Unveiling Unequivocal Charge Stripe Order in a Prototypical Cuprate Superconductor

In the cuprates, high-temperature superconductivity, spin-density-wave order, and charge-density-wave (CDW) order are intertwined, and symmetry determination is challenging due to domain formation. We investigated the CDW in the prototypical cuprate ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$ via x-ray diffraction employing uniaxial pressure as a domain-selective stimulus to establish the unidirectional nature of the CDW unambiguously. A fivefold enhancement of the CDW amplitude is found when homogeneous superconductivity is partially suppressed by magnetic field. This field-induced state provides an ideal search environment for a putative pair-density-wave state.

Figure 1

Intertwined nature of pair-density-wave (PDW) and charge-density-wave (CDW) order and their relation with x-ray diffraction (XRD). (a) A schematic representation of a unidirectional PDW state (yellow and blue stripes) in real space is shown together with the intertwined CDW state (white stripes). The PDW order parameter changes sign in alternate domains with a period ${\lambda }_{\mathrm{PDW}}$, and vanishes at the domain walls. In contrast, the local electronic density of states is enhanced at the domain walls resulting in an interwoven CDW pattern with a period ${\lambda }_{\mathrm{CDW}}=\frac{1}{2}{\lambda }_{\mathrm{PDW}}$. This illustrates that knowledge of the symmetry of the CDW is vital to identify PDW order. However, due to the fourfold symmetry of ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$, both checkerboard order in (b) as well as degenerate domains of uniaxial stripes [purple and cyan patches in (c)] are allowed. Notably, they result in identical diffraction patterns as illustrated in the circular insets of the respective panels. Note that for panel (c), we have color coded the XRD Bragg reflections in the inset with the color corresponding to the associated stripe domain. Here we use uniaxial pressure to break the fourfold symmetry. As shown in (d) this does not substantially alter the checkerboard XRD intensities. In contrast, it removes the degeneracy of uniaxial CDW stripe order, resulting in a monodomain state (e). As a result, the Bragg reflection associated with the extinct purple domain entirely vanishes as indicated with a black cross in the inset.

Figure 2

Effect of uniaxial pressure on the charge-density wave in ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$. (a) Schematic of the experimental scattering geometry. Uniaxial pressure and magnetic field are applied along the $b$ and $c$ axes, respectively. (b) Demonstration of how uniaxial pressure along the $b$ axis enhances the $a$-axis lattice parameter—smaller scattering angle $2\theta$ implies larger Cu-O-Cu distance. (c) Schematic illustration of the studied part of reciprocal space. Yellow, gray, and dark gray lines indicate, respectively, $h$, $k$, and $\ell$ scans through CDW reflections. Panels (d)–(f) show diffraction intensity of $h$, $k$, and $\ell$ scans through CDW ordering vectors without (purple) and with (cyan) application of uniaxial strain, respectively. For the $\ell$ scan (d), covering two Brillouin zones (BZs), background subtraction was performed. The $h$ and $k$ scans cover only a small fraction of a BZ and, hence, no subtraction is required. Error bars in (b) and (d)–(f), set by counting statistics, are smaller than the used symbols. Horizontal bars in (e) and (f) indicate the instrumental momentum resolution, and solid lines are fits to a Voigt profile.

Figure 3

Magnetic field and temperature dependence of strain-induced charge stripe order. (a),(b) Momentum $h$ and $k$ scans recorded in a $c$-axis magnetic field of 10 T and with compressive strain along the $b$-axis direction for $\ell =12.5$. Absence of charge order along the direction of applied strain reveals the unidirectional nature of the charge-density wave. Error bars are set by counting statistics. (c),(d) Diffraction intensity, displayed in false color scale, around, respectively, $(0,\delta ,12.5)$ and $(\delta ,0,12.5)$ recorded with temperature and magnetic field as indicated. (e),(f) CDW peak amplitude and correlation length as a function of temperature for magnetic field and strain conditions as indicated. Gray data points are adopted from Ref. [24]. Error bars reflect standard deviations of the fits. Gray dashed line indicates the superconducting transition temperature $T_c=27\mathrm{K}$.

Figure 4

Strain-induced stripe charge order detwinning. (a) Integrated charge-density wave intensity as a function of strain ${\epsilon }_a=\mathrm{\Delta }a/a$ and ${\epsilon }_b=\mathrm{\Delta }b/b$ along, respectively, the $a$ (tensile strain ${\mathcal{T}}_i^a$ direction) and $b$ (compressive strain ${\mathcal{C}}_i^b$ direction) axis. Insets illustrate the observed diffraction patterns consistent with stripe order (left) and biaxial or domains of stripe order (right), respectively. (b) Charge-density-wave incommensurability $\delta$ as a function of strain.