Softening of Cu-O Bond Stretching Phonons in Tetragonal ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$

Phonons in nearly optimally doped ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$ were studied by inelastic x-ray scattering. The dispersion of the low-energy modes is well described by a shell model, while the Cu-O bond stretching mode at high energy shows strong softening towards the zone boundary, which deviates strongly from the model. This seems to be common in the hole-doped high-$T_{\mathrm{c}}$ superconducting cuprates, and, based on this work, not related to a lattice distortion specific to each material.

Figure 1

(a) Magnetic susceptibility of the studied crystal ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$. (b) IXS data taken along the direction close to $[\xi 00]$ with experimental error bars (black), and the fitting curves (red) with several peaks (blue curves as an example for $\xi =3.11$). The phonon branches (peak positions) are indicated by the vertical bars (green).

Figure 2

Comparison of the experimental data (circles) with the calculation (lines). The Raman data in Ref. [19] are indicated by triangles.

Figure 3

(a) Enlarged spectra taken close to the bond stretching mode in a linear scale [see also Fig. 1b]. (b) The lines show the shell model calculation in which the interaction between the next-nearest neighbor oxygens in the ${\mathrm{CuO}}_2$ plane is added. The lines indicate (top to bottom) the $c$-polarized apical oxygen mode, the $a$-polarized Cu-O bond stretching mode, and the $a$-polarized in-plane Cu-O bending mode, respectively. The vertical bars indicate the FWHM of the peaks determined in fitting in Fig. 1b.

Figure 4

Comparison of peak position in hole-doped cuprates (YBCO [6], LSCO [9], and Hg1201). For the YBCO, the equivalent ${\Delta }_4$ symmetry modes are shown for simplicity [6].