Reconciliation of local and long-range tilt correlations in underdoped ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4(0\le x\le 0.155)$

A long-standing puzzle regarding the disparity of local and long-range ${\text{CuO}}_6$ octahedral tilt correlations in the underdoped regime of ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ is addressed by utilizing complementary neutron powder diffraction and inelastic neutron scattering (INS) approaches. This system is of interest because of the strong depression of the bulk superconducting transition at $x=1/8$ in association with charge and spin stripe order. The latter unidirectional order is tied to Cu-O bond-length anisotropy present in the so-called low-temperature tetragonal (LTT) phase. On warming, the lattice exhibits two sequential structural transitions, involving changes in the ${\text{CuO}}_6$ tilt pattern, first to the low-temperature orthorhombic (LTO) and then the high-temperature tetragonal (HTT) phase. Despite the changes in static order, inspection of the instantaneous local atomic structure suggests that the LTT-type tilts persist through the transitions. Analysis of the INS spectra for the $x=1/8$ composition reveals the dynamic nature of the LTT-like tilt fluctuations within the LTO and HTT phases. Within the low-temperature phase, the Cu-O bond-length splitting inferred from lattice symmetry and fitted atomic position parameters reaches a maximum of $0.3\%$ at $x=1/8$, suggesting that electron-phonon coupling may contribute to optimizing the structure to stabilize stripe order. This splitting is much too small to be resolved in the pair distribution function, and in fact we do not resolve any enhancement of the instantaneous bond-length distribution in association with stripe order. This study exemplifies the importance of a systematic approach using complementary techniques when investigating systems exhibiting a large degree of complexity and subtle structural responses.

Figure 1

Structural details of ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$. La/Ba are shown as large green spheres, Cu as intermediate blue spheres, and O as small red spheres. (a) Basic structural motif is shown for $I4/mmm$ (HTT) phase, featuring ${\text{CuO}}_6$ octahedral unit and ${\text{La/BaO}}_9$ cage. O(1) and O(2) denote planar and apical oxygen, respectively. Panels (b)–(e) highlight various aspects of the average crystal structures as follows: ${\text{CuO}}_6$ octahedral tilt symmetry (top row), in-plane bond-length distribution within ${\text{CuO}}_4$ plaquette (middle row), and dispersion of La/Ba-O distances within the ${\text{La/BaO}}_9$ cage (bottom row). Equal interatomic distances are represented by the same color. HTT denotes high-temperature tetragonal ($I4/mmm$), LTO is low-temperature orthorhombic ($Bmab$), LTT is low-temperature tetragonal ($P{4}_2/ncm$), and LTLO is low-temperature less orthorhombic ($Pccn$). The underlying in-plane symmetry is OE in HTT and LTO models, and OI in LTT and LTLO models, as indicated.

Figure 2

(a) Average structure ($x$, $T$) phase diagram of LBCO which has been replotted to highlight whether the structure is OE or OI. Open symbols and solid lines are from Hücker et al. [17], solid symbols are from the present study. Shaded are regions of interest for this study having average OE (HTT and LTO) and OI (LTLO/LTT) symmetries, as indicated. (b) Local structure phase diagram with phase designations based on the results presented in this study. Please see text for definitions of OE and OI.

Figure 3

Rietveld fits of the average structure models to LBCO data at 15 K. Closed blue symbols represent the data, solid red lines are the models, and solid green lines are the differences (offset for clarity). Vertical ticks mark reflections. (a) $x=0$ using $Bmab$ model, (b) $x=0.095$ using $Pccn$ model, (c) $x=0.125$ using $P{4}_2/ncm$ model, and (d) $x=0.155$ using $Pccn$ model.

Figure 4

Temperature evolution of in-plane lattice parameters of ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ obtained from Rietveld refinements for $x=0$ (a), $x=0.095$ (b), $x=0.125$ (c), and $x=0.155$ (d). HTT parameters are given in $F4/mmm$ setting. Vertical dashed lines indicate crystallographic phase transitions as specified in the text. Insets to (b)–(d) display temperature evolution of the lattice orthorhombicity $[\eta =2(a-b)/(a+b\left)\right]$ for doped samples, with their maxima marked by vertical arrows. Vertical dashed lines indicate low-temperature structural phase transitions. Sloping dashed straight lines are guides to the eye emphasizing anomalous trends seen in highlighted regions and discussed in the main text.

Figure 5

Temperature evolution of the average in-plane Cu-O bond lengths (solid blue circles) in ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ obtained from Rietveld refinements: $x=0.095$ (a), $x=0.125$ (b), and $x=0.155$ (c). In all three panels: solid black symbols show in-plane Cu-O bond for $x=0$ sample as a reference. Horizontal dashed lines are guide to the eye, vertical dashed lines mark crystallographic phase transitions. Vertical double arrows highlight the magnitude of changes discussed in the main text.

Figure 6

Temperature evolution of the average La-O(2) interatomic distances for (a) $x=0.095$, (b) $x=0.125$, and (c) $x=0.155$ composition. Bond multiplicity is indicated by a number where applicable. Short La-O(2) bond that connects two ${\text{LaO}}_2$ planes is shown in separate panels. Vertical dashed lines indicate structural phase transitions.

Figure 7

Temperature evolution of atomic displacement parameter (ADP) of apical oxygen (solid blue symbols) in ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ for $x=0.095$ (a), $x=0.125$ (b), and $x=0.155$ (c). Vertical dashed lines indicate transitions between crystallographic phases as labeled and specified in the text. In all panels ADP data for $x=0$ sample are presented by solid gray symbols, with solid red line representing a fit of the Debye model, as discussed in the text. Debye model for ADPs of doped samples has the same Debye temperature as for $x=0$, but different offset such as to provide a good fit to ADP in the low-temperature phases. Anomalous jumps in ADP discussed in the main text are indicated by double arrows. Insets focus on low-temperature transitions.

Figure 8

PDF fits of the average structure models to LBCO data at 15 K. Closed blue symbols represent the data, solid red lines are the models, and solid green lines are the differences (offset for clarity). (a) $x=0$ using $Bmab$ model, (b) $x=0.095$ using $Pccn$ model, (c) $x=0.125$ using $P{4}_2/ncm$ model, and (d) $x=0.155$ using $Pccn$ model.

Figure 9

PDF comparison of the average and local structure behavior in ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$. Top row: simulated PDFs calculated using parameters from fully converged Rietveld refinements reflecting average structure behavior. Low-$T$ profiles are shown in blue, higher-$T$ profiles are shown in red. (a) 15 K vs 50 K LTO structure PDFs for $x=0$; (b) 15 K LTLO vs 60 K LTO PDFs for $x=0.095$; (c) 15 K LTT vs 80 K LTO PDFs for $x=0.125$; (d) 15 K LTLO vs 70 K LTO PDFs for $x=0.155$. Bottom row: comparison of the raw experimental PDF data for the same respective temperatures as considered in (a)–(d); lower temperature data shown in blue, higher temperature data shown in red. Difference curves (high-T minus low-T PDF) are offset for clarity. Shaded areas represent span of the difference curves observed for $x=0$ composition. Changes observed in the average structure (marked by filled arrows) are not observed in the local structure (empty arrows), as experimental PDFs do not change across the OE/OI phase transitions in doped ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ samples.

Figure 10

OE versus OI PDF models for the local structure (solid gray lines). Data are shown as open gray symbols, difference curves (solid red lines) are offset for clarity. (a)–(d) 15 K data vs LTO model (OE symmetry) explains $x=0$ data well (as it should), fails for $x=0.095$, 0.125, and 0.155 (as it should). The largest discrepancies are marked by arrows. (e) LTO for $x=0$ sample at 50 K, (f)–(h) data at T of maximum orthorhombicity (60 K, 80 K, and 70 K for $x=0.095$, 0.125, and 0.155, respectively) vs LTO model (fails at same places as at 15 K); (i)–(k) same as (f)–(h) but with LTLO/LTT models (OI symmetry)—underlying data correspond to OI symmetry. Inset: La/Ba-O(2) distances (shaded area) contribute principally to the misfits marked by arrows in (b)–(d) and (f)–(h).

Figure 11

Semiquantitative exploration of the OI-ness of LBCO via assessment of $T$ dependence of apical oxygen isotropic atomic displacement parameter (ADP) as obtained by fitting different structural models [as indicated by arrows in (a)–(c)] to the PDF data over 4 nm $r$ range. (a) $x=0.095$, (b) $x=0.125$, and (c) $x=0.155$. Difference between the observed ADPs using LTO and LTLO/LTT models from panels (a)–(c) are displayed in (d)–(f) for $x=0.095$, $x=0.125$, and $x=0.155$, respectively, normalized by the square of the average transverse displacement of apical oxygen. See text for details.

Figure 12

Comparison of LBCO PDF data at 15 K (closed blue symbols) and at 100 K (solid red line) with difference curve (solid green line) offset for clarity. Horizontal gray dashed lines mark experimental uncertainty on 2 $\sigma$ level. Light-red solid line represents a 0.5 Å running average of the absolute value of the difference curve, multiplied by 2 and offset for clarity. (a) $x=0$, (b) $x=0.095$, (c) $x=0.125$, and (d) $x=0.155$. Difference in data for $x=0$ sample displays the change expected from canonical thermal evolution effects without symmetry changes. Differences in the data for all three doped samples display similar level of change as $x=0$ sample up to $\sim 9$ Å  with first substantial changes seen on longer length scale. Low $r$ assessment: green arrows mark changes seen in the data for all the samples, while the red arrows mark significant changes seen only in the doped samples, presumably associated with the change in average symmetry. Empty arrows around 3 Å mark places where changes are expected from the average structure, but not observed locally, as shown in Fig. 9 and discussed in the text. Local structure across the global OI to OE phase transition is preserved on subnanometer length scale.

Figure 13

(a) Diagram of the $(H,K,0)$ plane of reciprocal space indicating fundamental Bragg peaks (filled circles) and LTT superlattice peaks (open circles), with dot-dashed (dashed) line indicating the orientation of the data slice in (b) [(d)]. (b) Map of scattering intensity for $E$ vs $\mathbf{Q}=(3+\xi ,3-\xi ,0)$. (c) Intensity (integrated over $2\le E\le 4$ meV) vs $\mathbf{Q}=(3,3,L)$. (d) Intensity map for $E$ vs $\mathbf{Q}=(3+\zeta ,3+\zeta ,0)$. All measurements are at $T=180$ K, in the LTO phase.

Figure 14

Constant-energy slices, with signal integrated over $\pm 1$ meV, through the soft-phonon scattering around (330). (a), (d) $E=10$ meV; (b), (e) $E=7$ meV; (c), (f) $E=3$ meV. Data obtained at $T=60$ K for (a)–(c) and 180 K for (d)–(f).

Figure 15

Single-crystal results at the (330) LTT superlattice position for $x=0.125$. (a) Summary of the temperature dependence of the elastic (red circles) and inelastic (blue triangles) integrated intensities obtained from the following panels; vertical dashed lines denote phase boundaries, while dashed lines through data points are guides to the eye. (b)–(d) Elastic channel (integrated over $\pm 1.5$ meV) measured along the longitudinal direction at 120, 60, and 50 K, respectively. Solid lines are Gaussian-peak fits, used to determine the integrated intensity; weak, $T$-independent peaks are diffraction from the aluminum sample holder. (e)–(i) Inelastic signal from the soft-phonon fluctuations (2–5 meV integration) measured at 250, 180, 120, 60, and 50 K, respectively. Lines are Gaussian-peak fits.

Figure 16

(a), (b) Intensity map as a function of energy vs $\mathbf{Q}=(H,3,2)$, showing the transverse dispersion of tilt modes about the (032) superlattice peak position of the LTO phase, obtained at $T=180$ and 250 K, respectively. (c), (d) Intensity, integrated over a window of $\pm 1$ meV along the transverse direction, for $E=0$ (blue squares), 2.5 (red triangle), and 5 meV (green circles), at $T=180$ and 250 K, respectively.

Figure 17

(a) Normalized radial distribution function $R\left(r\right)$ for Cu-O(1) bonds comparing results from different analyses. Green lines show the Gaussian peaks obtained from PDF analysis for ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ with $x=0$ (solid) and $x=0.125$ (dashed). The effective profile for the latter sample from the Rietveld refinement (red dashed line) is not sensitive to correlated motion. The blue solid line indicates the model for ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ with $x=0.15$ and $T=15$ K from the XAFS study by Bianconi et al. [59]. (b) Pair distribution function for ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ with $x=0.125$ at 15 K (blue) and 100 K (red). (c) Results for Cu-O(1) ${\sigma }^2$ for $x=0.125$ from PDF (blue filled circles) and the EXAFS analysis (red open circles) of Lanzara et al. [60]. The solid line corresponds to an Einstein-model calculation, as described in the text.

Figure 18

Doping dependence of various system parameters of ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$. (a) Orthorhombicity at 100 K (dark blue solid symbols) from Rietveld. (b) Average planar Cu-O bond distance (solid black symbols) at 100 K temperature. (c) Average ${\text{CuO}}_6$ tilt angle extracted from apical oxygen (light red solid circles) and planar oxygen (dark red solid squares) positions at maximum orthorhombicity. (d) Orthorhombicity at 10 K after Hücker et al. [17] (light blue solid symbols). (e) Planar Cu-O bond length anisotropy at 15 K (olive solid symbols). (f) In-plane correlation lengths $\xi$ of charge ordering parallel (solid blue circles) and perpendicular (open blue circles) to the stripe direction at base temperature [17].

Figure 19

$(x,T)$ evolution of $\delta$ shown in Figs. 11–11. Solid lines mark structural transition temperature: $T_{\mathrm{HT}}$ (red) and $T_{\mathrm{LT}}$ (blue).