Phonon anomalies and lattice dynamics in superconducting oxychlorides Ca${}_{2-x}$CuO${}_2$Cl${}_2$

We observe a low energy flat mode (point 2 above), in-plane, mainly along the ${\Delta }_1$ and ${\Delta }_3$, directions. Along the ${\Sigma }_1$ direction the fitting of the central line is improved if we add some intensity in the same frequency region, but it is not well resolved, with perhaps the exception of the zone boundary. The difference is possibly due to the larger intensity of the elastic line in the geometry corresponding to ${\Sigma }_1$ scattering, which is accidental, and therefore, we can not consider this anomalous excitation anisotropic from our data. In the ${\Delta }_1$ and ${\Delta }_3$ directions it appears as a rather shallow intensity between the elastic line and the acoustic mode, but with an intensity well above the background. For a few scans only it appears with a well defined maximum. The origin of this mode is not understood at the moment, but also observed in closely related system, for which we are presently running experimental investigationsd’Astuto and Lamago (2012) in order to test different hypotheses of the nature of this additional mode, including defect vibrational states and phonon modes activated by symmetry breaking. We note also that several works using Raman scattering and infrared absorption in underdoped La${}_{2-x}$Sr${}_x$CuO${}_{4+\delta }$Tassini et al. (2005); Sugai et al. (2006); Caprara et al. (2005); Lucarelli et al. (2003); Venturini et al. (2002) found intensity at similar energy in under-doped sample, which is attributed to modes related to the stripes charge-modulations.

The anomaly on the ${\Sigma }_1$ acoustic mode close to the zone boundary concern only one point, while the rest of the dispersion fit quite well with our simulations, both for the frequencies and the intensities. It would be nevertheless interesting to further study this mode, by complementary ab-initio calculation, to verify our findings, as well as with more detailed measurements close to the zone boundary.

The calculated frequency of the Cu-O bond bending mode at the zone center $\Gamma$ is very close to the experimental IR position for the corresponding Eu(2) mode, but our calculation shows an upward dispersion, which does not seems to be confirmed in our data along ${\Delta }_3$. Indeed, the experimental intensity along ${\Delta }_3$ between 30 and 40 meV, seems to lie at lower energy with respect to the calculated one. However, the Cu-O bond bending mode has a relatively low yield, along both ${\Delta }_1$ and ${\Delta }_3$ comparable to the bond-stretching one, while being at lower energy. They are therefore measured on top of the tails of the much stronger modes at lower energy, as described above, fact that complicates their detection. A further investigation of this dispersion along ${\Delta }_1$ and ${\Delta }_3$ would be possible only at low temperature, as discussed above. Note that, on the contrary, for the corresponding Eu(2) mode along ${\Sigma }_1$ the mode is reasonably well resolved, and match very well both the calculated frequency dispersion and the intensity.

Concerning the Cu-O bond stretching along the bond direction, ($q$, 0, 0), we calculate a symmetrical dispersion in the extended zone, similarly to other cuprate with T-structure. Frequencies at both zone center $q$=0 and extended zone boundary $q$=2$\frac{\pi }{a}$ are the same, and do not change with doping, in the systems where doping dependence is known (see, for example, Ref. Reznik, 2010 and Pintschovius et al., 2006) . For such branch, the mode has a vanishing small dynamical structure factor, so we can not compare directly the dispersion close to zone center and $q$=2$\frac{\pi }{a}$. On the other hand, IR data shows well defined phonons mode at the zone center, but only in the insulating phases. We remark that the IR data assigned to the EU(1) mode on heavily underdoped samples, corresponds very well to the frequency we found at the zone boundary for both ${\Delta }_1$ and ${\Delta }_3$ Cu-O bond stretching modes. This suggests that the physics for this mode is very similar to the one of La${}_{2-x}$Sr${}_x$CuO${}_{4+\delta }$Pintschovius et al. (2006); Reznik (2010), despite the present lack of dispersion data in the undoped parent compound, and of mode frequencies an the Zone Center $\Gamma$ in the doped one.

In Fig. 7, top panels, we shows the detail of the Cu-O bond stretching for the direction with ${\Delta }_1$ (half-breathing) and ${\Sigma }_1$ (full-breathing) character and compared to our lattice dynamics simulations, with different screening wave number ${\kappa }_s$.

In the same figure, bottom panel, we can follow the corresponding broadening of the mode, which starts at a value of less than 1 meV, once instrumental resolution subtracted, to end up at more than 12 meV. Note that the softening and simultaneous broadening is observed only for the mode with character ${\Delta }_1$, propagating along $\left(q00\right)$ (Fig. 7, left, top and bottom). The same mode propagating along the diagonal (${\Sigma }_1$, Fig. 7, left) is dispersing upward, with a line-shape that is resolution limited, as in other cuprates Reznik (2010).

STM experiments in oxychlorides Hanaguri et al. (2004) report a modulation of the Fourier transform of the differential conductance map $g(r,E)$ at wave-vectors of ($nq$, 0 0) with $n$ integer and $q\sim$ 0.25, i.e. for modulations with a periodicity corresponding to 4$a$. This is reminiscent of what happens in cuprates Reznik et al. (2008); Reznik (2010) and other perovskite-like ternary transition metal oxides Weber et al. (2009), with the difference that in oxychlorides, there is no observation of static diffraction by charge density modulation. We can then confirm that the observation of STM has a dynamical precursor at room temperature, as for the stripes in other cuprates. Note that the pattern observed by STM where first labelled as ”checkerboard” Hanaguri et al. (2004), although this was not confirmed by further studiesKohsaka et al. (2007), that maintained, however, a Cu-O-Cu bond-centered electronic pattern without long-range order but with 4-lattice-constant-wide unidirectional electronic domains Kohsaka et al. (2007). Indeed, for a ”checkerboard” pattern, we should have observed the same phonon anomaly along both the ${\Delta }_1$ and the ${\Sigma }_1$ directions, as it is the case in those transition metal oxides where such a modulation is presentWeber et al. (2009). Here we observe the anomaly only for the mode with ${\Delta }_1$ character in agreement with other cuprates showing a stripe charge modulation pattern Reznik et al. (2008); Reznik (2010).

Regardless of the exact pattern of the charge modulation, we note that such anomaly can not be accounted for in simple models based on Landau liquid band type as DFT ones Reznik et al. (2008), as the charge modulations themselves can not be simulated in such theoretical framework. We remark also that the propagation vector ($n$0.25, 0 0) of these unidirectional electronic domains, where we observe the ”sharp dip” anomaly (Fig. 7), would corresponds to a nesting vector of the Fermi Surface, as reported for Na-doped Ca${}_2$CuO${}_2$Cl${}_2$ Shen et al. (2005), although the region of the Fermi Surface connected by this vector does not show well defined quasi-particles.

Apart from this peculiar anomaly close to the charge modulation wave-vector ($\pi /2a$ 0 0), it has been claimed that the overall bending down of the mode with ${\Delta }_1$ character is well reproduced by DFT calculations Bohnen et al. (2003); Falter et al. (2006); Giustino et al. (2008) that reproduce a smooth cosine function shape. These calculations found that such mode has one of the highest coupling with charge carriers. Nevertheless, the calculated electron-phonon coupling was still too low to account for the strong kink in ARPES dispersion Giustino et al. (2008), and the very high $T_c$ of cuprates Bohnen et al. (2003). Moreover, similar calculations showed a concurrent strong downward bending of the ${\Sigma }_1$ mode, that can lead to discrepancies at the zone boundary X ($\pi /a$ $\pi /a$ 0) between calculated and experimental frequencies in optimal or slightly under-doped compounds up to 10 meV Bohnen et al. (2003); Falter et al. (2006). It was suggested that such a difference originates from unrelated effects over the two directions ($q$, 0, 0) and ($q$, $q$, 0)Bohnen et al. (2003). Note that Giustino and co-workers in Ref. Giustino et al., 2008 use the label half-breathing and full-breathing to distinguish Cu-O bond-stretching modes with different character, as in other worksPintschovius et al. (2006), but the full-breathing label is not consistent throughout literature, as it indicates modes with ${\Sigma }_1$ character in Ref. Pintschovius et al., 2006, while misleadingly label a dispersion along ($q$ 0 0) with ${\Delta }_3$ character in Ref. Giustino et al., 2008. It is worthwhile to notice that the later dispersion has been found to be flat for all dopings, either experimentally and theoretically. At the same time, the authors of Ref. Giustino et al., 2008 do not show their results along ${\Sigma }_1$, where a very bad agreement is found in other DFT calculations Bohnen et al. (2003); Falter et al. (2006) and experimental data. We therefore avoid the ”breathing” notation.

Contrary to DFT simulations, our simple lattice dynamical calculation based on a shell model, reproduces well the dispersion of the mode with ${\Sigma }_1$ character, while missing the softening of the one with ${\Delta }_1$ character, when using a screening wave number of ${\kappa }_s=\frac{\pi }{2a}$ (see Fig. 7). When increasing the doping, and consequently the screening wave number, we observe a modest softening for both bands. The effect is relatively weak in our calculations, as we have kept all other parameters fixed, in order to allow a direct comparison between the two cases. The important point here is that the softening affects exclusively the modes with ${\Delta }_1$ and ${\Sigma }_1$ character, with a trend that reproduces the experimental one with increasing doping. This suggests that overestimated screening effects can lead to simulations that reproduce the dispersion at much higher doping. If compared to optimal or slightly under-doped compounds, this gives a dispersion which is accidentally similar to the one for the mode with ${\Delta }_1$ character, while giving a large difference between calculation and experimental dispersion for the mode with ${\Sigma }_1$ character, and accordingly underestimate the electron-phonon coupling. Indeed, the dispersion found in Ref. Bohnen et al., 2003; Giustino et al., 2008 for the Cu-O bond-stretching modes with ${\Delta }_1$ and ${\Sigma }_1$ character are strikingly similar to the experimental dispersion found only in highly overdoped cuprates La${}_{2-x}$M${}_x$CuO${}_4$ (M=Sr,Ba)Pintschovius et al. (2006); d’Astuto et al. (2008). Therefore, a possible unique origin for the softening of modes with both ${\Delta }_1$ and ${\Sigma }_1$ character in DFT calculation can not be ruled out. This would imply that the phonon simulations by DFT correspond to a much higher doping, that, because of the larger screening, have a lower electron-phonon coupling. To confirm such conclusion also in oxychlorides, it would be important to compare with DFT calculations, and to have data on the un-doped parent compound.