Stripe order in superconducting La${}_{2-x}$Ba${}_x$CuO${}_4$ ($0.095⩽x⩽0.155$)

The correlations between stripe order, superconductivity, and crystal structure in ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ single crystals have been studied by means of x-ray and neutron diffraction as well as static magnetization measurements. The derived phase diagram shows that charge stripe order (CO) coexists with bulk superconductivity in a broad range of doping around $x=1/8$, although the CO order parameter and correlation length fall off quickly for $x\ne 1/8$. Except for $x=0.155$, the onset of CO always coincides with the transition between the orthorhombic and the tetragonal or less orthorhombic low-temperature structures. The CO transition evolves from a sharp drop at low $x$ to a more gradual transition at higher $x$, eventually falling below the structural phase boundary for optimum doping. With respect to the interlayer CO correlations, we find no qualitative change of the stripe stacking order as a function of doping, and in-plane and out-of-plane correlations disappear simultaneously at the transition. Similarly to the CO, the spin stripe order (SO) is also most pronounced at $x=1/8$. Truly static SO sets in below the CO and coincides with the first appearance of in-plane superconducting correlations at temperatures significantly above the bulk transition to superconductivity (SC). Indications that bulk SC causes a reduction of the spin or charge stripe order could not be identified. We argue that CO is the dominant order that is compatible with SC pairing but competes with SC phase coherence. Comparing our results with data from the literature, we find good agreement if all results are plotted as a function of $x^{\prime }$ instead of the nominal $x$, where $x^{\prime }$ represents an estimate of the actual Ba content, extracted from the doping dependence of the structural transition between the orthorhombic phase and the tetragonal high-temperature phase.

Figure 1

Temperature vs hole-doping phase diagram of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ single crystals. Onset temperatures: $T_c$ of bulk superconductivity (SC), $T_{\mathrm{CO}}$ of charge stripe order (CO), $T_{\mathrm{SO}}$ of spin stripe order (SO), and $T_{\mathrm{LT}}$ of the low-temperature structural phases LTT and LTLO. At base temperature CO, SO, and SC coexist at least in the crystals with $0.095⩽x⩽0.135$. For $x=0.155$ we identified CO but not SO, and observe a mixed LTT and LTLO phase. In the case of $x=0.095$, very weak orthorhombic strain persists at low $T$. For $x=0.165$ we have measured $T_c$ only, before the crystal decomposed. Solid and dashed lines are guides to the eye. Although $T_{\mathrm{CO}}$, $T_{\mathrm{SO}}$, and $T_{\mathrm{LT}}$ for several $x$ were also determined with XRD and ND, most data points in this figure are from magnetic susceptibility measurements. Here, only $T_{\mathrm{SO}}$ for $x=0.095$ is from ND and $T_{\mathrm{CO}}$ and $T_{\mathrm{LT}}$ for $x=0.155$ from XRD.

Figure 2

Crystal structure and reciprocal lattice of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$. (a) Unit cell in the HTT phase ($I_4/\mathit{mmm}$). Tilt directions of the CuO${}_6$ octahedra in (b) the LTO phase ($\mathit{Bmab}$) and (c) the LTT phase ($P{4}_2/\mathit{ncm}$). Note that in the LTT phase the tilt direction alternates between [100]${}_t$ and [010]${}_t$ in adjacent layers. The same is true for the stripe direction. Reciprocal lattice in terms of the HTT unit cell for (d) the LTT phase and (e) the LTO phase, projected along $\ell$ onto the $(h,k)$ plane. Only reflections relevant to this work are shown. Fundamental Bragg reflections are indicated by black bullets and circles, CO reflections by blue squares, SO reflections by red diamonds, and superstructure reflections for $\ell =0$ that are only allowed in the LTT and LTLO phases by gray bullets. In (e) we also indicate the reciprocal lattice of the orthorhombic phase with its two twin domains A (closed symbols) and B (open symbols). The trajectories of typical scans are indicated by arrows, along with the value of $\ell$. The HTT phase compares to (d) with only the fundamental Bragg reflections present.

Figure 3

Structural properties of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ from XRD. (a) Orthorhombic strain $s$ vs temperature as a function of Ba doping. $s$ was determined from transverse scans through the (2,0,0)/(0,2,0) Bragg reflections that are simultaneously present owing to twin domains; see Fig. 2e. For $x=0.155$ the LTO$\leftrightarrow$LTLO transition of the majority phase is shown, although a significant volume fraction of 20% turns LTT at $T_{\mathrm{LT}}$; see Fig. 14. Inset: $s$ vs $x$ at 60 K. The solid line is a guide to the eye. (b) Temperature of the HTT$\leftrightarrow$LTO transition vs doping. The solid line describes $T_{\mathrm{HT}}(x)$ using $T_{\mathrm{HT}}(0)\times (1-x/x_c)$. $T_{\mathrm{HT}}(0)$ and $x_c$ were chosen such that at $x=0$ the line intercepts at 525 K for ${\mathrm{La}}_2{\mathrm{CuO}}_4$ and goes through 235 K at $x=0.125$, which is the most accurately known $T_{\mathrm{HT}}$ for Ba-doped compounds. (c) Integrated intensity from $k$ scans through the (1,0,0) superstructure peak. For (a) and (c) error bars are within symbol size.

Figure 4

In-plane CO correlations. $h$ scans (left) and $k$ scans (right) through the CO peak at ($2+2\delta$,0,5.5) for different dopings after subtraction of a linear background. Except for the $k$ scan for $x=0.115$, all intensities are normalized to the integrated intensity of the (2,0,6) Bragg reflection, as explained in Sec. 2, and are directly comparable. Error bars are not shown if within symbol size. The data for $x=0.155$ has been multiplied by a factor of 5. The dashed line marks the CO-peak position for $x=0.125$ and emphasizes its shift with doping. All scans were collected in the same scattering geometry. The resolution functions along $h$ and $k$ were measured at the (2,0,6) Bragg reflection and are indicated by black solid lines for $x=0.125$.

Figure 5

Integrated intensity vs temperature and doping from $k$ scans through the (1,0,0) peak and $h$ scans through the CO peak. Note that the data in this figure was not measured for all dopings in the same scattering geometry and in the case of the CO peak not always at the same $(h,k,l)$ position. Therefore, presented intensities are normalized at low temperature. Error bars for the (1,0,0) intensity are within symbol size. The green dashed-dotted lines indicate the onset of bulk SC at $T_c$ and the red dotted lines the onset of SO at $T_{\mathrm{SO}}$.

Figure 6

Interlayer CO correlations. $\ell$ scans along $\mathbf{Q}=(2+2\delta$,0,$\ell$) at base temperature for different dopings after background subtraction. Intensities are normalized to the integrated intensity of the (2,0,6) Bragg reflection, as explained in Sec. 2, and are directly comparable. The data for $x=0.125$ has been taken from Ref. 34 and was measured in a pressure cell, which explains the low counting statistic. The data has been corrected for the absorption of the pressure cell. All scans were collected in the same scattering geometry. Error bars are not shown if within symbol size. The resolution function was measured at the (2,0,6) Bragg reflection and is indicated by the black solid line for $x=0.125$. Because of a significant doping and temperature dependence of the background, there is no unique way to subtract it. In most cases the background was either measured along the same $\mathbf{Q}$ at $T\gtrsim T_{\mathrm{LT}}$, or along $\mathbf{Q}=(2+2\delta$, 0.03, $\ell$) at base temperature.

Figure 7

Melting of the CO. (a) $h$ scans and (b) $\ell$ scans through the (2$\delta$,0,8.5) CO peak for $x=0.125$ at different temperatures. The remaining profile in the $\ell$ scans for $T>T_{\mathrm{CO}}$ originates from diffuse scattering around (0,0,8). Curves for $T<T_{\mathrm{CO}}$ are shifted for clarity. The black solid lines indicate the resolution function measured at the (0,0,8) Bragg reflection. The mosaic of the (0,0,8) reflection reveals the presence of two domains contributing to the CO peak. The orientation matrix was determined based on the peak positions of the larger domain. (c)–(e) Integrated intensities from corresponding $h$ and $\ell$ scans for $x=0.115$, 0.125, and 0.135. The data in (c)–(e) was normalized at a base temperature of $~$3 K where additional scans were performed.

Figure 8

In-plane SO correlations. $h$ scans through the SO peak at ($0.5+\delta$,0.5,0) for different dopings. The horizontal bar at the bottom indicates the instrumental resolution full width at half maximum (FWHM) of 0.0078 r.l.u. The intensities have been normalized to the crystal volume in the neutron beam and for $x⩾0.115$ are directly comparable; see text and Sec. 2.

Figure 9

Peak intensity vs temperature and doping of (a) the (1,0,0) superstructure peak normalized at low temperature and (b) the ($0.5+\delta$,0.5,0) SO peak normalized to the crystal volume in the neutron beam. The data in (b) are directly comparable only for $x⩾0.115$; see text and Sec. 2. (c) Resolution corrected half width at half maximum (HWHM) of $h$ scans vs temperature for $x=0.115$ and $x=0.125$. The dashed lines in (a)–(c) indicate the onset temperatures for $x=0.125$ of the LTO$\leftrightarrow$LTT transition, the SO peak intensity, and the peak broadening. Data for $x=0.125$ previously presented in Ref. 36.

Figure 10

Field-cooled static susceptibility $\chi =M/H$ of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ vs temperature for different dopings and magnetic fields applied parallel to the $c$ axis (top) and parallel to the $\mathit{ab}$ plane (bottom). Similar to Ref. 51, small deviations between curves for same $x$ but different fields, owing to experimental error ($\pm 0.007\times {10}^{-4}$ emu/mol), were corrected by shifts in $\chi$, so that curves match for $T>T_{\mathrm{LT}}$ where no field dependence was observed. CO leads to anomalies most pronounced for $H\parallel c$. SO is identified by means of a weak ferromagnetic type transition for $H\parallel \mathit{ab}$. One can see that $T_{\mathrm{SO}}$ also coincides with the onset of weak diamagnetism from superconducting correlations for $H\parallel c$. Furthermore, for $x⩽0.135$ $T_{\mathrm{CO}}$ coincides with $T_{\mathrm{LT}}$. For $x=0.155$ no anomalies are observed and $T_{\mathrm{CO}}$ and $T_{\mathrm{LT}}$ are from XRD. For $x=0.095$ only one transition at $T_{\mathrm{CO}}=T_{\mathrm{LT}}$ is observed. We have limited the data for $H\parallel \mathit{ab}$ to fields below the spin-flop transition (Ref. 51), which will be the focus of a future publication.

Figure 11

Critical temperatures of the SO transition in ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ single crystals. ($▵$) Onset of SO peak intensity and ($\nabla$) saturation temperature of SO-peak width as measured with ND, ($•$) onset of weak ferromagnetic (WFM) type signal for $H\parallel \mathit{ab}$ in static magnetic susceptibility, which we associate with $T_{\mathrm{SO}}$. $(\blacksquare )$ $T_{\mathrm{SO}}$ as measured with $\mu$SR; taken from Ref. 69.

Figure 12

Superconductivity in ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$. Normalized FC (top) and ZFC (bottom) susceptibility vs temperature for a magnetic field of $H=100$ G applied parallel to the $c$ axis. Data in (a) and (c) are for $x⩽1/8$ and in (b) and (d) for $x⩾1/8$.

Figure 13

Superconductivity and stripe order in ${\mathrm{La}}_{1.905}{\mathrm{Ba}}_{0.095}{\mathrm{CuO}}_4$. (a) Integrated intensity of the (1,0,0) superstructure reflection. (b) FC signal for $H=100$ G applied parallel to the $c$ axis and the $\mathit{ab}$ plane. $T_c$ indicates the onset of bulk SC in the LTO phase. $T_{\mathrm{LT}}^{\mathrm{onset}}$ and $T_{\mathrm{LT}}^{\mathrm{end}}$ denote the onset and the completion of the LTO$\to$LTLO transition, respectively.

Figure 14

Low-temperature phase transition in ${\mathrm{La}}_{1.845}{\mathrm{Ba}}_{0.155}{\mathrm{CuO}}_4$. $\omega$ scans through the (2,0,0)/(0,2,0) Bragg reflections that are simultaneously present owing to twin domains; see Fig. 2e. (a) In the LTO phase just above the phase transition, (b) in the mixed LTLO and LTT phase below the transition, and (c) at base temperature. Error bars are within symbol size.

Figure 15

Crucial parameters of stripe phase and crystal structure in${\mathrm{n}}_{\mathrm{La}}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ vs nominal Ba content. (a) HTT$\leftrightarrow$LTO transition temperature $T_{\mathrm{HT}}$ and octahedral tilt angle $\Phi$ at 60 K of average structure. $\Phi$ was calculated using ${\Phi }^2=f(b_o-a_o)$ with $f=380$ (deg)${}^2/$Å for the tilt of the apical oxygen (Refs. 30 and76). (b) LTO$\leftrightarrow$LTT/LTLO transition temperature $T_{\mathrm{LT}}$ and integrated intensity of the (1,0,0) peak at base temperature. The solid line through $I_{(100)}$ is a scaled fit to the square of the $\Phi$ data in (a). (c) Charge stripe order temperature $T_{\mathrm{CO}}$ as well as peak intensity of CO and SO-peaks. (d) In-plane correlation lengths $\xi$ of CO and SO parallel ($\parallel$) and perpendicular ($\perp$) to the stripe direction at base temperature. At $x=0.125$ there is an additional point for ${\xi }_{\mathrm{SO}}^{\parallel }$ at $~600$ Å. (e) Incommensurability $\delta$ extracted with XRD from the CO-peaks and with ND from the SO-peaks. In the case of ${\delta }_{\mathrm{CO}}$ we have averaged the values extracted from the $h$ scans in Figs. 4 and 16. The solid lines $\delta =x$ and $\delta =1/8$ describe the low and high $x$ reference of the stripe model. Solid and dashed lines in (b), (c), and (d) are guides to the eye, except for the solid line for ${\Phi }^2$ in (b). Error bars are not shown if within symbol size.

Figure 16

Comparison of CO-peak widths from transverse scans at base temperature for different dopings. Intensities are normalized and shown after subtraction of a linear background. Top: $h$ scans through the (2$\delta$,0,8.5) peak. Bottom: $k$ scans through the ($2+2\delta$,0,5.5) peak. The lines through the data are fitted Lorentzians. The resolution functions, indicated by black solid lines, were determined for each crystal and both scattering geometries by performing transverse scans through the (0,0,8) and (2,0,6) Bragg reflections, respectively. Note that in the $(h,0,\ell )$ plane (top) the crystal with $x=0.125$ reveals the presence of two domains. The orientation matrix was determined based on the peak positions of the larger domain. Apart from the normalization, the $k$ scans in the bottom panels are the same as in Fig. 4.

Figure 17

Comparison of selected parameters of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ from current work and literature plotted vs nominal ($x$) and calculated ($x^{\prime }$) Ba content. (a) Experimental data for $T_{\mathrm{HT}}$ and theoretical curve used to estimate actual Ba content. Small dots in (a)–(c) represent data plotted vs nominal Ba content, large symbols those vs calculated Ba content. (b) Structural and bulk SC transition temperatures $T_{\mathrm{LT}}$ and $T_c$. Solid lines are guides to the eye. (c) Incommensurability $\delta$ extracted with XRD from the CO peak and with ND from the SO peak. The solid lines describe the low and high $x$ reference of the stripe model. (a)–(c) Where available, error bars are shown. The literature data were taken from Dunsiger et al. (Ref. 47), Adachi et al. (Ref. 42), and Fujita et al. (Ref. 18).