Resonant elastic soft x-ray scattering in oxygen-ordered YBa${}_2$Cu${}_3$O${}_{6+\delta }$

Static charge-density-wave (CDW) and spin-density-wave (SDW) order has been convincingly observed in La-based cuprates for some time. However, more recently it has been suggested by quantum oscillation, transport, and thermodynamic measurements that density-wave order is generic to underdoped cuprates and plays a significant role in YBa${}_2$Cu${}_3$O${}_{6+\delta }$ (YBCO). We use resonant soft x-ray scattering at the Cu $L$ and O $K$ edges to search for evidence of density-wave order in ortho-II and ortho-VIII oxygen-ordered YBCO. We report a null result—no evidence for static CDW order—in both ortho-II and ortho-VIII ordered YBCO. While this does not rule out static CDW order in the CuO${}_2$ planes of YBCO, these measurements place limits on the parameter space (temperature, magnetic field, scattering vector) in which static CDW order may exist. In addition, we present a detailed analysis of the energy and polarization dependence of the ortho-II superstructure Bragg reflection [0.5 0 0] at the Cu $L$ edge. The intensity of this peak, which is due to the valence modulations of Cu in the chain layer, is compared with calculations using atomic scattering form factors deduced from x-ray absorption measurements. The calculated energy and polarization dependence of the scattering intensity is shown to agree very well with the measurement, validating the approach and providing a framework for analyzing future resonant soft x-ray scattering measurements.

Figure 1

The crystal structure of ortho-II ordered YBa${}_2$Cu${}_3$O${}_{6.5}$. YBCO has Cu sites in both the CuO${}_2$ planes (Cu2) and the chain layer (Cu1). In ortho-II ordered YBCO, the dopant oxygen atoms in the chain layer are ordered forming alternating rows of full and empty chains. This induces an alternation in the valence of Cu1 from Cu${}^{2+}$ in the full chains (Cu1f, red) to Cu${}^{1+}$ in the empty chains (Cu1e, blue).

Figure 2

The x-ray absorption (total fluorescence yield) of YBa${}_2$Cu${}_3$O${}_{6+\delta }$ at the Cu $L_3$ absorption edge for $\vec{\epsilon }\parallel \vec{a}$, $\vec{\epsilon }\parallel \vec{b}$, and $\vec{\epsilon }\parallel \vec{c}$. Three samples are shown with oxygen doping $\delta =0$ (black), 0.5 (red), and 1 (blue). The average of the XAS of the undoped ($\delta =0$) and fully doped ($\delta =1$) samples shown in green agrees well with the measured spectra of ortho-II ordered YBCO ($\delta =0.5$), supporting the premise that ortho-II ordered YBCO consists of alternating full and empty chains.

Figure 3

Oxygen $K$ edge x-ray absorption for undoped ($\delta =0$), fully doped ($\delta =1$), and ortho-II ordered YBCO ($\delta =0.5$) measured by total fluorescence yield for $\vec{\epsilon }\parallel \vec{a}$, $\vec{b}$, and $\vec{c}$. For $\vec{\epsilon }\parallel \vec{a}$, the XAS pre-edge probes the unoccupied $p_x$ states that are hybridized with the Cu 3$d_{x^2-y^2}$ states of the CuO${}_2$ planes. For $\vec{\epsilon }\parallel \vec{c}$, the XAS pre-edge probes the unoccupied $p_z$ states that are hybridized with the Cu 3$d_{y^2-z^2}$ states of the chain layer. Measurements with $\vec{\epsilon }\parallel \vec{b}$ probe both the CuO${}_2$ planes and the chain layer. The peak around 529.5 eV for $\vec{\epsilon }\parallel \vec{a}$ is equivalent to the peak where the stripe scattering intensity is enhanced in RSXS measurements of La-based cuprates.[29, 34]

Figure 4

Cu $L$ edge searches for stripe ordering in ortho-VIII YBCO. (a),(b) The measured intensity through [$H$ 0 1] and [$H$ 0 1.3] with $\vec{\epsilon }\parallel b$ at several temperatures for different photon energies. (c),(d) The measured intensity through [$H$ 0 0] and [$H$ 0 0.3] with $\vec{\epsilon }\parallel b$ at several temperatures for different photon energies. At 931.6 eV, the atomic scattering form factor for Cu2 in the CuO${}_2$ planes should be enhanced, whereas for 933.6 eV, the enhancement is more likely associated with the Cu1 site in the chain layer. In all cases, no superlattice peak is observed near $H$ $=$ 1/4. The sample geometry is depicted in the insets.

Figure 5

Oxygen $K$ edge searches for CDW order in ortho-VIII YBCO. (a)–(c) The measured intensity through [$H$ 0 0] and [$H$ 0 0.3]. Sample geometry is depicted in the insets. No superlattice peak is observed near $H$ $=$ 1/4. The sloping background in (a) and (b) is attributed to angle-dependent x-ray fluorescence.

Figure 6

Resonant x-ray scattering from the ortho-II oxygen superstructure of YBa${}_2$Cu${}_3$O${}_{6.5}$ at the Cu $L_{2,3}$ absorption edges. (a) The intensity of the [0.5 0 0] superstructure peak vs incident photon energy for $\sigma$ (blue) and $\pi$ (red) incident polarization. (b) and (c) The scattering intensity through [0.5 0 0] vs $H$ and $K$, respectively. The lines are the result of fitting the data to Eq. (1).

Figure 7

The real ($f_1$) and imaginary ($f_2$) components of the scattering form factor calculated from the x-ray absorption of undoped and fully doped YBCO for $\vec{\epsilon }\parallel \vec{a}$, $\vec{\epsilon }\parallel \vec{b}$, and $\vec{\epsilon }\parallel \vec{c}$. $f$ is determined for Cu1e and Cu1f from undoped ($\delta$ $=$ 0) and fully doped ($\delta$ $=$ 1) YBCO, respectively. Since the XAS is the sum of contributions from both the chain and plane sites, the calculated scattering form factors are also the sum of Cu1 and Cu2 sites.

Figure 8

A comparison of the measured and calculated scattering intensities as a function of incident photon energy through the Cu $L$ edges for both $\vec{\epsilon }\parallel \vec{c}$ ($\sigma {\sigma }^{\prime }$) and $\vec{\epsilon }\perp \vec{c}$ ($\pi {\pi }^{\prime }$) scattering geometries. The measured $\vec{\epsilon }\parallel \vec{c}$ spectrum was scaled in intensity to roughly match the calculated spectrum, which is expressed in units of $e^2$ [following Eq. (2)]. The $\vec{\epsilon }\perp \vec{c}$ spectrum was scaled by the same amount as the $\vec{\epsilon }\parallel \vec{c}$ spectrum and offset for clarity. This scaling parameter is the only free parameter. The calculation agrees well with the data, capturing the relative intensities of the $L_3$ and $L_2$ edges, as well as the relative intensity of the $\vec{\epsilon }\parallel \vec{c}$ and $\vec{\epsilon }\perp \vec{c}$ scattering geometries.

Figure 9

(a) The measured intensity vs photon energy at the [0.35 0 0] superlattice peak of ortho-VIII ordered YBCO. A large background (black) from x-ray fluorescence measured away from the superlattice peak contributes to the measured intensity at [0.35 0 0]. (b) Subtracting out the x-ray fluorescence reveals the measured scattering intensity vs photon energy through the Cu $L$ edge. Apart from a constant offset, the measured scattering intensity (blue) agrees well with the calculated intensity (red).