Stripe order, depinning, and fluctuations in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$ and ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.075}{\mathrm{Sr}}_{0.050}{\mathrm{CuO}}_4$

We present a neutron scattering study of stripe correlations measured on a single crystal of ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. Within the low-temperature-tetragonal (LTT) phase, superlattice peaks indicative of spin and charge stripe order are observed below $50\mathrm{K}$. For excitation energies $\hslash \omega ⩽12\mathrm{meV}$, we have characterized the magnetic excitations that emerge from the incommensurate magnetic superlattice peaks. In the ordered state, these excitations are similar to spin waves. Following these excitations as a function of temperature, we find that there is relatively little change in the $\mathbf{Q}$-integrated dynamical spin susceptibility for $\hslash \omega \sim 10\mathrm{meV}$ as stripe order disappears and then as the structure transforms from LTT to the low-temperature-orthorhombic phase. The $\mathbf{Q}$-integrated signal at lower energies changes more dramatically through these transitions, as it must in a transformation from an ordered to a disordered state. We argue that the continuous evolution through the transitions provides direct evidence that the incommensurate spin excitations in the disordered state are an indicator of dynamical charge stripes. An interesting feature of the thermal evolution is a variation in the incommensurability of the magnetic scattering. Similar behavior is observed in measurements on a single crystal of ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.075}{\mathrm{Sr}}_{0.050}{\mathrm{CuO}}_4$; maps of the scattered intensity in a region centered on the antiferromagnetic wave vector and measured at $\hslash \omega =4\mathrm{meV}$ are well reproduced by a model of disordered stripes with a temperature-dependent mixture of stripe spacings. We discuss the relevance of our results to understanding the magnetic excitations in cuprate superconductors.

Figure 1

Zero-field-cooled (ZFC) and field-cooled (FC) susceptibility of ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$ single crystal measured at $10\mathrm{Oe}$. ZFC susceptibility of ${\mathrm{La}}_{1.89}{\mathrm{Ba}}_{0.11}{\mathrm{CuO}}_4$, taken from Ref. 37, is shown as a reference.

Figure 2

Scan geometry in the $\left(hk0\right)$ tetragonal plane. Solid squares show nuclear Bragg peaks; open and solid circles denote nuclear and magnetic IC superlattice peaks, respectively.

Figure 3

IC peaks from (a) SDW order (measured along the path labeled A in Fig. 2) and (b) CDW order (measured along path B) in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. Solid (open) circles indicate measurements below (above) $T_{st}$.

Figure 4

Temperature dependences of (a) (100), (b) CDW, and (c) SDW superlattice peak intensities in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$.

Figure 5

Inelastic neutron scattering spectra of ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$ at (a) $30\mathrm{K}$, (b) $48\mathrm{K}$, (c) $65\mathrm{K}$ and (d) $100\mathrm{K}$ at a constant energy of $3\mathrm{meV}$. The solid lines are fits assuming four equivalent peaks at $(0.5\pm \delta ,0.5,0)$ and $(0.5,0.5\pm \delta ,0)$.

Figure 6

Inelastic neutron scattering spectra of ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$ at (a) $30\mathrm{K}$, (b) $65\mathrm{K}$, (c) $100\mathrm{K}$ and (d) $200\mathrm{K}$ at a constant energy of $6\mathrm{meV}$. The solid lines are fits assuming four equivalent peaks at $(0.5\pm \delta ,0.5,0)$ and $(0.5,0.5\pm \delta ,0)$.

Figure 7

Local spin susceptibility as a function of $\omega$ in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$ at (a) $8\mathrm{K}$, (b) $30\mathrm{K}$, (c) $65\mathrm{K}$, (d) $100\mathrm{K}$, and (e) $200\mathrm{K}$. Dashed lines are guides to the eye.

Figure 8

Temperature dependences of (a) local spin susceptibility ${\chi }^{″}$, (b) peak width (half width at half maximum) $\kappa$, and (c) incommensurability $\delta$ at the energy transfers of 3 and $6\mathrm{meV}$ in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. Vertical lines indicate $T_{st}$ and $T_{d2}$. Dashed lines are guides to the eye.

Figure 9

$\omega$ dependence of (a) incommensurability $\delta$ and (b) resolution-corrected peak width (half width at half maximum) $\kappa$ of IC peaks for ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. Open circles denote $30\text{−}\mathrm{K}$ data; solid circles, $65\mathrm{K}$; open squares, $200\mathrm{K}$.

Figure 10

Mesh scans about ${\mathbf{Q}}_{\mathrm{AF}}$ for $\hslash \omega =4\mathrm{meV}$. (a), (c), and (e) are raw data, after subtraction of background (see text) and correction for the Cu magnetic form factor, at $T=45$, 100, and $200\mathrm{K}$, respectively; panels on the right [(b),(d),(f)] were obtained from those on the left by averaging over four-fold symmetry operations. The white circles indicate the positions of the elastic magnetic peaks that appear below $37\mathrm{K}$.

Figure 11

Simulations of the $4\mathrm{meV}$ scans, as described in the text. (a), (c), and (e) are simulations of the mesh scans at $T=45$, 100, and $200\mathrm{K}$, respectively. (b), (d), and (f) compare the calculated curves with the symmetrized data along $\mathbf{Q}=(1+h,h,0)$. The white circles are the same as in Fig. 10.