Crystal Structures of Highly Hole-Doped Layered Perovskite Nickelate Pr_2−xSr_xNiO₄ Studied by Neutron Diffraction

Figures 1(a) and 1(b) show the powder diffraction patterns of Pr_1.3Sr_0.7NiO₄ and Pr_1.1Sr_0.9NiO₄, respectively, measured at 9 K. According to our Rietveld refinements, both diffraction patterns were well reproduced by tetragonal structures with the space group \(I4/mmm\). Because the magnetic Bragg peaks were too weak to observe in the powder diffraction measurements, they were not considered in the refinements. Even with our high-resolution diffraction measurements, we did not observe peak splitting of peaks in any of the diffraction patterns, indicating that the crystal structures of Pr_1.3Sr_0.7NiO₄ and Pr_1.1Sr_0.9NiO₄ remain tetragonal in the temperature range studied. For both samples, we found that the peaks whose indices are dominated by h and k have tails on their left sides, while the peaks characterized by dominant l indices have tails on their right sides. The tails cannot be reproduced by anisotropic peak broadening, but they can be interpreted as the existence of minority phases with larger c axes and shorter a axes. The differences in the lattice parameters suggest that the minority phases originate from domains with hole concentrations lower than the nominal values. The coexistence of minority phases has also been reported for La\(_{2-x}\)Sr_xNiO₄ with \(x\sim 1\).^20,21) We successfully fit the powder diffraction patterns to two (majority and minority) tetragonal phases, both characterized by \(I4/mmm\) symmetry [Fig. 1(d)]. The mass fractions of the minority phases were 24 and 13% for Pr_1.3Sr_0.7NiO₄ and Pr_1.1Sr_0.9NiO₄, respectively. We discuss the crystal structure of the majority phases hereinafter. We should note that the hole concentrations (or x values) for the majority phases are slightly larger than the nominal concentrations because of the existence of minority phases. We estimated the x values in the majority phases from the \(c/a\) ratios²⁰⁾ and found that their deviations from the nominal values are less than 0.01 for both samples.

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Figure 1. (Color online) Powder diffraction patterns of (a) Pr_1.3Sr_0.7NiO₄ and (b) Pr_1.1Sr_0.9NiO₄ measured at 9 K and shown here as a function of lattice spacing d. The solid lines are calculated intensities obtained by Rietveld refinement. The vertical lines are calculated peak positions of the majority (top rows) phase and minority (bottom rows) phase. The solid lines below the peak positions represent the differences between the observed and calculated intensities. (c) Schematic of the crystal structure of Pr\(_{1-x}\)Sr_xNiO₄. The arrows indicate the directions of the atomic displacements corresponding to the ADPs. (d) Enlarged view of the diffraction pattern of Pr_1.1Sr_0.9NiO₄ near the 006 (\(d\sim 2.06\) Å) and 105 (\(d\sim 2.07\) Å) peaks.

Table I lists the crystal structure parameters for the majority phases of both samples at 9 K and ∼290 K.³²⁾ In Pr_1.1Sr_0.9NiO₄, the lattice parameters a and c are 0.3% longer and 1% shorter, respectively, when compared with those of Pr_1.3Sr_0.7NiO₄. This concentration dependence of the lattice parameters is similar to that of La\(_{2-x}\)Sr_xNiO₄.^20,23) Meanwhile, the lattice parameters of Pr\(_{2-x}\)Sr_xNiO₄ are 1% smaller than those of La\(_{2-x}\)Sr_xNiO₄ at the same concentrations.^20,23) The latter result reflects the smaller ionic radius of Pr³⁺ compared with La³⁺.³³⁾

»View table Table I. Crystal structure parameters and bond lengths of Pr1.3Sr0.7NiO4 and Pr1.1Sr0.9NiO4 obtained by Rietveld refinement. The space group is \(I4/mmm\) with the following atomic positions: Pr/Sr at 4e \((0,0,z)\), Ni at 2a \((0,0,0)\), O1 at 4e \((0,0,z)\), and O2 4c \((0,1/2,0)\).

Figures 2(a) and 2(b) show the temperature dependences of the lattice parameters of Pr_1.3Sr_0.7NiO₄ and Pr_1.1Sr_0.9NiO₄, respectively. The lattice parameters for both materials decrease as the temperature decreases. For La\(_{3/2}\)Sr\(_{1/2}\)NiO₄, it has been reported that the lattice parameter a increases below 120 K in accordance with a decrease in the optical gap,³⁴⁾ which was attributed to the development of stripe charge ordering.¹⁷⁾ However, such an increase in a was not observed in Pr_1.3Sr_0.7NiO₄ or Pr_1.1Sr_0.9NiO₄, which suggests that the temperature-dependent variations in the hole concentration and orbital character accompanied by a transition from checkerboard to stripe charge ordering are less significant compared with those at \(x=1/2\). This finding is consistent with the fact that the loss of the optical-conductivity spectral intensity due to the appearance of stripe charge ordering is negligible at \(x=0.7\).¹⁷⁾ At high temperatures, for both compounds, the c axis shrinks considerably faster than the a axis as the temperature decreases. The temperature dependence of c saturates at low temperatures, resulting in a minimum ratio of the two lattice parameters, \(c/a\). The minimum of \(c/a\) occurs at ∼60 K for Pr_1.3Sr_0.7NiO₄, while the minimum occurs at ∼80 K for Pr_1.1Sr_0.9NiO₄ [insets in Figs. 2(a) and 2(b)]. Because these temperatures are close to the spin ordering temperatures (\(T_{\text{s}}\)),^16,17,35) it would be interesting if the minima of \(c/a\) correspond to the spin orderings. However, \(T_{\text{s}}\) is higher in the case of \(x=0.7\), which is inconsistent with the difference in the temperatures at the minimum \(c/a\). Furthermore, the minimum \(c/a\) value occurs at a similar temperature for \(x=1/4\) and 1/3 in La\(_{2-x}\)Sr_xNiO₄,²⁶⁾ although their \(T_{\text{s}}\)s differ from those at \(x=0.7\) and 0.9. Therefore, the temperature dependence of \(c/a\) is not related to the magnetic ordering, but instead can be ascribed to the greater compressibility of the c axis compared with that of the a axis at high temperatures due to the layered perovskite structure. The absence of anomalous features in the temperature dependence of the lattice parameters can be confirmed by the fact that the temperature dependence of the unit cell volume (V) is well described by the Debye model. Figure 2(c) shows the temperature dependence of V, where the solid lines present a fit to the Debye model: \begin{equation} V \approx V(T = 0) + \frac{9\gamma N k_{\text{B}}}{B} T \left(\frac{T}{\Theta_{\text{D}}}\right)^{3} \int^{\Theta_{\text{D}}/T}_{0} \frac{x^{3}}{e^{x} - 1}\,dx. \end{equation} (1) Here γ, B, \(k_{\text{B}}\), \(\Theta_{\text{D}}\), and N are the Grüneisen parameter, bulk modulus, Boltzmann constant, Debye temperature, and number of atoms in the unit cell, respectively.³⁶⁾ The observed values show no significant deviations from the calculated values throughout the temperature range studied. This finding suggests that the temperature dependence of the unit cell volume is dominated by phonon contributions. The fitting parameters obtained from these fits are summarized in Table II, exhibiting values similar to those for La\(_{7/4}\)Sr\(_{1/4}\)NiO₄ and La\(_{5/3}\)Sr\(_{1/3}\)NiO₄.²⁶⁾

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Figure 2. (a, b) Temperature dependence of the lattice parameters of (a) Pr_1.3Sr_0.7NiO₄ and (b) Pr_1.1Sr_0.9NiO₄. The insets show the temperature dependence of the \(c/a\) ratio. (c) Temperature dependence of the unit cell volume of Pr_1.3Sr_0.7NiO₄ (open circles) and Pr_1.1Sr_0.9NiO₄ (closed circles). The solid lines represent fits to the Debye model [Eq. (1)].

»View table Table II. Parameters obtained by fitting the unit cell volumes to the Debye function.

The Ni–O bond lengths should more directly reflect the electronic state of the Ni ions than the lattice parameters. Figure 3 shows the temperature dependence of the two types of Ni–O bonds in Pr_1.3Sr_0.7NiO₄ and Pr_1.1Sr_0.9NiO₄. The in-plane Ni–O bond length (Ni–O2) is exactly half the lattice parameter a, and its error bars are very small. This parameter decreases smoothly with decreasing temperature for both compounds. In contrast, the out-of-plane Ni–O bond length (Ni–O1) depends on the z position of O1, and large error bars originating from the ambiguity in atomic position can be seen. Nevertheless, there is a qualitative difference between the behaviors of the Ni–O1 bonds of the two compounds: The Ni–O1 bond length for Pr_1.3Sr_0.7NiO₄ decreases uniformly as the temperature decreases, whereas that of Pr_1.1Sr_0.9NiO₄ decreases more rapidly at \(T\gtrsim 150\) K than at \(T\lesssim 150\) K. This difference can be more clearly observed in the ratios of the two Ni–O bonds (insets in Fig. 3) and should, at least partly, contribute to the difference in temperatures corresponding to the minimum \(c/a\) (Fig. 2). This change in the bond contraction rate of Ni–O1 at \(x=0.9\) also affects the atomic displacement parameter (ADP) of O1 along the out-of-plane direction (\(U_{33}\)). As shown in Fig. 4(a), the slope of the temperature dependence of O1 \(U_{33}\) changes at approximately 150 K.

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Figure 3. Temperature dependence of the Ni–O bond length for (a) Pr_1.3Sr_0.7NiO₄ and (b) Pr_1.1Sr_0.9NiO₄. The open and solid symbols denote the out-of-plane (Ni–O1) and in-plane (Ni–O2) bonds, respectively. The insets show the temperature dependence of the ratio of the two Ni–O bond lengths, Ni–O1/Ni–O2. The solid lines serve as a guide for the eye.

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Figure 4. (Color online) Temperature dependence of the ADP \(U_{ij}\) for (a) O1, (b) O2, (c) Pr/Sr, and (d) Ni. The open and closed symbols denote values for Pr_1.3Sr_0.7NiO₄ and Pr_1.1Sr_0.9NiO₄, respectively. The circles, diamonds, and squares denote \(U_{11}\), \(U_{22}\), and \(U_{33}\), respectively. In (a) and (b), the values of the O atoms for La\(_{5/3}\)Sr\(_{1/3}\)NiO₄ reproduced from Ref. 26 are shown as broken (\(U_{11}\)), dashed-dotted (\(U_{22}\)), and solid (\(U_{33}\)) lines.³⁸⁾

Notably, Ni–O1 in Pr_1.3Sr_0.7NiO₄ exhibits a temperature dependence similar to the monotonic temperature dependence observed at \(x=1/3\) and the convex temperature dependence observed at \(x=1/4\).²⁶⁾ Therefore, only the \(x=0.9\) sample shows qualitatively different behavior compared with the lower concentrations. We suggest that this difference in the behavior of Ni–O1 is related to the difference in charge itinerancy. \(R_{2-x}\)Sr_xNiO₄ shows insulating behavior with charge ordering at \(x < 1\). However, as the hole concentration increases, the itinerancy increases, and the material becomes an anomalous metal with checkerboard-type charge correlations at \(x\gtrsim 1\).^22,25) The \(x=0.9\) sample is located near the boundary of the insulator–metal transition. Its electrical conductivity is metallic at high temperatures, and it transitions to an insulator at low temperatures.²⁵⁾ At this point, the development of checkerboard-type charge correlations in the NiO₂ planes as a function of temperature should sweep out the excess holes into the out-of-plane \(d_{3z^{2}-r^{2}}\) orbitals, which should result in a more rapid decrease in the out-of-plane Ni–O bond lengths at higher temperatures. The bond contraction rate finally decreases and becomes similar to that of the insulating lower-concentration compounds at lower temperatures, where the checkerboard-type charge correlations are well developed.

The lattice parameters and bond lengths described above were obtained by structural refinements based on a tetragonal structure with \(I4/mmm\) symmetry. The structural modulations associated with the doped holes and modified orbital occupations are averaged with respect to this symmetry and are obscured. In contrast, the difference between the modulated atomic positions and the atomic positions of the average crystal structure are incorporated in the ADPs in the structural refinement. Therefore, a precise determination of the ADPs, particularly those of the O atoms, provides useful information for investigating the local charge and orbital occupations of the Ni ions.^27,37)

Figure 4 shows the temperature dependence of the ADPs for Pr_1.3Sr_0.7NiO₄ (open symbols) and Pr_1.1Sr_0.9NiO₄ (closed symbols). The displacement directions corresponding to the respective ADPs are drawn schematically in Fig. 1(c). In Figs. 4(a) and 4(b), the values for the O atoms in La\(_{5/3}\)Sr\(_{1/3}\)NiO₄ reported in Ref. 26 are plotted as well (lines). The ADPs of the O atoms in Pr_1.3Sr_0.7NiO₄ are similar to those of the O atoms in La\(_{5/3}\)Sr\(_{1/3}\)NiO₄ [Figs. 4(a) and 4(b)]. \(U_{11}\) is larger than \(U_{33}\) for O1, whereas \(U_{33}\) is the largest and \(U_{22}\) is the smallest for O2. These ADP magnitudes suggest that it is difficult for the O atoms to vibrate along the Ni–O bonds, and the displacements of the O atoms are dominated by rotation of the NiO₆ octahedra. For Pr_1.1Sr_0.9NiO₄, the ADPs of O2 are similar to those for Pr_1.3Sr_0.7NiO₄ and La\(_{5/3}\)Sr\(_{1/3}\)NiO₄, although the values are slightly larger [Fig. 4(b)]. This result holds for the ADPs of Ni [Fig. 4(d)]. Abeykoon et al. reported that the ADPs of the in-plane O and Ni atoms along the Ni–O bond directions deviate from the normal Debye behavior due to the development of stripe charge correlations.²⁷⁾ In the present study, we did not observe any anomalies associated with spin or charge ordering in the ADPs of O2 or Ni. The absence of anomalies in the in-plane ADPs may originate from the fact that the two-dimensional character of the checkerboard-type charge ordering restricts the degrees of freedom of the associated atomic displacements compared with the one-dimensional stripe charge ordering. In contrast, the ADPs of O1 in Pr_1.1Sr_0.9NiO₄ clearly show different behavior compared with those for Pr_1.3Sr_0.7NiO₄ and La\(_{5/3}\)Sr\(_{1/3}\)NiO₄ [Fig. 4(a)]. \(U_{33}\) is considerably larger than that for Pr_1.3Sr_0.7NiO₄, while \(U_{11}\) possesses similar values to Pr_1.3Sr_0.7NiO₄. As a result, the magnitude relationship between \(U_{11}\) and \(U_{33}\) is reversed with respect to that for Pr_1.3Sr_0.7NiO₄ and La\(_{5/3}\)Sr\(_{1/3}\)NiO₄. The ADPs of Pr/Sr for Pr_1.1Sr_0.9NiO₄ show a similar anomalous behavior [Fig. 4(d)]; again, \(U_{33}\) is larger than \(U_{11}\), in contrast to the results for Pr_1.3Sr_0.7NiO₄.

The significantly larger O1 \(U_{33}\) indicates exceptionally large displacements of the apical oxygen atoms along the out-of-plane direction around their average positions in Pr_1.1Sr_0.9NiO₄. A recent XAS study²²⁾ revealed that the Ni³⁺ sites created by hole doping are divided into two sites at \(x > 1/2\), while the number of Ni²⁺ sites continues to decrease. The occupancy of Ni³⁺ sites with \(d_{x^{2}-y^{2}}\) orbitals (holes in the \(d_{3z^{2}-r^{2}}\) orbitals) increases as x increases, whereas that of Ni³⁺ sites with \(d_{3z^{2}-r^{2}}\) orbitals (holes in the \(d_{x^{2}-y^{2}}\) orbitals) remains constant at 0.5. If the occupancies of the Ni³⁺ sites with \(d_{x^{2}-y^{2}}\) orbitals and the Ni²⁺ sites change as functions of \(x-0.5\) and \(-x+1\), respectively, the Ni sites will consist of 50% \(d_{x^{2}-y^{2}}\) Ni³⁺ sites and 50% \(d_{3z^{2}-r^{2}}\) Ni³⁺ sites at \(x=1\). In other words, the checkerboard-type charge ordering is accompanied by orbital ordering of the \(d_{x^{2}-y^{2}}/d_{3z^{2}-r^{2}}\) orbitals. This orbital ordering results in a mixture of long and short out-of-plane Ni–O bonds and induces a large O1 \(U_{33}\). Because the Pr and Sr atoms are located above or below the O1 atoms [Fig. 1(d)], displacement of the O1 atoms along the out-of-plane direction should affect the positions of the Pr/Sr atoms. Therefore, the ADPs of Pr/Sr in Pr_1.1Sr_0.9NiO₄ show tendencies similar to those of O1.