Unidirectional diagonal order and three-dimensional stacking of charge stripes in orthorhombic ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ and ${\mathrm{Nd}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$

The interplay between crystal symmetry and charge stripe order in ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ and ${\mathrm{Nd}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ has been studied by means of single crystal x-ray diffraction. In contrast to tetragonal ${\mathrm{La}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$, these crystals are orthorhombic. The corresponding distortion of the $\mathrm{Ni}{\mathrm{O}}_2$ planes is found to dictate the direction of the charge stripes, similar to the case of diagonal spin stripes in the insulating phase of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$. In particular, diagonal stripes seem to always run along the short $a$ axis, which is the direction of the octahedral tilt axis. In contrast, no influence of the crystal symmetry on the charge stripe ordering temperature itself was observed, with $T_{\mathrm{CO}}\sim 240\mathrm{K}$ for La, Pr, and Nd. The coupling between lattice and stripe degrees of freedom allows one to produce macroscopic samples with unidirectional stripe order. In samples with stoichiometric oxygen content and a hole concentration of exactly $1∕3$, charge stripes exhibit a staggered stacking order with a period of three $\mathrm{Ni}{\mathrm{O}}_2$ layers, previously only observed with electron microscopy in domains of mesoscopic dimensions. Remarkably, this stacking order starts to melt about $40\mathrm{K}$ below $T_{\mathrm{CO}}$. The melting process can be described by mixing the ground state, which has a three-layer stacking period, with an increasing volume fraction with a two-layer stacking period.

Figure 1

(Color online) (a) Phase diagram of ${\mathrm{Ln}}_{2-x}{\mathrm{Sr}}_x\mathrm{Ni}{\mathrm{O}}_4$ with $\mathrm{Ln}=\mathrm{La}$, Pr, and Nd as obtained from x-ray powder diffraction on polycrystalline specimens (Refs. 35, 41). For Pr and Nd the HTT/LTO phase boundary is shifted to higher $x$ and $T$ with respect to La. Data points marked with an “e” are linear extrapolations of measurements up to $800\mathrm{K}$. Top: Lattice parameters $a$, $b$, and $c$ as well as orthorhombic strain at room temperature. Data for the single crystals at $x=0.33$ are indicated by triangular symbols. (e) Tilt directions of the $\mathrm{Ni}{\mathrm{O}}_6$ octahedra in the different structural phases.

Figure 2

(Color online) Projection along $l$ of the $(h,k)$ zone. Domain $A$: round symbols. Domain $B$: quadratic symbols. Tilt between the two domains exaggerated. Fundamental reflections are black, LTO superlattice peaks grey, LTLO superlattice peaks green (with dot) and white. The charge stripe peaks (CO) are red and blue (with cross), and appear at incommensurate positions, displaced by $2ϵ$ along $k$ from Bragg peaks. Reflection conditions for $l$ are indicated in the legend. $\Delta l$ depends on the stacking order, which we discuss in Sec. 4E. The three enumerated scans are those in Figs. 3c, 3d.

Figure 3

(Color online) Scans performed on ${\mathrm{Nd}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ at $T=200\mathrm{K}$ through possible charge stripe peak positions $(0,4–2ϵ,3)$ and $(4–2ϵ,0,3)$ of domains $A$ (round symbols) and $B$ (quadratic symbols) in the orientation matrix of domain $A$. Arrows indicate expected peak positions in the case that stripes would also run along the $b$ axis. Enumerated scans in (c), (d) are indicated in Fig. 2. The horizontal bars in (c) and (d) indicate the instrumental resolution full width.

Figure 4

Temperature dependence of the normalized intensities of (a) the LTO superstructure peak, (b) the LTLO superstructure peak, and (c) the charge stripe order peak for the Ar-annealed ${\mathrm{Nd}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ crystal. Data collected in transmission geometry $\left(100\mathrm{keV}\right)$.

Figure 5

Temperature dependence of the normalized intensities of (a), (c) the LTO superstructure peak and (b), (d) the charge stripe order peak in ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$. Data on left side in transmission geometry $\left(100\mathrm{keV}\right)$ on as-prepared-in-air crystal, and on right side in reflection geometry $\left(8.1\mathrm{keV}\right)$ after Ar annealing.

Figure 6

(Color online) Comparison of $k$ scans through $(h,4–2ϵ,3)$ and $(h,4+2ϵ,3)$ at $10\mathrm{K}$ for ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ (PSNO) (a)–(c) and ${\mathrm{Nd}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ (NSNO) (d). Scans in (c) performed at $h=4$ (triangular symbols). All other scans at $h=0$. Data collected (a) in reflection and (b) in transmission on as-prepared sample, (c) in reflection and (d) in transmission on Ar-annealed samples. The intensities of the peaks at $k=4+2ϵ$ were normalized by same factors as their counterparts at $k=4–2ϵ$. The horizontal bar in (d) indicates typical instrumental resolution full width.

Figure 7

(Color online) (a) Comparison of $l$ scans at $10\mathrm{K}$ through charge peak at $(h,4–2ϵ,3)$ for ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ and ${\mathrm{Nd}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$. For triangular symbols scans were performed at $h=4$. All other scans at $h=0$. Typical instrumental resolution full width is indicated by horizontal bar. (b) Incommensurability $ϵ$ as a function of temperature. Error for $ϵ$ indicated by vertical bar. Symbols and color code for samples and experiments are the same as in Fig. 6.

Figure 8

(Color online) Diagrams indicating the positions of the charge stripe peaks (red diamonds) as well as corresponding ordering wave vectors for samples with (a) three-layer stripe stacking period and (b) two-layer period or strong stacking disorder. Full symbols indicate fundamental reflections, open symbols LTO superstructure reflections. In (b) we assume a two-layer period similar to Fig. 10c. Additional charge peaks would appear for $l$ even, assuming the stacking depicted in Fig. 10b (Ref. 50).

Figure 9

(Color online) (a) $l$ scans through $(4,4–2ϵ,3)$ of Ar-annealed ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ for different temperatures. Instrumental resolution full width is indicated by horizontal bar. (b) Peak positions as a function of $T$ from fit with three peaks. The position of the central peak was kept fixed at $l=3$.

Figure 10

Stripe stacking patterns. (a) Three-layer period comparable to cubic stacking, (b) two-layer period comparable to hexagonal stacking, (c) two-layer period due to body centered stacking with alternating layers of Ni-O-Ni-bond centered (black symbols) and Ni-site centered (grey symbols) stripes.

Figure 11

(Color online) (a) Calculated scattering intensity for $l$ scans as a function of the probability $z$, where $z=0$ for hexagonal close packing and $z=\infty$ for cubic close packing (after Ref. 57). The curve for $z=1$ (dashed) represents complete disorder. The curve for $z=3.5$ represents a system with a fairly developed three-layer stacking period. (b) $l$ positions of the maxima as a function of the volume fraction of ABCA stacking segments.

Figure 12

(Color online) $l$ scans through $(4,4–2ϵ,3)$ charge peak of Ar-annealed ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ plus fits (red solid line) as described in the text. The dash-dot line (blue) represents the structure factor, the dotted line (green) a $T$-independent constant background. Instrumental resolution full width is indicated by horizontal bar in (a).

Figure 13

(Color online) Top: temperature dependence of various properties of Ar-annealed ${\mathrm{Pr}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ close to the charge stripe order transition. (a) $l$-integrated intensity of $(4,4–2ϵ,3)$ charge peak, (b) parameter $z$ from fits of $l$ scans (see Fig. 12), (c) inverse in-plane correlation length from $k$ scans (d) incommensurability $ϵ$ (see Fig. 2), (e) measured specific heat (inset) and specific heat anomaly obtained by subtraction of fourth-order polynomial; solid line (red) in inset. Bottom: $ab$-plane resistivity and its second derivative of the Ar-annealed ${\mathrm{Nd}}_{1.67}{\mathrm{Sr}}_{0.33}\mathrm{Ni}{\mathrm{O}}_4$ crystal.