Synchrotron x-ray scattering study of charge-density-wave order in ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$

We present a detailed synchrotron x-ray scattering study of the charge-density-wave (CDW) order in simple tetragonal ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$ (Hg1201). Resonant soft x-ray scattering measurements reveal that short-range order appears at a temperature that is distinctly lower than the pseudogap temperature and in excellent agreement with a prior transient reflectivity result. Despite considerable structural differences between Hg1201 and ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$, the CDW correlations exhibit similar doping dependencies, and we demonstrate a universal relationship between the CDW wave vector and the size of the reconstructed Fermi pocket observed in quantum oscillation experiments. The CDW correlations in Hg1201 vanish already below optimal doping, once the correlation length is comparable to the CDW modulation period, and they appear to be limited by the disorder potential from unit cells hosting two interstitial oxygen atoms. A complementary hard x-ray diffraction measurement, performed on an underdoped Hg1201 sample in magnetic fields along the crystallographic $c$ axis of up to 16 T, provides information on the form factor of the CDW order. As expected from the single-${\mathrm{CuO}}_2$-layer structure of Hg1201, the CDW correlations vanish at half-integer values of $L$ and appear to be peaked at integer $L$. We conclude that the atomic displacements associated with the short-range CDW order are mainly planar, within the ${\mathrm{CuO}}_2$ layers.

Figure 1

Phase diagram of Hg1201 ($p>0.04$), extended to $p=0$ based on data for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$ (YBCO) [12, 42]. Solid blue line: Doping dependence of the superconducting (SC) transition temperature $T_c\left(p\right)$ [43]. Dark gray region: Deviation (not to scale) of $T_c$ from estimated parabolic doping dependence (dashed blue line). $T$*: Pseudogap (PG) temperature, estimated from deviation of $T$-linear planar resistivity in the strange-metal (SM) state [12, 15]. $T$**: Temperature below which Fermi-liquid (FL) behavior is observed in the PG state [12, 13, 14, 15]. $T_{q=0}$ and $T_{AF}$: Characteristic temperatures below which $q=0$ magnetic order [3, 44] and an enhancement of antiferromagnetic (AF) fluctuations [4, 5] are observed. $T_{CDW}$: Onset of short-range CDW order estimated from RXS (blue circles; $p=0.09$ results from Ref. [45]) and XRD (blue diamond). $T_{\mathrm{opt}}$: Characteristic temperature observed in time-resolved optical reflectivity measurements [38].

Figure 2

Momentum dependence of the CDW peak observed via RXS in seven underdoped samples along [100]. Left: Raw spectra collected at indicated temperatures. Right: Intensity difference of data in left panels. The high-temperature data constitute an estimate of the background scattering. Although the location of the CDW peak is not clear from the raw data, the background-subtracted scans clearly reveal CDW peaks. Blue lines: Gaussian fits to the data.

Figure 3

Momentum scans across the CDW wave vector at the Cu $L_3$ resonance at various temperatures for two Hg1201 samples with (a) $T_c=55$ K ($p=0.064$) and (b) $T_c=94$ K ($p=0.126$). Due to the very low intensity of the CDW peak in the first sample, systematic subtraction of high-temperature “background” from the low-temperature scans did not result in a reliable estimate of the temperature dependence. Instead, the onset temperature of the CDW order was determined from the temperature dependence of the integrated intensity (between $H=0.2$ and 0.35 r.l.u.) after introducing an offset to match the background level at 200 K; the result is shown in the inset of (a). The onset temperature $T_{CDW}=140\left(20\right)$ K is the temperature at which the integrated intensity saturates. No intensity change with temperature is observed in the nearly optimally doped sample ($T_c=94$ K), consistent with the absence of CDW order. (c) Background-subtracted momentum scan across $q_{CDW}$ for a sample with $T_c=76$ K ($p=0.105$) along the equivalent directions [100] and [010]. The solid lines are Gaussian fits (with linear background) to the data, yielding equivalent peak positions, amplitudes, and widths, within the experimental uncertainty.

Figure 4

(a)–(d) Temperature dependence of the background-subtracted CDW peak for four Hg1201 samples observed by RXS. The peak intensity decreases above $T_c$ and saturates at the characteristic doping-dependent temperature $T_{CDW}$. In order to obtain the most accurate temperature dependence of the CDW peak parameters, high-temperature data were subtracted from the low-temperature scans (i.e., data at 200 K, 240 K, 200 K, and 155 K were selected as reference background for the samples with $T_c=65$ K, 69 K, 79 K, and 88 K, respectively). The solid lines are the result of subsequent Gaussian-function fits. (e)–(h) Temperature dependence of the peak amplitude and width (FWHM). Blue areas indicate the superconducting state for each sample. The thick gray lines are guides to the eye.

Figure 5

CDW peak in a Hg1201 sample, with $p=0.098$ ($T_c=75$ K), observed via 80 keV XRD in various Brillouin zones, in an effective 14 T or 16 T $c$-axis magnetic field, as indicated. The high energy of the hard x rays allowed access a large portion of momentum space. The corresponding Brillouin zone is marked above each panel. (a) Momentum scans across the CDW wave vector at 40 K and 200 K. (b) Intensity difference of the scans collected at the indicated temperatures. The solid lines are the result of subsequent fits to a Gaussian function. (c) Two representative momentum scans across the CDW wave vector at 60 K and 200 K. The solid lines represent the estimated linear background. (d) Corresponding intensity difference of the 40 K and 200 K data. These temperatures lie well below and above $T_{CDW}$, respectively. No clear evidence of CDW intensity is seen at half integer $L$. (e) Linear-background-subtracted data at the indicated temperatures. The solid lines are the results of fits to a Gaussian function. (f), (g) Temperature dependence of the CDW peak amplitude and width (FWHM) from the fit in (e). Green area indicates the temperature region of the superconducting state, in the absence of the magnetic field. Thick gray line in (f) is a guide to the eye. The data in (a)–(c) and (e) are offset for clarity.

Figure 6

(a) CDW correlation length (in units of the planar lattice parameter $a$) near $T_c$ for Hg1201. Open and filled blue triangles indicate RXS and XRD (at 14 T) results, respectively; the $p\approx 0.09$ result is from Ref. [45]. For comparison, the data for YBCO from Ref. [49] are shown as red squares. For Hg1201, $\xi /a$ estimated about 20 K below $T_{CDW}$ is shown as well (blue crosses). Blue dashed line: Simulated characteristic size of CDW domains, as discussed in the text. (b) $q_{CDW}$ for Hg1201 (blue circles) and YBCO (red diamonds). Black lines are linear fits (see text). Inset compares the correlation length of the CDW order in Hg1201 (obtained near $T_c$) with the corresponding wavelength ${\lambda }_{CDW}$, expressed in lattice units. The red and blue bands for $\xi \left(p\right)/a$ are guides to the eye.

Figure 7

(a) Noninteracting tight-binding FS of Hg1201 at $p=0.07$ (light blue) and $p=0.11$ (light red) with hopping parameters from Ref. [72]. Thick lines: Reconstructed electron pocket; double arrows: $q_{CDW}$ (see main text). (b) Pocket size, as a fraction of the Brillouin zone, and associated quantum-oscillation frequency ($F$) for Hg1201 and YBCO as a function of $q_{CDW}$. Blue square (Hg1201) [20, 45] and red triangles (YBCO) [61] represent the doping levels for which both $q_{CDW}$ and $F$ have been determined experimentally. $F$ is related to the Fermi pocket size $S$ via the Onsager relation, $F=\hslash S/2\pi e$. Blue circles: Pocket size estimated from tight-binding calculation, with $q_{CDW}$ obtained from the linear fit in Fig. 6. Horizontal error bars: Experimental uncertainty in $q_{CDW}$. Blue triangles: Recent high-precision experimental result for $F$ [21]. Inset: Doping dependence of the electron pocket in the reduced Brillouin zone (increments: $\mathrm{\Delta }p=0.01$). Predicted Lifshitz-type change in FS topology at $p\approx 0.105$ (black arrow) due to overlap of reconstructed pockets [see result for $p\approx 0.11$ in panel (a)].

Figure 8

Representation of the two ${\mathrm{HgO}}_{\delta }$ layers adjacent to a ${\mathrm{CuO}}_2$ layer in Hg1201. (a) Red and blue dots represent randomly distributed i-O atoms with a density that corresponds to $p=0.083$ (Refs. [27, 28, 29]), the doping level at which the CDW correlation length is the largest (Fig. 6). The red dots indicate unit cells with one i-O, either in the top or the bottom ${\mathrm{HgO}}_{\delta }$ layer. The dark blue dots indicate unit cells with two i-O, one in each of the two ${\mathrm{HgO}}_{\delta }$ layers. In the model considered here (see also Appendix pp3), CDW patches (light blue circles) nucleate in these relatively rare unit cells and extend to the closest doubly occupied site. This results in a CDW correlation length of $\xi /a\approx 7.7$, in good agreement with the experimental value $\xi /a=8.0\left(7\right)$ at this doping level. With increasing doping, the calculated correlation length decreases and follows the experimental values, as shown in Fig. 6. Displayed area: ${\left(100a\right)}^2$. (b) Same as in (a), but on a larger scale [displayed area: ${\left(500a\right)}^2$] for better visualization of the CDW patch distribution.

Figure 9

Illustration of the average atomic displacements proposed for the CDW order in Hg1201. The red, green, and black dashed lines correspond to the ${\mathrm{CuO}}_2$, BaO, and ${\mathrm{HgO}}_{\delta }$ layers, respectively. Black (blue) arrows represent the longitudinal (transverse) atomic displacements. Small and large displacements are indicated by the length of the arrows (not to scale). See the text for further details.

Figure 10

(a) CDW patches (yellow circles) in the ${\mathrm{CuO}}_2$ planes formed in regions between i-O atoms (red dots) in the adjacent ${\mathrm{HgO}}_{\delta }$ layers. Displayed area: ${\left(100a\right)}^2$. The i-O density corresponds to $p=0.083$ [27, 28, 29]. (b) CDW peak, calculated from the distribution of charge-order patches in (a); the correlation length $\xi /a=2.6$ is obtained from a Gaussian fit. (c) Same as in (a), but CDW patches develop around i-O and extend to nearest-neighbor i-O. In both (a) and (c), the density of the charge-ordered patches has been decreased for clarity.