ ⎢⎣ X = 1 N ⎥⎦

(6) 49

where N is the number of cations in the unit cell. Minimizing G, which is simply the ionic valence mismatch averaged over the structure, has proven to be effective in structure prediction in a wide variety of materials, demonstrating its relation to binding energy. A pertinent example is provided by the software package SPuDS,50,51 which has been shown to reliably predict the crystal structures of perovskite oxides of arbitrary composition. This is achieved via minimization of G, thus predicting which of the octahedral tilt systems elaborated by Glazer52 is preferred in a given perovskite. One step beyond the use of G minimization in structure prediction are the attempts made to quantitatively relate valence mismatch directly to energy. In the context of Li ion conductors, for example, relations of the form

(5)

This is just ε = ΔVmax − ΔVX, the relative valence mismatch, averaged over all cations. The outcome (Figure 5b) is the central result of this work, a universal relation, for all known substitutions in SrTiO3, between Ta and a simple, easy to calculate parameter (⟨ε4⟩). All 34 data points from the literature collapse to a single straight line on this plot, along with Mg2+ [after accounting for O vacancies (green arrow)], and the points determined here for SrTi1−xNbxO3 (also highlighted in the inset). We note that the intercept at Ta = 0 simply corresponds to the ⟨ε4⟩ value required to completely destabilize the tetragonal phase. It is important to emphasize at this stage that while the choice of reference point ΔVmax has some influence over the exact g value obtained in Figure 5a, the collapse of Ta to a single universal linear relationship to some power of ⟨ε⟩ is always possible. In our case, we have provided multiple simple arguments for ΔVmax = 3.98 (i.e., close to 4), which we find results in a g very close to 4. We also note that the correlation shown in Figure 5b is obtained without accounting for any average or local structural distortions due

N

E = E1

∑ (ΔVX)g X=1

+ E2

(7)

have been postulated,53 where E1 and E2 are constants and the exponent g has special significance. Interaction potentials such as the Morse potential have been explicitly shown to be consistent with eq 7 with g = 2, although other g values are certainly conceivable, particularly even numbers. On the basis 7979

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Chemistry of Materials of these concepts, Adams et al.53 accurately reproduced Li ion transport pathways in various solid electrolytes, validating this approach, although the limits of applicability appear to be unknown. In light of panels a and b of Figure 5, we propose that eq 7 indeed applies to SrTiO3 (apparently with g ≈ 4), kBTa, where kB is Boltzmann’s constant, being proportional to the binding energy of the tetragonal phase. Using ε = ΔVmax − ΔVX rather than ΔVX in eq 7 then simply gives this energy relative to the maximally destabilized tetragonal case, directly yielding Ta ∝ ∑ΝX=1ε4, thus explaining the correlation shown in Figure 5b. It is then simple to show that dTa/dx ∝ ε4, in turn explaining the correlation shown in Figure 5a. It should be noted here that the argument that kBTa is proportional to the binding energy of the tetragonal phase requires some specific assumptions, including that the equivalent energy of the cubic phase and the entropy associated with the transition remain approximately constant with substitution. The latter is directly supported by the data depicted in Figure 3c, at least over the doping range probed. Proportionality between the transition temperature and the binding energy of the low-T phase may also be predicated on a second-order nature for the transition. As a final comment, we note that further work along the lines of the analysis presented here is clearly worthwhile. Similar methods directly relating transition temperatures to valence mismatch parameters could be applicable to other systems exhibiting second-order structural phase transformations, while future theoretical work could elucidate the applicability of eq 7 to SrTiO3, including the fundamental origin of g = 4. While the latter is clearly evidenced by the data depicted in Figure 5, further theoretical understanding is needed.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by the U.S. Department of Energy through the University of Minnesota Center for Quantum Materials, under Grants DE-FG02-06ER46275 and DE-SC0016371. E.M. additionally thanks the Fonds de Recherche du Québec, Nature et Technologies, and the Natural Science and Engineering Research Council of Canada for financial support. Parts of this work were performed on equipment funded by the University of Minnesota NSF MRSEC. We thank Boris Shklovskii, Kostya Reich, Ian Fisher, and Ram Seshadri for valuable discussions, as well as Dan Frisbie for a critical reading of the manuscript.

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REFERENCES

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CONCLUSIONS Subsequent to presenting heat capacity measurements probing the thermodynamics of the antiferrodistortive structural phase transformation in Nb-doped SrTiO3, we have offered the first theoretical analysis capable of a unified view of the transition temperature in substituted SrTiO3. This analysis recognizes that previous attempts were frustrated by a nonlinear dependence of the rate of change of transition temperature with tolerance factor. Ionic valence mismatch is then thought to provide the best correlation with this rate of change in transition temperature, revealing a number of new insights into this long-studied phase transition. These lead directly to a new parameter, ⟨ε4⟩, easily computed from the ionic valence mismatch in the tetragonal phase using bond valence sums. The antiferrodistortive transition temperature was shown to follow a universal linear relation with this parameter for all known cation substitutions in the literature on SrTiO3. This provides the first unified view of the substituent-dependent transition temperature in SrTiO3, deepened understanding of several aspects of the phase transition, and demonstrated predictive power based on a simply computed parameter.

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thermodynamic parameters), and a tabular summary of parameters used in the bond valence sum calculations in the tetragonal phase (PDF)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.6b03667. Transition temperature versus ionic radius variance, additional details about heat capacity measurements (including background subtraction issues, short vs long heat-pulse results, and a summary of extracted 7980

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