Incommensurate Phonon Anomaly and the Nature of Charge Density Waves in Cuprates

While charge density wave (CDW) instabilities are ubiquitous to superconducting cuprates, the different ordering wave vectors in various cuprate families have hampered a unified description of the CDW formation mechanism. Here, we investigate the temperature dependence of the low-energy phonons in the canonical CDW-ordered cuprate ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. We discover that the phonon softening wave vector associated with CDW correlations becomes temperature dependent in the high-temperature precursor phase and changes from a wave vector of 0.238 reciprocal lattice units (r.l.u.) below the ordering transition temperature to 0.3 r.l.u. at 300 K. This high-temperature behavior shows that “214”-type cuprates can host CDW correlations at a similar wave vector to previously reported CDW correlations in non-214-type cuprates such as ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$. This indicates that cuprate CDWs may arise from the same underlying instability despite their apparently different low-temperature ordering wave vectors.

Popular Summary

In most metals, the itinerant free electrons are spread uniformly throughout the material. But in some substances, such as high-temperature superconductors (HTSCs), the electrons can form standing waves known as charge density waves (CDWs), which deform the underlying atomic crystal lattice of the material. In 2012, researchers discovered this electronic state in the HTSC ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$, triggering intense debate about whether this behavior has the same origin as charge correlations discovered in another HTSC family, ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$, over 20 years ago. Here, we examine the CDW correlations in ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ through its interaction with collective lattice motions and find strong evidence to support a unified CDW mechanism in high-temperature superconductors.

We use state-of-the-art x-ray techniques to precisely measure the lattice vibrations in ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ and find how they couple to the CDW. We discovered that the fluctuating CDW correlations that exist at high temperature have a different periodicity than the static ordered CDW but the same periodicity as ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$, which may arise from coupling between the CDW and spin correlations. This reconciles the puzzling wave-vector difference between ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$ and ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$, thus providing strong evidence that CDWs in different cuprates are likely to arise from the same underlying instability despite their different ordering wave vectors.

Our results provide a vital new characterization of the electronic state from which high-temperature superconductivity emerges.

Figure 1

Room-temperature phonon dynamic structure factor $S(\mathit{Q},\omega )$ of ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. (a) Color map around $(0,K,14.5)$ for $K=0\to 0.5$ r.l.u., showing two modes labeled $M1$ and $M2$ that match theoretical predictions shown as dashed lines. (b)–(g) Representative IXS spectra at $K=0.09$, 0.15, 0.2, 0.23, 0.31, and 0.34 r.l.u., respectively.

Figure 2

Temperature dependence of the ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$ phonon spectra. (a)–(c) Elastic-line-subtracted and Bose-factor-corrected IXS spectra at $K=0.09$, 0.23, and 0.35 r.l.u., respectively. Purple, green, and red circles correspond to data at 12, 130, and 300 K, respectively. Strong temperature dependence is seen at 0.23 r.l.u., whereas the $K=0.09$ and 0.35 r.l.u. show no obvious temperature dependence.

Figure 3

Phonon line-shape analysis in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. (a)–(c) Example fits of spectra at $K=0.09$, 0.23, and 0.31 r.l.u. at temperatures of 12, 130, and 300 K, respectively. The individual phonon and elastic components are shown as dashed gray lines. Spectra at $K=0.23$ and 0.31 r.l.u. are vertically shifted for clarity. (d)–(h) Evolution of fitting parameters. Modes $M1$ and $M2$ are shown as circles and squares, respectively, and different colors denote different temperatures. (d) The energy dispersion along $(0,K,14.5)$ through ${\mathit{Q}}_{\mathrm{CDW}}$ (vertical line). (e,f) Temperature-dependent peak energy and width, respectively, at $K=0.23$ r.l.u. (${\mathit{Q}}_{\mathrm{CDW}}$). The vertical line marks $T_{\mathrm{CDW}}$. (g,h) The $\mathit{Q}$-dependent peak widths at 12 and 130 K, respectively. The shaded gray region at ${\mathit{Q}}_{\mathrm{CDW}}$ shows the resolution-limited width of the CDW Bragg peak at 12 K. The horizontal red line denotes the experimental energy resolution of 1.5-meV FWHM. Note that width values do not include the experimental energy resolution, as this is separately accounted for by the convolution in Eq. (1) of Ref. [35].

Figure 4

Comparison between phonon softening and diffraction. Purple circles plot CDW-related phonon softening. The main panel plots the energy difference between 300 and 12 K for ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$, whereas the inset shows the softening between 300 and 5 K for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$. Dashed purple lines are guides to the eye. These are compared to diffraction data in green. For ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$ [11], the phonon softening mirrors the CDW Bragg peak. In ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$, the phonon softening does not match the ordered CDW Bragg peak taken at 23 K; rather, it matches precursor CDW measured at 59 K [24].

Figure 5

Temperature-dependent phonon incommensurability in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_4$. Extracted phonon dispersions of $M2$ at 12, 130, and $300K$ along ($0K$ 14.5) are shown as purple, green, and red squares. Dashed lines are guides to the eye, and arrows indicate the $\mathit{Q}$ associated with the CDW-related phonon softening. Dispersions at 130 and 300 K are offset by 1.5 and 3 meV, respectively, for clarity.