Evidence of Lattice Strain as a Precursor to Superconductivity in BaPb_0.75Bi_0.25O₃

Figure 1(a) shows the crystal structure of the compound under study; also shown are the Bi(Pb)-O-Bi(Pb) angles of the apical and basal planes of the two phases. Panels (b) and (c) for the same figure (1) show FE-SEM micrographs of the BaPb_0.75Bi_0.25O₃ pellet. The average grain size was estimated to be $\sim$8 $\mu$m.

In figure 2, we show the typical Rietveld fitting of the xrd patterns of BaPb_0.75Bi_0.25O₃ collected at 10 K and 300 K. A two-phase Rietveld refinement, consisting of tetragonal I4/mcm phase and orthorhombic Ibmm phase, has been performed to fit the experimental data; the crystal structure and the room temperature lattice parameters are in line with the earlier worksMarx et al. (1992); Climent-Pascual et al. (2011); Giraldo-Gallo et al. (2015); Nicoletti et al. (2017). The absence of unindexed peaks points to the purity of the sample. The lattice parameters and atomic positions obtained from the refinement of the patterns collected at 300 K and 10 K are shown in Table 1.

The quality of fit, when modelled using a single phase – either tetragonal I4/mcm or orthorhombic Ibmm – is significantly lower as compared to the two-phase fit. The inset (b1) of figure 2(b) shows the fit of the peaks in the range $76^{\circ}\leq 2\theta\leq 78^{\circ}$ when modelled using a single tetragonal phase. A similar fit is observed when the xrd patterns are refined using a single orthorhombic phase (not shown). Inset (b2) of figure 2(b) shows the peaks in the range when fit using a diphasic model. A sample with tetragonal symmetry shows a single pair of (404) peak (corresponding to Cu K_α1 and Cu K_α2)Hashimoto and Kawazoe (1993). On the other hand, in case of orthorhombic symmetry, this peak splits into (404) and (044). To ensure a proper fit, it is necessary to model data using two phases, which result in the decomposition of twin peaks into its 6 constituent peaks – one pair corresponding to (404) reflection from tetragonal phase, one pair each corresponding to (404) and (044) reflections from the orthorhombic phase – thus indicating the coexistence of I4/mcm and Ibmm phases.

In figure 3(a), we show the resistivity versus temperature data for the compound under study. As the temperature drops, the resistivity gradually increases till $\sim$250 K, below which a change of slope is observed. The resistivity increases rapidly until $\sim$10.5 K ($T_{C}$), below which it drops sharply indicating the onset of superconductivity, as seen in the inset of figure 3(a). The figure 3(b) shows the field-cooled (FC) magnetic moment of the sample as function of temperature. As reported in the literatureUchida et al. (1988); Hashimoto et al. (1994), the sample is diamagnetic in the entire temperature range. The magnetic moment shows sharp drop at around $\sim$11 K indicating the onset of the superconducting transition. The combined resistivity and dc magnetic moment measurements suggest the onset of superconductivity at $T_{C}\approx 11$ K.

Figures 4(a)-(b) illustrate the fit of the resistivity data in two temperature ranges, using different models. The resistivity data in figure 3(a) is roughly divided into 3 regions based on this. In region 1, ($\sim 250$ K - 300 K), the electrical transport is dominated by activated behaviour, which is appreciable when sufficient thermal energy is available for the electron to be excited across the band gapSchnakenberg (1968); Greaves (1973); Kumar et al. (2007). It is important to note that the studies by Uchida et al.Uchida et al. (1987), indicate that temperature dependence of normal-state resistivity of BaPb_1-xBi_xO₃ is like that of semiconductors for $x>0.20$. Figure 4(a) shows the variation of $\log\rho$ as a function of $1/T$, modelled by equation 1.

where, $\rho_{0}$ is a pre-exponential factor, $E_{G}$ is the activation energy, and $k_{B}=1.38\times 10^{-23}$ JK^-1 is the Boltzmann’s constant. From the straight line fit of the data, we obtain an activation energy of $E_{g}=7.6$ meV.

As the temperature drops below 250 K, it’s not possible to model the data using Arrhenius-type behaviour. We observe a transition of the resistivity behaviour from Arrhenius to variable range hopping (VRH). Mott variable-range hopping model has a characteristic temperature dependence of resistivityMott (1972); Mott and Davis (2012) given by

where $p$ is a parameter that describes the dimensionality of the conduction in the system, and $\rho_{0}$ is a pre-exponential constant. $T_{M}$ is the characteristic Mott temperature, which describes the energy barrier of an electron hopping from one localized state to another, given by the equation 3.

Here, it is important to note that it was not possible to fit the entire temperature range below 250 K, till the $T_{C}$, with a single curve. It was necessary to split them into separate regions with distinct characteristic temperatures and localization lengths. However, while we were able to fit the data using equation 2 mathematically, we found that the ratio of mean hopping distance to electron localization length, given by equation 4, is much less than 1, which is an indication that equation 2 is not applicable in our caseRosenbaum (1991).

In the intermediate temperature range, the resistivity data can be fit using Greaves VRHGreaves (1973). The acoustic phonon contribution to the resistivitySchnakenberg (1968); Greaves (1973), takes the form as given in equation 5,

where, $A$ is the pre-exponential constant and $T_{G}$ is the characteristic temperature of Greaves VRH.

For Greaves VRH to be valid, the plot of $\ln(\rho/\sqrt{T})$ versus $T^{-1/4}$ should be linear. In this intermediate temperature range ($\Theta_{D}/4<T<\Theta_{D}/2$) where the deviation from the fit is observed has been taken to be $\Theta_{D}/2$ in the higher temperature limit, and $\Theta_{D}/4$ in the lower temperature limitSakata et al. (1999); Kumar et al. (2007). Below this temperature, we have been unable to model the resistivity using this equation. Thus, the best fit was observed in the temperature range $\sim$76 K to 166 K, and we obtained the characteristic temperature of $T_{G}=27563$ K. Using the value of $N\left(E_{F}\right)=2.2\times 10^{21}$ states/cm³, as reported by Kitazawa, et al.Kitazawa et al. (1985), we obtain the electron localization length of $\xi=15.49$ Å. The deviation of the resistivity from the fit begins at $\sim$76 K at the lower end, and $\sim$166 K at the higher end. Thus, we may estimate that the Debye temperature of the compound lies between $\sim$304 K to 332 K.

To understand the structural link with the transport properties and understand the behaviour of phase separation, we have carried out temperature dependent xrd on the compound. Figures 5-7 show the variation of the lattice parameters, bond lengths, bond angles, lattice strain, and unit cell volume with temperature. The xrd patterns were analysed using Rietveld profile refinement and the structural parameters were obtained as a function of temperature. The goodness of fit obtained is in the range 1.45 – 1.63 for all temperatures. As the temperature drops from 300 K to 10 K, we observe that the number of peaks remain the same. This suggests the absence of any structural transitions at lower temperatures, in contrast to some of the earlier studiesOda et al. (1985); Asano et al. (1988). Moreover, we observe that the fraction of the tetragonal phase is $\sim$70%, and is largely temperature independent (see figure 7(a)).

Keeping these results in mind, revisiting the resistivity studies of the previous section gives an opportunity to understand the high transition temperature of VRH from Arrhenius type behaviour. Firstly, at low temperature, the thermal energy available for excitation of electron is insufficient, thus leading the electrical transport behaviour away from activated behaviour. In such a scenario, it is favorable for the electron to hop to a site farther than the nearest neighbour with a lower potential, giving rise to observed VRH behaviourKumar et al. (2007). Furthermore, we hypothesize that this high transition temperature is due to the role of disorder in the system. Two leading contributions to disorder are (a) the structural phase coexistence of orthorhombic and tetragonal phases at all temperatures, with fraction of the tetragonal phase constant at $\sim$70%, and largely temperature independent, and (b) the composition lying within the transition region between metal and semiconducting regions. The work on BaPb_1-xBi_xO₃ by Uchida et al.Uchida et al. (1987) suggests that the compound under study lies in the region where both metal and semiconducting regions co-exist.

Figures 5(a) and 5(b) illustrate the variation of the lattice parameters of I4/mcm and Ibmm phases with temperature. In order to facilitate the comparison of the structural data with resistivity measurements, the entire temperature range have been divided into three regions, identical to that in figure 3(a).

In the activated region, the $a$ parameter of the tetragonal phase, decreases by 0.0014%, whereas $c$ increases by 0.001%. As the temperature drops to region II, the decrement in $a$ is 0.0022%, and the increment in $c$ is 0.001%. In region III, the lattice parameters are nearly constant within the range of experimental error.

The antiphase Bi(Pb)-O₆ octahedral rotations about the $c$-axis increase with decreasing temperature. The decreasing Bi(Pb)-O₂-Bi(Pb) bond angle results in the contraction of the $a$ parameter. This reduction in the bond angle effectively contracts the interstitial volume around the Barium atom, thus increasing the O₁-Ba-O₂ angle, which drives the increase in the $c$ parameter with decreasing temperature.

The lattice parameters of the Ibmm phase, however, paint a more complex picture. The figure 5(a) and (b) respectively show the variation of $a$, $b$, and $c$ parameters of the Ibmm phase. The lattice parameters $a$ and $c$ reduce monotonically until $\sim$75 K, dictated by the reducing bond lengths and bond angles; further, $b$ decreases monotonically until the beginning of region II, and remains constant in region II. Below $\sim$75 K, $a$ increases, while $b$ decreases; $c$ remains nearly constant. The interplay between the bond lengths and bond angles leading to the observed behaviour of the lattice parameters are easily understood from graphs in panels (e) and (f) of figure 5.

From figures 6(a) and (b), we clearly observe that the nature of Bi(Pb)-O₆ octahedra is different in the two phases. In the case of tetragonal phase, there are two long apical Bi-O bonds and four short basal Bi-O bonds, while in the case of orthorhombic phase, the trend is reversed. It is interesting to note that, Gallo et al.Giraldo-Gallo et al. (2015), have observed that the phase separation takes the form of partially disordered nanoscale stripes. It is possible that the difference in the nature of distortion could be one of the reasons for such stripes. These experiments were carried out by the authors on single crystalline BaPb_0.75Bi_0.25O₃ and have observed nanoscale structural phase separation.

To understand the nature of phase separation in the present work, one can propose two scenariosChmaissem et al. (2003); Sharma et al. (2022) (a) two phases coexisting in the same grain; (b) two phases lying in two different grains. In the first scenario, it is expected that one phase grows at the cost of the other while in the second case the phase fraction remains the same. Towards this, we have calculated the average size of the structural phases as a function of temperature using Scherrer formulaPatterson (1939) (equation 6).

where, $K=0.9$ is a constant, $\lambda=1.5405$ Å is the x-ray wavelength, $\beta$ is the FWHM of the peak at angle $2\theta$. Our results show that the size of the structural phases remains the same through temperature range of study. This behaviour suggests the second scenario proposed above. However, temperature dependent transmission electron microscopy (TEM) studies will be helpful to unravel the nature of phase separation in this polycrystalline compound.

Graphs in figure 6(c) and (d) show the temperature variation of tetragonal and orthorhombic strain. In the forth coming section, we will be using the term lattice strain synonymous to tetragonal/orthorhombic strain. In the tetragonal phase (figure 6(c)), we observe the lattice strain increases with decrease in temperature and in the orthorhombic phase (figure 6(d)), the lattice strain decreases until around 125 K and below this temperature it is found to increase. It is interesting to note that in the temperature range close to $T_{C}$, the lattice strain is maximum possibly playing a role as precursor for superconductivity. Such behaviour has been observed in the case of YBa₂Cu₃O₇ compoundHorn et al. (1987), where the orthorhombic strain is found to be maximum around $T_{C}$ with little to no anomaly in the unit cell volume. However, in the present study we observe an anomaly in the unit cell volume around 35 K and 65 K, in the tetragonal and orthorhombic phases, respectively, (see figure 7(c)). In this figure, the anharmonic part of the lattice contribution to the unit cell volumes was obtained by fitting with Debye model, as given by equation 7.

where, $V_{0}$ is the volume at absolute zero, $N$ is the number of atoms per unit cell, $\gamma$ is the Grünesian parameter and $\Theta_{D}$ is the Debye temperature. The three fitting parameters – $V_{0}$, $\left(9\gamma Nk_{B}\right)/B$ and $\Theta_{D}$ – were determined using the least square fitting, are shown in table 2. We readily see that the $\Theta_{D}^{\rm tetra}=330.4$ K, the Debye temperature of the majority phase fraction of the compound, obtained from the fit matches well with the estimate obtained from the resistivity data, as previously discussed.

Panels (a)–(e) of figure 8 show the core level spectra of Ba 3$d_{5/2}$, Ba 4$d_{5/2}$, O 1$s$, Bi 4$f_{7/2}$, and Pb 4$f_{7/2}$, respectively, at 300 K and 30 K. Our results show that with decrease in temperature there is a decrement in the width of the core level peaks. This is very clearly observed in the case of Ba and O 1s core levels. The Bi and Pb 4$f_{7/2}$ core levels exhibits two features labelled as A and A*. Such features have been observed in Bi core level spectra by Prachi et al.Telang et al. (2022) in the case of Bi based pyrochlore iridate where the sample is metallic and have attributed the features A and A* to be of poorly and well screened features. Such two features have also been observed in the case of Pb 4$f_{7/2}$ by Payne et al.Payne et al. (2007), in the case of PbO₂ sample that is metallic. It is interesting to note that in the case of BaBiO₃, that is semi conducting, only poorly screened feature is observedBharath et al. (2019); Plumb et al. (2016). The well screened feature arises when the Bi/Pb 4$f_{7/2}$ core hole is screened by the transfer of electrons from the ligand and the poorly screened feature arises when no such transfer occurs.

The comparison of xps and He_II valence band spectra collected at 300 K is shown in figure 9. Finite DOS at $E_{F}$ is clearly visible in both xps and He_II spectra. We observe six features in the spectra labelled as A, B₁, B₂, B₃, C and D. The xps valence band represents the bulk features of the sample. To identify the features, band structure calculations were carried out on BaPb_0.75Bi_0.25O₃, in the tetragonal phase using DFT method, as shown in figure 10. Our results show that the total density of states (TDOS) of TB-mBJ is shifted towards lower energy as compared to the LDA calculations, as seen in panel (a) of figure 10. The partial density of states (PDOS) are plotted in panels (b)–(f) of figure 10. While LDA calculations generally sufficient in case of metallic compounds, it has been well documented that LDA underestimates the band gap at the fermi levelPerdew and Levy (1983); Sham and Schlüter (1983) of semiconducting or insulating compounds. This was the case of the parent compound, BaBiO₃, where the gap obtained in the case of LDA calculations was underestimated as compared to the TB-mBJ calculationsBharath et al. (2019). However, the compound under study, BaPb_0.75Bi_0.25O₃, exhibits both metallic and semiconducting behaviours, as discussed previously. Further, we’re also probing the effects of Pb doping in BaBiO₃ to understand how the band gap observed in BaBiO₃ closes upon Pb doing. Thus, TB-mBJ calculations have been employed to understand the electronic structure of BaPb_0.75Bi_0.25O₃. The experimental spectra were matched with the TB-mBJ calculations with a finite energy shift. Towards this, in the present work, a rigid shift of -0.8 eV was given to the DOS obtained from the TB-mBJ calculation to match with the experimental spectrum.

Our results show that there is finite DOS at the Fermi level that is in line with the experimental spectra. The calculated DOS can be divided into five regions. In region I (-14 to -12 eV), there is significant contribution from Ba $5p$ states and weak contributions from Bi/Pb $6s$ and Pb $6d$ states. Region II (-12 to -7 eV) covers dominant contributions from Bi/Pb $6s$ and weak contributions from rest of the DOS. Region III (-7eV to -1.5 eV) has contributions from Bi/Pb $6s$, $6p$, $6d$, $5f$ and O $2p$ states. In region IV (-1.5 to $E_{F}$) there is dominant contribution from Bi $6s$ and weak contributions from other states. The region V (above $E_{F}$ to 2 eV) has dominant contribution from Pb $6s$ and weak contributions from rest of the states. We now identify the different features based on the above results. The feature $D$ represents surface cleanliness. In the case of xps, the intensity of this feature is low as compared to the He_II spectra due to higher sensitivity to surface oxygen. The features B₁, B₂ and B₃ covers region III and the feature $A$ encompasses region II. The photoionisation cross section for Al K_α source is more for Bi/Pb $6s$ as compared to the photoionisation cross section of O $2p$, while in the case of He_II spectra, the reverse is true. Hence, the higher intensity of feature $A$ observed in xps as compared to He_II spectra suggests the contributions from Bi/Pb $6s$ states.

Pb doping introduces a significant change in the electronic structure near $E_{F}$ as compared to BaBiO₃. An insulating gap at $E_{F}$ observed in BaBiO₃Bharath et al. (2019) closes on 75% Pb doping. The region between -3 to -1.4 eV which was identified as feature $B$ in BaBiO₃Bharath et al. (2019) splits into 3 features upon Pb doping, namely, B₁, B₂ and B₃. All the three features have dominant contributions from O $2p$ states. In figure 10 (g), we have presented the TB-mBJ calculations of monoclinic BaBiO₃ (labelled as BBO (Mono)), BaBiO₃ using the structural parameters of BaPb_0.75Bi_0.25O₃ (labelled BBO (Tetra)) and BaPb_0.75Bi_0.25O₃. By performing TB-mBJ calculations on BaBiO₃ using the structural parameters of BaPb_0.75Bi_0.25O₃, we intend to understand the origin of states at $E_{F}$. Our results show that there is finite DOS at $E_{F}$ in this case. This suggests that the finite DOS appearing at $E_{F}$ is mainly driven by the crystal structure. Further, the DOS at Fermi level was observed to decrease upon Pb doping, as seen clearly in the inset of figure 10 (g). To ascertain that the position of bismuth does not play an important role in the electronic structure of BaPb_0.75Bi_0.25O₃, we repeated the TB-mBJ calculations by substituting the bismuth atom in each of the different positions of lead. The total DOS was observed to be identical in all the cases.

Having identified the states that contribute to the Fermi level ($E_{F}$), we now look into the evolution of the states close to $E_{F}$. In figure 11(a), we show the temperature dependent valence band collected using HeI spectra. All the spectra are normalised at 100 meV. Our results show that all the spectra show a finite intensity at $E_{F}$ and this intensity is found to decrease on lowering the temperature. To observe more clearly the temperature behaviour of the states close to $E_{F}$, the spectral density of states was obtained by dividing the experimental spectra by the Fermi Dirac distribution and symmetrising it, figure 11(b). The figure shows significant reduction in the intensity as a dip in the vicinity of $E_{F}$ with decrease in temperature. This dip increases as the temperature drops and a systematic increase in the spectral DOS above 100 meV is observed with decrease in temperature suggesting the stabilization of pseudogap.

It is important to note that Matsuyama et al.Matsuyama et al. (1989) have collected He I and II spectra on BaPb_0.85Bi_0.15O₃ at around 300 K and 180 K. Based on the Fermi edge observed in this compound, they proposed that the superconductivity would be driven by cooper pairing of electrons in Fermi liquid states. To check for this possibility, the SDOS was plotted as a function of $(E-E_{F})^{n}$. Our results show that the value of $n$ thus obtained was 0.5 suggesting the role of disorder existing in the compound, as seen in figure 12(a). Earlier studiesSarma et al. (1998); Kobayashi et al. (2007) have shown that such decrement in the intensity could arise due to the localization of the states at $E_{F}$ induced by disorder. Under this situation, it is expected that the DOS at $E_{F}$ follows $N(E_{F})=a+b\sqrt{T}$, where $T$ is the temperature; such a behaviour can be clearly observed in panel (b) of figure 12, thus suggesting that the compound is a disordered metal for $T>T_{C}$. However, to observe whether there is any characteristic peak observed close to $E_{F}$ representing the superconducting state, experiments below $T_{C}$ are required.