Analogies of phonon anomalies and electronic gap features in the infrared response of Sr$_{14-x}$Ca_xCu₂₄O₄₁ and underdoped YBa₂Cu₃O$_{6+x}$

1.1.1.  c-axis response.

1.1.2. In-plane response.

1.2.1.  a-axis response along the rungs of the ladders.

1.2.2.  c-axis response along the legs of the ladders.

High-quality Sr₂Ca₁₂Cu₂₄O₄₁ single crystals were grown at PSI using a modified mirror floating zone furnace (Model: FZ-T-10 000-HVI-VP-PC, Crystal Systems Corp. Japan) with oxygen pressure up to 35 bar as described in [81]. The crystals were oriented with an x-ray Laue diffractometer and millimeter-sized ac- and ab-surfaces cut with a diamond wire-saw. These surfaces have been polished to optical grade using oil-based diamond paste. The Sr$_{14-x}$Ca_x Cu₂₄O₄₁ crystals with a lower Ca content of x = 0 and 8 were grown as described in [82], their surfaces were also cut and polished as described above.

The oxygen isotope exchange of the SCCO crystal with x = 12 was performed using an experimental set-up as reported previously [83]. The exchange was performed in 98% enriched ¹⁸O₂ gas (1.2 bar) at 550 ^∘C for 200 h, followed by cooling with a rate of 20 ^∘C h⁻¹ down to room temperature. The same crystal was first measured in its pristine state with ¹⁶O, was subsequently exchanged with ¹⁸O and measured, and finally back exchanged to ¹⁶O and remeasured.

A pair of strongly underdoped YBa₂Cu₃O_6.5 crystals with either ¹⁶O or ¹⁸O isotopes was prepared by annealing under identical conditions in isotope enriched oxygen gas atmosphere. The samples were taken through six cycles of 24 h annealing in 0.9 atm of either 98% enriched ¹⁸O₂ or natural ¹⁶O₂ at 786 ^∘C, corresponding to an equilibrium oxygen content of x = 0.5 [84]. After each annealing step, the sample chamber was air quenched and the 98% enriched ¹⁸O₂ gas was replaced. At the end of the final annealing treatment the sample chamber was rapid-quenched in water to avoid any further oxygen uptake. The overall mass change of the ¹⁸O crystal, before and after the ¹⁸O treatment, was consistent with a 95% exchange of all oxygen sites.

For the Sr$_{14-x}$Ca_x Cu₂₄O₄₁ single crystals with x = 0, 8 and 12 the infrared reflectivity spectra were measured at a near-normal incidence angle with a Bruker VERTEX 70v Fourier-transform infrared (FTIR) spectrometer. The incident light was polarized with a free-standing wire grid polarizer, that was oriented either along the a-, b-, or c-axis direction, and the reflectivity spectra were normalized with an in situ gold overfilling technique [85]. The spectra from 30 to 20 000 cm⁻¹ were collected at temperatures between 5 and 300 K with the sample mounted in an ARS-Helitran cryostat. In the near-infrared to ultraviolet range (4000–50 000 cm⁻¹) the complex optical response function at room temperature was measured with a commercial ellipsometer (Woollam VASE). The optical conductivity and related response functions and constants were obtained by performing a Kramers–Kronig analysis of the $R(\omega)$ spectra [76]. For the low-frequency extrapolation below 30 cm⁻¹, we used a Hagen-Rubens function ($R = 1 - A\sqrt{\omega}$) for the conducting samples and a constant for the insulating ones. On the high-frequency side, the extrapolation was anchored with the room-temperature ellipsometry data.

The measurements of the far-infrared c-axis response (100–650 cm⁻¹) of the pair of strongly underdoped YBa₂Cu₃O_6.5 crystals with either ¹⁶O or ¹⁸O oxygen isotopes ($T_c = 52$ K) were performed at the infrared beamline of the KARA (formerly ANKA) synchrotron at the Karlsruhe Institute of Technology (KIT) using a homebuilt ellipsometer that is attached to a Bruker 66v Fourier transform infrared (FTIR) spectrometer as described in [86]. The ellipsometric response was measured on an ac-plane of the plate-like crystals with the plane of incidence of the infrared light parallel to the c-axis and the incidence angle set to 75 degree. The obtained spectra of the c-axis pseudo-dielectric function have not been corrected for anisotropy effects, since they are known to cause only small changes to the absolute values of the c-axis response and, in particular, not to cause artificial spectroscopic features [86].

The a-axis response of a detwinned YBa₂Cu₃O_6.6 single crystal with $T_c = 61$ K was measured with reflectivity using the reflectivity setup described above for the Sr$_{14-x}$Ca_x Cu₂₄O₄₁ crystals. For the Kramers–Kronig analysis of the measured $R(\omega)$ spectra we used the reffit program [87]. In addition, its complex a-axis dielectric function was determined with infrared ellipsometry at the infrared beamline of the KARA Synchrotron in the range from 100 to 4000 cm⁻¹ at an incidence angle of 80 degree.

The SC transition temperatures of the YBa₂Cu₃O$_{6+x}$ crystals were determined from dc susceptibility measurements where T_c was defined as the midpoint of the onset of the diamagnetic signal.

We start with the discussion of the optical response of the Sr₂Ca₁₂Cu₂₄O₄₁ crystal (x = 12) along the a-axis parallel to the rungs of the two-leg ladders. Figure 2(a) shows, for representative temperatures, the spectra of the measured reflectivity, $R_a(\omega)$. Figure 2(b) displays the corresponding spectra of the real part of the optical conductivity, $\sigma_{1a}(\omega)$, that have been derived with a Kramers–Kronig transformation, as described in the experimental section above.

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Figure 2(c) shows a magnified view of the spectral range of the narrow infrared active modes some of which exhibit a very anomalous temperature dependence. The modes, whose origin is further discussed in sections 2.3 and 3, are labeled from A to G* in the order of increasing frequency. The modes near 400 and 700 cm⁻¹ are especially marked as C* and G* since they exhibit a very anomalous temperature dependence, i.e. they are very weak above 200 K and become pronounced features only in the PG state below $T^{\,\ast}$. These anomalous modes were previously noted in [80], but their origin remained unclear. In particular, it is not known whether they are due to infrared-active phonons or rather related to some electronic excitations. The overall electronic contribution to the $\sigma_{1a}(\omega)$ spectra is almost frequency independent and featureless between 300 and 200 K. This changes below an onset temperature $T^{\,\ast} \approx 180$–200 K where a partial suppression of the conductivity develops at low frequencies ($\omega \lt$ 600 cm⁻¹). Figure 2(d) shows the conductivity spectra for the frequency range up to 3000 cm⁻¹ which reveal that the SW loss below the PG energy $E_{pg} \approx 600$ cm⁻¹ is compensated by a broad overshoot band above E_pg that extends to about 2200 cm⁻¹. This is evident from the inset of figure 2(d) which displays the integrated SW, $S(\omega) = \int_0^{\omega}\sigma_1(\omega^{^{\prime}})d\omega^{^{\prime}}$, at 200 K and 10 K and confirms that they are matched above 2200 cm⁻¹.

Figures 2(e) and (f) show the corresponding c-axis conductivity spectra of a weakly underdoped NdBa₂Cu₃O_6.9 crystal that have been adopted from [55]. They highlight that the electronic conductivity spectra have a surprisingly similar shape and exhibit the same kind of PG phenomenon with a suppression of the conductivity below $E_{pg} \approx 800$ cm⁻¹ and a transfer of the missing low-frequency SW to a broad overshoot band above E_pg that develops already far above T_c . The c-axis spectra of NdBa₂Cu₃O_6.9 also contain infrared-active phonon modes some of which exhibit an anomalous temperature dependence and a transverse plasma mode (tPM) around 450 cm⁻¹. The latter becomes a pronounced feature only below T_c and seems to have a resemblance to the C* feature at 400 cm⁻¹ in the a-axis response of Sr₂Ca₁₂Cu₂₄O₄₁, which is also strongest at low temperature and becomes very weak toward high temperatures.

Figure 2(g) details the temperature dependence of the PG-like, partial suppression of the electronic a-axis conductivity of Sr₂Ca₁₂Cu₂₄O₄₁ in terms of the integrated low-energy SW, $SW_0^{360} = \int_0^{360\textrm{cm}^{-1}}\sigma_{1a}(\omega)d\omega$. Whereas $SW_0^{360}$ increases slightly between 300 and 200 K, it exhibits a pronounced decrease below an onset temperature $T^{\,\ast} \approx 180$–200 K. Figures 2(h) and (i) show the temperature dependencies of the oscillator strengths of the features C* and G*, respectively, that were obtained by fitting the spectral structures with Lorentzian oscillators as detailed in the appendix A. Both exhibit an anomalous increase that also starts around $T^{\,\ast}$. This coincidence suggests that the modes C* and G*, if of phononic origin, are activated by the coupling to some electronic excitations that appear (or are strongly enhanced) in the PG state below $T^{\,\ast}$.

Figure 3 shows the corresponding a-axis infrared spectra of a Sr₆Ca₈Cu₂₄O₄₁ (x = 8) crystal for which the hole doping of the two-leg ladders is slightly reduced as compared with Sr₂Ca₁₂Cu₂₄O₄₁ (x = 12) [39]. The spectra in figure 3(b) reveal the same kind of PG effect as those of the x = 12 sample in figures 2(b) and (d). The onset frequency of the gap-like suppression, E_pg , and the growth of the broad overshoot band above the gap edge are somewhat obscured here by an overall decrease of the electronic conductivity that starts already at 300 K and is weakly frequency dependent. The latter is likely related to a redistribution of some of the holes from the ladder layers to the chain layers that occurs as the temperature is reduced, as discussed, e.g. in [39]. In order to highlight the characteristic PG features, we have normalized the conductivity spectra to the value at $\omega = 2\,500$ cm^-1, a frequency that is above the energy scale of the PG. The result shown in figure 3(c) reveals an onset of the conductivity suppression at $E_{pg} \approx 820$ cm⁻¹ and a clear overshoot band that starts above the gap edge and extends up to about 2300 cm⁻¹. Figure 3(d) displays the temperature dependence of the PG effect in terms of the low-frequency SW, $SW_0^{360} = \int_0^{360{\text{ cm}^{-1}}}\sigma_{1a}(\omega)d\omega$. It reveals an onset of the conductivity suppression around $T^{\,\ast}$ ≈ 200–220 K. The temperature and energy scales of the PG, $T^{\,\ast}$ and E_pg , thus tend to increase as the hole doping of the ladders is reduced. Note that this parallels the trend of the PG in the c-axis response of underdoped YBCO for which the onset temperature and gap energy are also increasing toward the underdoped side of the hole doping phase diagram [55, 56]. Figures 3(e) and (f) display the corresponding temperature dependencies of the oscillator strengths of the features C* and G* around 400 cm⁻¹ and 700 cm⁻¹, respectively, that were obtained by fitting the spectral structures with Lorentzian oscillators as detailed in appendix A. For both features, again the oscillator strength exhibits an anomalous increase with an onset close to $T^{\,\ast}$.

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Figure 4 shows a comparison of the far-infrared active modes that appear in the a-axis conductivity spectra at 300 K for the Sr$_{14-x}$Ca_x Cu₂₄O₄₁ crystals with x = 0, 8 and 12. Similar spectra have been obtained in earlier studies [39, 79, 80, 88]. The notation of the features for the spectrum at x = 12 has been adopted from figure 2(c). The peaks A and B are strongly blue-shifted upon Ca substitution, as indicated by the dotted lines, and are therefore assigned to phonon modes that involve mainly vibrations of the Sr/Ca ions. The phonon modes with a predominant contribution of the oxygen ions are expected at considerably higher frequencies in the range from about 500–650 cm⁻¹. The lattice-dynamical calculations for the ladder plane of [89] predict two a-axis polarized oxygen modes, the leg oxygen mode at 570 cm⁻¹ and the rung oxygen mode at 640 cm⁻¹, respectively. For the corresponding displacement patterns, see figure 5. A corresponding chain oxygen mode, that is not included in the calculations of [89], is expected at an intermediate frequency. Accordingly, we preliminarily assign the structures D, E, and F to the leg-oxygen, chain-oxygen, and rung-oxygen modes, respectively. The splitting between the peaks D and E is the largest at x = 12. For x = 0 the coupling between the oscillations of the leg oxygens and those of the chain oxygens seems to give rise to only one pronounced peak around 555 cm⁻¹. Especially for the peaks D and F, it can be seen that their oscillator strengths, positions and asymmetries exhibit a strong variation as a function of the Ca content. In particular, the rung-oxygen mode F is a sharp and pronounced feature at x = 0 whereas it becomes broadened and much weaker at x = 12. Likewise, the leg-oxygen mode D for x = 12 is strongly red-shifted and highly asymmetric as compared to x = 0. In section 3 we will show that these trends can be understood in terms of a coupling of these phonons to electronic excitations within the ladders.

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This leaves us with the assignment of the modes C* and G* that are very weak at 300 K but increase anomalously in strength below $T^{\,\ast}$. The lattice origin of these anomalous modes has been confirmed by a study of the oxygen isotope-effect on the a-axis infrared conductivity spectra for which the result is displayed in figure 6(a). Here the modes C* and G* can be seen to exhibit an oxygen-isotope shift on the order of 10 cm⁻¹ (indicated by the dashed lines) that is of a similar magnitude as that of the regular, ladder-related phonon modes D and F. The circumstance that the magnitudes of these isotope shifts of about 10 cm⁻¹ or 1.5%–2% are considerably lower than the maximal theoretical value (for which we neglect the contributions of the cations that are expected to be small for such high-frequency modes) of $\Delta\omega = \omega_{16}(1-\sqrt{\frac{16}{18})}$ can be understood in terms of an only partial (about 50%) substitution of ¹⁸O for ¹⁶O. The somewhat smaller isotope shift of the chain related phonon mode E of about 6 cm⁻¹ may indicate that the ¹⁶O to ¹⁸O exchange rate in the chain layers is even somewhat lower than in the ladder layers. Irrespective of these partial exchange rates of the oxygen isotopes, the similar isotope shifts of the anomalous C* and G* modes on one hand and those of the regular D and F phonons on the other hand, establish that the former modes are also predominantly of phononic origin. While they cannot be assigned to regular infrared active phonon modes from the Brillouin-zone center, we outline in section 3 that they correspond to off-center phonon modes that are activated by a strong coupling to electronic excitations of a charge ordered state that develops below $T^{\,\ast}$ within the two-leg ladders. Note that the very weak structures above $T^{\,\ast}$ are likely activated by the disorder due to the randomly distributed Ca and Sr ions.

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Figure 6(b) shows the result of a corresponding oxygen isotope effect study for the c-axis conductivity of strongly underdoped YBa₂Cu₃O_6.5. Here the highest IR-active phonon mode exhibits an isotope shift of about 30 cm⁻¹ that is close to the theoretical value for a 100% exchange rate and a pure oxygen character of the phonon mode. Despite the large oxygen isotope shifts of the high energy infrared-active phonon modes of about 30 cm⁻¹, it is evident that the broader mode around 410 cm⁻¹ shows no sign of a corresponding isotope shift. This lack of an oxygen isotope effect highlights that this mode has a predominantly electronic origin, in agreement with its assignment to an electronic transverse plasma mode (tPM) [61–63]. Its similar location to the C* mode of SCCO at x = 12, which is a feature clearly due to a phonon, thus appears to be accidental.

For the interpretation of the peaks C* and G*, we adopt the following rough picture of the charge dynamics of the ladder planes that is based on model calculations and supported by experimental data (for reviews, see [28, 39, 90]).

We assume that at high temperatures and low hole concentrations the holes behave approximately independently. The a-axis charge dynamics involves effective intra-ladder and inter-ladder hoppings, see figure 3 of [91] for a schematic representation. The inter-ladder hopping matrix elements are considerably smaller than the main intra-ladder one, $|t_{\perp,1}|$ in the notation of Müller et al [91]. In a first approximation, the inter-ladder matrix elements thus can be neglected. The matrix element $t_{\perp,1}$ gives rise to two bands—bonding and antibonding—with the energy difference (at the simplest tight binding level) of 2 $\left|t_{\perp,1}\right|$ and the corresponding electronic transitions from the bonding to the antibonding band. However, the correlation effects are known to strongly reduce the magnitude of the interband splitting and thus the energy of the interband transitions. For example, recent density matrix renormalization group (DMRG) calculations for the problem of one doped hole in a two-leg t-J ladder performed by White et al [92] yield values of the interband splitting which are an order of magnitude lower than 2 $|t_{\perp,1}|$. We therefore assume that for concentrations of 10%–25% doped holes, i.e. in the range that is relevant to the Ca doped ladders, the bonding–antibonding interband transition occurs at a rather low energy and is overdamped. Accordingly, at high temperatures the a-axis conductivity is codetermined by this overdamped transition and by an incoherent or weakly coherent inter-ladder hopping.

At low temperatures, the holes are expected to form vertical pairs [27, 93–96], i.e. pairs with one hole in the bottom leg of a ladder and the other hole nearby in the upper leg. For a schematic representation of the hole pair of a t-J ladder, see figure B.1. The spins will be ordered accordingly, in the limit of weak inter-rung coupling they form rung singlets. The PG in the a-axis conductivity, i.e. the decrease of $\sigma_{a}(\omega)$ at low frequencies with decreasing temperature, that results in a (partial) gap feature, is most likely determined by the hole pairing and/or by the spin correlations.

Note that a PG of comparable size occurs also in the low-temperature a-axis conductivity spectra computed using the t-J model. Figure 7 shows our calculated spectra of the a-axis (along-the-rung) conductivity of a single ladder and of the a-axis inter-ladder conductivity. The latter has been computed assuming a weak inter-ladder coupling. Both of them can be seen to display a PG of about 0.2 eV. The macroscopic a-axis conductivity can be expressed as a weighted average of the two components with model- and frequency-dependent weights. For details of the calculations and the relation between the PG and the quasiparticle spectral function, see appendix B.3.

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The experimental finding, that the infrared active modes C* and G* originate from off-zone-center lattice vibrations strongly suggests that the hole pairs develop an ordered pattern. This conclusion agrees with earlier resonant soft x-ray scattering [43, 44] and Raman scattering [97] experiments which indicate that the hole pairs form a crystal with a spacing (along the legs) of five or three lattice parameters. To some extent, the pair ordering can also be identified in hole correlation functions, see appendix B.2.

The origin of the peaks C* and G* can now be qualitatively understood using the schematic representation of the hole pattern shown in figure 8. The thick blue lines indicate the locations of the hole pairs. The electromagnetic radiation induces (hole) charge oscillations between the legs. The resulting dynamic charge redistribution is indicated by the ± signs in the figure. It modifies the electric field driving the lattice vibrations, such that the leg-oxygen phonon mode with the displacement pattern represented by the red arrows is excited. The same coupling mechanism applies to the analogous rung-oxygen phonon mode. According to the calculations by Nunner et al [89], the frequency of the relevant leg (rung) oxygen mode is considerably lower (slightly higher) than the corresponding zone-center frequency. We thus propose that the off-center leg and rung modes correspond to the peaks C* and G*, respectively, in figures 2(c), 3(b), 4 and 6(a).

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In the high-temperature state above $T^{\,\ast}$ the hole density of the ladder planes can be expected to be uniform. As discussed in section 3.1, the matrix element of the intra-ladder hopping (along-the-rungs) is considerably larger than the inter-ladder hopping matrix elements. For this reason, electromagnetic radiation polarized along the a-axis brings about a charging of the legs. In order to describe the related physics in simple terms, we approximate the legs by homogeneously charged cylinders of radius c, c = 0.8 Å, for a schematic representation of the geometry of the model, see figure 9(a). Note that the results are not significantly affected by moderate changes of c. The field-induced (linear) charge densities are denoted by $\pm \rho$. They are related to the intra-ladder and inter-ladder current densities (defined as currents per unit area perpendicular to the rungs), $j_{\mathrm r}$ and $j_{\mathrm{int}}$, respectively, via the continuity equation:

$$\begin{equation} -\mathrm{i}\omega\rho/\ell =j_{\mathrm{int}}-j_{\mathrm r}, \end{equation} \tag{ 1 }$$

here $\ell$ is the distance between neighboring ladder planes, see figure 9(a).

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The current densities are assumed to be proportional to the a-axis-components $E_{\mathrm r}$ and $E_{\mathrm{int}}$ of the intra-ladder and inter-ladder local electric fields ${\mathbf{E}}_{\mathrm{r}}$ and ${\mathbf{E}}_{\mathrm{int}}$:

$$\begin{equation} j_{\mathrm r}=\sigma_{\mathrm{r}}E_{\mathrm{r}},\,\, j_{\mathrm{int}}=\sigma_{\mathrm{int}}E_{\mathrm{int}}.\end{equation} \tag{ 2 }$$

The quantities $\sigma_{\mathrm{r}}$ and $\sigma_{\mathrm{int}}$ will be denoted as local conductivities. The local fields $E_{\mathrm{r}}$ and $E_{\mathrm{int}}$ are defined as the averages of the total electric field over the lines connecting the centers of the cylinders, that are represented by the thick arrows in figure 9(a). The densities $\pm \rho$ influence both the average electric field ${\mathbf{E}}_{\mathrm{a}}$ (a-axis-component labeled $E_{\mathrm{a}}$) and the fields ${\mathbf{E}}_{\mathrm r}$ and $\mathbf{E}_{\mathrm{int}}$,

$$\begin{equation} E_{\mathrm{a}}\,/\,E_{\mathrm{int}}\,/\,E_{\mathrm{r}}=E_{\mathrm{ext}}+ \alpha\,/\,\beta^{\mathrm{h}}\,/\,\gamma^{\mathrm{h}}\, \times \frac{\rho/\ell}{2\epsilon_0\epsilon_\infty}, \end{equation} \tag{ 3 }$$

here $E_{\mathrm{ext}}$ is the a-axis component of the part of the average internal field that is not due to the effects of $\pm \rho$, and α, $\beta^{\mathrm{h}}$, $\gamma^{\mathrm{h}}$ are model parameters. Their numerical values have been obtained using electrostatic calculations described in appendix C and are given in table 1. Further, $\epsilon_{\infty}$ describes the screening by high-energy excitations that are not included explicitly in the present effective model. Finally, the macroscopic conductivity σ_a is given by:

$$\begin{equation} \sigma_{\mathrm{a}}E_{\mathrm{a}}=(a_1 j_{\mathrm{r}}+a_2j_{\mathrm{int}})/(a_1+a_2), \end{equation} \tag{ 4 }$$

where $a_1,\,a_2$ are lattice parameters, see figure 9(a). The expression on the right hand side represents the volume average of the a-axis current density. This closed set of equations [98, 99] allows us to express $\sigma_{\mathrm{a}}(\omega)$ in terms of the local conductivities $\sigma_{\mathrm{r}}(\omega)$ and $\sigma_{\mathrm{int}}(\omega)$.

Table 1. Numerical values of the parameters entering the model equations of section 3.2 ('homogeneous model') and of section 3.3 ('dipole model'). They have been obtained by electrostatic calculations described in appendices C and D. The values of α, $\delta_\mathrm{rung}$, and $\delta_\mathrm{leg}$ are model independent. For fitting the experimental data, different values of some of the parameters were used. They are given in the brackets. For the present setup (i.e. $\sigma_{int}=0$), the values of $\beta^\mathrm{h},\,\beta^\mathrm{d},\,\nu^\mathrm{h},\,\nu^\pm$ have no impact on the results, those of the dipole model are therefore not included. The values of the lattice parameters used in our calculations are $a_1=c_1=3.931$ Å, $a_2=1.804$ Å, $\ell=6.685$ Å.

	α = 0.685	$\delta_{\mathrm{rung}} =-1.00$	$\delta_{\mathrm{leg}}=-1.00$
homog. model	$\beta^{\mathrm{h}} =-2.02$	$\mu_{\mathrm{rung}}^{\mathrm{h}} =-2.84$	$\mu_{\mathrm{leg}}^{\mathrm{h}}=0.125(0.438)$
	$\gamma^{\mathrm{h}} =1.93(1.5)$	$\nu_{\mathrm{rung}}^{\mathrm{h}}=3.00$	$\nu_{\mathrm{leg}}^{\mathrm{h}} =-3.45$
		$\xi_{\mathrm{rung}}^{\mathrm{h}} =1.95$	$\xi_{\mathrm{leg}}^{\mathrm{h}}=-0.086(-0.3)$
dipole model	$\gamma^{\mathrm{d}}=-0.031(1.5)$	$\mu^+_{\mathrm{rung}}=-2.52$	$\mu^+_{\mathrm{leg}}=0.459$
	[$\gamma^\mathrm{d}=1$ used in figure 12]	$\xi^+_{\mathrm{rung}}=1.72$	$\xi^+_{\mathrm{leg}}=-0.315$
		$\mu^-_{\mathrm{rung}}=-2.94$	$\mu^-_{\mathrm{leg}}=-1.58(-2.19)$
		$\xi^-_{\mathrm{rung}}=2.02$	$\xi^-_{\mathrm{leg}}=1.09(1.5)$

Next, we present an extension of the above equations that includes the rung-oxygen mode and the leg-oxygen mode. For the sake of simplicity, the chains of rung oxygens belonging to one ladder and the chains of leg oxygens belonging to one leg are approximated by solid cylinders of radius c that are homogeneously charged, see figure 9(c). The advantage of the above approximations is that all contributions to the internal electric field, that are induced by the incoming a-axis polarized electromagnetic radiation, are also a-axis polarized and depend on the a-axis coordinate only. Whereas the (a-axis components of the) fields driving the oxygen ions, i.e. the cylinders in figure 9(c), $E_{\mathrm{ph-rung}}$ and $E_{\mathrm{ph-leg}}$, are influenced by $\pm \rho$, the oxygen displacements have impact on the fields $E_{\mathrm{a}}$, $E_{\mathrm{r}}$, and $E_{\mathrm{int}}$. The dynamics of the combined system of holes and lattice vibrations is described by the following set of equations:

$$\begin{equation} E_{\mathrm{a}} = E_{\mathrm{ext}} + \alpha\frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} + \sum\limits_{i\in\{\mathrm{rung,leg}\}}\frac{\delta_i}{2\epsilon_0\epsilon_\infty} P_i, \end{equation} \tag{ 5 }$$

$$\begin{equation} E_{\mathrm{r}} = E_{\mathrm{ext}} + \gamma^{\mathrm{h}}\frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} + \sum\limits_{i\in\{\mathrm{rung,leg}\}} \frac{\mu_i^{\mathrm{h}}}{2\epsilon_0\epsilon_\infty} P_i, \end{equation} \tag{ 6 }$$

$$\begin{equation} E_{\mathrm{int}} = E_{\mathrm{ext}} + \beta^{\mathrm{h}}\frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} +\sum\limits_{i\in\{\mathrm{rung,leg}\}} \frac{\nu_i^{\mathrm{h}}}{2\epsilon_0\epsilon_\infty}P_i, \end{equation} \tag{ 7 }$$

$$\begin{equation} E_{\mathrm{ph-rung}} = E_{\mathrm{ext}} + \xi_{\mathrm{rung}}^{\mathrm{h}} \frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} + \frac{\lambda}{2\epsilon_0\epsilon_\infty}P_{\mathrm{leg}}, \end{equation} \tag{ 8 }$$

$$\begin{equation} E_{\mathrm{ph-leg}} = E_{\mathrm{ext}} + \xi_{\mathrm{leg}}^{\mathrm{h}} \frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} + \frac{\lambda}{2\epsilon_0\epsilon_\infty}P_{\mathrm{rung}}, \end{equation} \tag{ 9 }$$

$$\begin{equation} \sigma_{\mathrm{a}} E_{\mathrm{a}} = \frac{a_1}{a_1+a_2}j_{\mathrm{r}}+\frac{a_2}{a_1+a_2}j_{\mathrm{int}} -\mathrm{i}\omega \sum\limits_{i\in\{\mathrm{rung,leg}\}}P_i. \end{equation} \tag{ 10 }$$

Here $P_{\mathrm{rung}}$ and $P_{\mathrm{leg}}$ are the macroscopic polarizations due to the rung and the leg mode, respectively, that are connected to the fields $E_{\mathrm{ph-rung}}$ and $E_{\mathrm{ph-leg}}$ as follows: $P_{i} = r_{i}n_{i}Q_{i} = \epsilon_{0}\chi_{i}(\omega)E_{\mathrm{ph}-i}$, $i\in\{\mathrm{rung},\mathrm{leg}\}$. The symbols r_i , n_i , Q_i , χ_i denote the a-axis displacements of the cylinders representing the oxygen chains (see figure 9(c)), densities of the (rung/leg) oxygens, effective charges of the (rung/leg) oxygens, and the phonon polarizabilities, respectively. The polarizabilities have been expressed in terms of Lorentzian functions: $\chi_{i}(\omega) = S_{i}\omega_{i}^{2}/(\omega_{i}^{2}-\omega^{2}-\mathrm{i}\omega\gamma_{i})$. The values of Q_i have been set to $Q = -2|e|$, where e is the elementary charge, and the values of oxygen plasma frequencies $S_{i}\omega_{i}^{2}$ have been obtained as $n_{i}Q^{2}/(\epsilon_{0}M_{\mathrm{O}})$, where $M_{\mathrm{O}}$ is the oxygen mass. Appendix C details our electrostatic calculations of the parameters $\alpha,\beta^\mathrm{h},\gamma^\mathrm{h},\delta_i,\,\mu_i^\mathrm{h},\,\nu_i^\mathrm{h},\,\xi_i^\mathrm{h}$. Their numerical values are given in table 1. In our calculations we do not consider direct coupling between the leg-oxygen and the rung-oxygen vibrations, i.e. we set λ = 0.

Next, we demonstrate that the model allows us to reproduce and interpret the doping-induced suppression of the rung-oxygen phonon peak (feature F in figures 2(c), 4 and 6(a)) and the asymmetry and broadening of the leg-oxygen phonon peak (feature D in figures 2(c), 4 and 6(a)). Figure 10 shows the spectra of the real part of the macroscopic conductivity $\sigma_{\mathrm{a}}$, calculated using the model equations with $\sigma_{\mathrm{int}}(\omega) = 0$, $\sigma_{\mathrm{r}}$ given by a frequency independent constant (details of the ω-dependence were found to be of minor importance) and $\varepsilon_\infty$=3.5. Panels (a)–(c) present results of the calculations which take into account only the rung-oxygen mode ($S_{\mathrm{rung}} = 0.21$, $\omega_{\mathrm{rung}} = 625$ cm⁻¹, $\gamma_{\mathrm{rung}} = 10$ cm⁻¹), only the leg-oxygen mode ($S_{\mathrm{leg}} = 0.54$, $\omega_{\mathrm{leg}} = 550$ cm⁻¹, $\gamma_{\mathrm{leg}} = 10$ cm⁻¹), or both phonon modes, respectively.

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Panel (a) shows that the SW of the rung-oxygen resonance decreases strongly with increasing $\sigma_{\mathrm{r}}$. This trend agrees with the doping dependence of the phonon feature F in the experimental spectra of figure 4 which becomes much weaker as the Ca content (and thus the conductivity) increases. This behavior can be qualitatively interpreted using figure 9(c). Here the (hole) charge oscillations between the legs ($\pm \rho$ in the figure) give rise to a pronounced suppression of the phonon driving field $E_{\mathrm{ph-rung}}$ [determined by the second term on the right hand side of equation (8)] that can be expected to decrease the SW of the phonon-related structure. This interpretation is supported by our analysis of the spectra of $\sigma_{\mathrm{a}}$ in Figure 11(a) that shows, for the case of $\sigma_{\mathrm{r}} = 75\, \Omega^{-1}$cm⁻¹, the spectra of the real part of $\sigma_{\mathrm{a}}$ (solid blue line), the corresponding hole- and phonon-contributions (solid orange and green lines, respectively), and, as a reference, the corresponding spectrum of $\mathrm{Re}\,\sigma_{\mathrm{a}}$ for $\sigma_{\mathrm{r}} = 0$ (dotted red line). Figure 11(b) displays the real and imaginary components of the ratio $E_{\mathrm{ph-rung}}/E_{\mathrm{a}}$. It confirms that the magnitude of the local field is suppressed and that its maximum is shifted below the frequency of the resonance in the polarizability $\chi_{\mathrm{rung}}$.

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For the leg phonon peak (D) the following changes are predicted to take place with increasing $\sigma_{\mathrm{r}}$ (see figure 10(b)): A shift toward lower frequencies, a SW increase, and an increased broadening and asymmetry. The redshift as well as the increase of the broadening and the asymmetry are consistent with the observed doping dependence reported in figure 4. The origin of these changes can be understood in terms of the change of the phonon-driving local field that is documented in figure 11(d).

As discussed in section 3.1, the large SWs of the phonon peaks C* and G* in the low-temperature conductivity spectra indicate that the ground state exhibits a charge ordering. Based on results of previous theoretical [100, 101] and experimental studies [41, 43, 44, 97], we conjecture that at low temperatures holes are forming vertical pairs that develop a pattern as shown in figure 8. The lattice spacing of these hole pairs may differ, they may also fluctuate both in space and in time (at time scales longer than those of the phonons). For concreteness and for the sake of simplicity, we adopt here the order of figure 8 with a period of five lattice spacings along the legs. Accordingly, we have modified the model presented in section 3.2 for the high-temperature phase with a homogeneous charge distribution, as described in the following.

• (1)  
  The field-induced charge densities reside on spheres of radius c, see figure 9(b). The spheres are assumed to be homogeneously charged. The corresponding (linear, averaged over a leg) charge densities are denoted by $\pm\rho$, as in the homogeneous case. The intra-ladder current density is assumed to be concentrated on the rungs surrounded by the charged spheres, and the driving field $E_{\mathrm{r}}$ is obtained as the average of the total electric field over the line connecting the sphere centers, that is represented by the black arrow in figure 9(b). The contribution of the two spheres to $E_\mathrm{r}$ includes a self-interaction component to be discussed below.
• (2)  
  Leg-oxygen- and rung-oxygen-phonons with wave vector components $q_{c} = \pm 2\pi/(5c_1)$, $q_{a} = \pi/(a_1+a_2)$ are included. The displacement pattern of the leg-oxygen phonon under consideration is shown in figure 8. The displacement of the oxygens at the n_c -th rung, in the n_a -th ladder is given by $u(n_{c},n_{a},t) = u(t)\cos [q_{c}n_{c}c_1 + q_{a}n_{a}$ $(a_1+a_2)]$. The pattern of the rung-oxygen mode is analogous. Modulation along the b direction (perpendicular to the ladder planes) does not affect the results significantly. The modes are excited due to the charge modulation by the relevant Fourier components $E_{\mathrm{ph}-i}^{-}$, $i\in\{\mathrm{rung},\mathrm{leg}\}$, of the total electric field. The modes do not contribute to the macroscopic polarization. In order to simplify the notation, however, we define the corresponding auxiliary polarizations as $P_{i}^{-} = u_{i}n_{i}Q_{i} = \epsilon_{0}\chi_{i}^{-}(\omega)E_{\mathrm{ph}-i}^{-}$.
• (3)  
  In order to clarify the notation, the quantities related to the zone-center phonons, originally P_i , $E_{\mathrm{ph}-i}$, χ_i , etc $i\in\{\mathrm{rung},\mathrm{leg}\}$, will be denoted with the superscript +, i.e. $P_{i}^{+}$, $E_{\mathrm{ph}-i}^{+}$, $\chi_{i}^{+}$, etc. The quantities related to the off-center modes will be denoted with the superscript −.
• (4)  
  The values of the model parameters γ^d , $\xi^\pm$, $\mu^\pm$, etc. have been obtained using electrostatic calculations involving the present geometry (see appendix D for details) and are given in table 1.

This system is described by the following set of equations for the rung direction components of the vectors:

$$\begin{equation} E_{\mathrm{a}} = E_{\mathrm{ext}} + \alpha\frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} + \sum\limits_{i}\frac{\delta_i}{2\epsilon_0\epsilon_\infty} P_i^+, \end{equation} \tag{ 11 }$$

$$\begin{equation} E_{\mathrm{r}} = E_{\mathrm{ext}} + \gamma^{\mathrm{d}}\frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} + \sum\limits_{i,j}\frac{\mu_i^{\,j}}{2\epsilon_0\epsilon_\infty} P_i^{\,j}, \end{equation} \tag{ 12 }$$

$$\begin{equation} E_{\mathrm{int}} = E_{\mathrm{ext}} + \beta^{\mathrm{d}}\frac{\rho/\ell}{2\epsilon_0\epsilon_\infty} + \sum\limits_{i,j}\frac{\nu_i^{\,j}}{2\epsilon_0\epsilon_\infty}P_i^{\,j}, \end{equation} \tag{ 13 }$$

$$\begin{equation} E_{\mathrm{ph}-i}^+ = E_{\mathrm{ext}} + \xi_{i}^+ \frac{\rho/\ell}{2\epsilon_0\epsilon_\infty}, \end{equation} \tag{ 14 }$$

$$\begin{equation} E_{\mathrm{ph}-i}^- = \xi_{i}^- \frac{\rho/\ell}{2\epsilon_0\epsilon_\infty}, \end{equation} \tag{ 15 }$$

$$\begin{equation} \sigma_{\mathrm{a}} E_{\mathrm{a}} = \frac{a_1}{a_1+a_2}j_{\mathrm{r}}+\frac{a_2}{a_1+a_2}j_{\mathrm{int}} -\mathrm{i}\omega \sum\limits_{i} P_i^+, \end{equation} \tag{ 16 }$$

where i runs over {rung, leg} and j over $\{+,-\}$; the quantities $\rho,\,j_r,\,j_{int}$ stand for averages over the c-axis. The continuity equation (1) is formally unchanged. In this model, the intra-ladder conductivity will be denoted by a superscript $\mathrm{d}$ ('dipole'), $\sigma_\mathrm{r}^\mathrm{d}$, to avoid confusion.

The value of the parameter $\gamma^{\mathrm{d}}$ of equation (12) is very high and almost entirely due to the contribution of the two spheres mentioned in (1). This contribution to the field, however, should not be included (at least not fully) because it is due (at least partially) to an unphysical self-interaction term. If we exclude this contribution, we obtain the very small and negative value of $\gamma^{\mathrm{d}}$ presented in table 1. The parameter $\gamma^{\mathrm{d}}$ is sensitive to many details that are, within the present model, not under control (extent of the pairs, degree of modulation, etc). We have, therefore, adjusted the value of this particular parameter considering the experimental data. A reasonable qualitative agreement with the data is obtained for $\gamma^{\mathrm{d}} = 1$ (see appendix D for details) and this is the value which has been used in the following.

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Figure 12(a) shows the spectra of the real part of the macroscopic conductivity $\sigma_{\mathrm{a}}$, calculated using the model equations with $\sigma_{\mathrm{int}}(\omega) = 0$ and $\sigma_{\mathrm{r}}^\mathrm{d}(\omega)$ given by $t\sigma_{0}^{\mathrm{d}}(\omega)$. Here t is a real constant, and $\sigma_{0}^{\mathrm{d}}(\omega)$ is a broad Lorentzian with a resonance in the mid-infrared, for which the real part is shown as the brown line in figure 12(a). The following values of the parameters of $\chi_{\mathrm{rung}}^{-}$ and $\chi_{\mathrm{leg}}^{-}$ have been used: $S^-_{\mathrm{rung}} = 0.16$ and $S^-_{\mathrm{leg}} = 0.65$, both consistent with the oxygen charge of $-2|e|$, $\omega^-_{\mathrm{rung}} = 710\,\mathrm{cm}^{-1}$, $\omega^-_{\mathrm{leg}} = 500\,\mathrm{cm}^{-1}$ and $\gamma^-_{\mathrm{rung}} = \gamma^-_{\mathrm{leg}} = 10\,\mathrm{cm}^{-1}$. It can be seen that in the presence of a nonzero spatially modulated component $\sigma_{\mathrm{r}}^\mathrm{d}(\omega)$ of the intra-ladder conductivity, the spectra display additional peaks due to the off-zone-center oxygen vibrations. Moreover, it is evident that the SWs of the peaks increase with increasing magnitude of $\sigma_{\mathrm{r}}^\mathrm{d}(\omega)$, reaching values comparable to those of the experimental data for $\sigma_{\mathrm{r}}^\mathrm{d}(\omega) = \sigma_{0}^{d}(\omega)$.

In order to reproduce (at a qualitative level) the temperature dependence of the spectra, we combine the two models. The total intra-ladder conductivity is then given by $\sigma_{\mathrm{r}}(\omega) = \sigma_{\mathrm{r}}^{\mathrm{h}}(\omega)+\sigma_{\mathrm{r}}^{\mathrm{d}}(\omega)$, with homogeneous and modulated components in the form $\sigma_{\mathrm{r}}^{\mathrm{h}}(\omega) = t\sigma_{0}^{\mathrm{h}}(\omega)$ and $\sigma_{\mathrm{r}}^{\mathrm{d}}(\omega) = (1-t)\sigma_{0}^{\mathrm{d}}(\omega)$, respectively. The temperature is represented by a real parameter t, equal to 1 and 0 for the high-temperature normal state and the low-temperature ordered state, respectively. The equations of the combined model are presented in appendix E. Figure 12(b) shows the results of the calculations with $\sigma_{0}^{\mathrm{h}}(\omega) = 100\,\Omega^{-1}{\mathrm{cm}}^{-1}$ and $\sigma_{0}^{\mathrm{d}}(\omega)$ of figure 12(a).

The behavior of the off-zone-center modes can be explained as follows: Consider first the case of a purely imaginary local conductivity $\sigma_{\mathrm{r}}^{\mathrm{d}} = -\mathrm{i}\omega\epsilon_0\epsilon^*$ and a single phonon mode. The accumulated charge density ρ can be shown, by combining equations (1), (12) and (15), to follow the Lorentzian profile of the phonon with the resonance frequency shifted to $\sqrt{\omega_i^2+\frac{\mu_i^-\xi_i^- S_i\omega_{i}^{2}\epsilon^*}{2\epsilon_\infty(2\epsilon_\infty+\gamma^{\mathrm{d}}\epsilon^*)}}$ and the SW rescaled by a factor $\propto\frac{\epsilon^*}{(2\epsilon_\infty+\gamma^{\mathrm{d}}\epsilon^*)}$. In the calculation of the conductivity, ρ acts (up to a geometric factor) exactly as a phonon polarization would, i.e. $\sigma_{\mathrm{a}}E_{\mathrm{a}} = \frac{a_1}{a_1+a_2}j_{\mathrm{r}}\propto -\mathrm{i}\omega\rho$. For a nonzero real part of $\sigma_{\mathrm{r}}^{\mathrm{d}}(\omega)$, the Lorentzian profile of ρ gets broadened, and deviations from the form $\sigma_{\mathrm{r}}^{\mathrm{d}} = const. -\mathrm{i}\omega\epsilon_0\epsilon^*$ cause an asymmetry. Adding a second phonon mode affects the original response similarly to a change in $\epsilon^*$, as long as the frequencies of the phonon peaks are sufficiently far from each other.

The results presented in sections 3.2 and 3.3 demonstrate that (a) The model of the homogeneous charge distribution allows one to reproduce and interpret the observed doping-induced suppression of the rung-oxygen mode as well as the broadening and asymmetry of the leg-oxygen mode; (b) The model of the charge ordered case clarifies the origin of the pronounced peaks C* and G* in the low-temperature conductivity data and provides values of SW of these peaks that are comparable to those of the data. Here we show that it is even possible to achieve reasonable fits of the data by tuning the input values of the conductivities, the phonon polarizabilities, and some of the model parameters, including the chain oxygen mode and an additional component $\sigma_\mathrm{int}^\ast$ of the electronic conductivity, that describes an incoherent inter-ladder transport. So far we have considered two local current densities, the intra-ladder one $j_\mathrm{r}$ and the inter-ladder one $j_\mathrm{int}$, and the corresponding local conductivities, $\sigma_\mathrm{r}$ and $\sigma_\mathrm{int}$, see equation (2). The former current density is determined by intra-ladder hopping terms of the single particle Hamiltonian (matrix elements $t_{\perp1}$, $t_{\perp4}$, and $t_{\perp6}$ of figure 3 of [91]), and the latter by hopping terms between the nearby legs of neighboring ladders ($t_{\perp2}$, $t_{\perp5}$ of figure 3 of [91]). The interladder transport, however, is complicated, involving significant longer-range hopping terms, e.g. between the bottom leg of a bottom ladder and the bottom leg of the upper ladder ($t_{\perp3}$, $t_{\perp7}$ of figure 3 of [91]), and its description in terms of $\sigma_\mathrm{int}$ is thus not sufficient. In order to improve the present phenomenological approach, we consider a component of the electronic current density $j^\ast$ that is driven by the average field $E_\mathrm{a}$, rather than by the local fields, and denote the corresponding conductivity as $\sigma_\mathrm{int}^\ast$: $j^\ast = \sigma_\mathrm{int}^\ast E_\mathrm{a}$. The current density $j^\ast$ does not interfere with $\pm \rho$ and the phonons, it is simply added to the right-hand side of equations (10) and (16).

Figure 13(a) compares the measured 300 K spectrum with the fit obtained using the model of section 3.2 and figure 13(b) the measured 10 K spectrum with the fit obtained using the model of section 3.3. The dashed green and the dash-dotted violet lines represent the spectra of the real parts of $\sigma_\mathrm{r}$ and $\sigma_\mathrm{int}^*$, respectively. Since the fitting with variable $\sigma_{\mathrm{r}}$, $\sigma_{\mathrm{int}}$, and $\sigma_\mathrm{int}^*$ yields highly correlated results, we have set $\sigma_{\mathrm{int}} = 0$ for simplicity. The chain mode is included in such a way that it is driven by the average field (not including the depolarization field due to the mode itself), given by equation (11). This corresponds to $\xi_{\mathrm{chain}}^{\mathrm{h}} = \xi_{\mathrm{chain}}^+ = \xi_{\mathrm{chain}}^- = \alpha$ and $\lambda_{\mathrm{chain-leg}} = \lambda_{\mathrm{chain-rung}} = \delta$ (the meaning of the parameter $\lambda_{\mathrm{chain-leg}}$ follows from the structure of equations (8) and (9)). The off-zone-center chain mode is not considered. The effective charge of the chain oxygen is set to $-1.6|e|$ (those of the leg- and rung-oxygens are fixed at $-2.0|e|$ as above). The obtained values of the input phonon frequencies are 655 (720) cm⁻¹, 562 (550) cm⁻¹, and 598 cm⁻¹ for the rung, leg, and chain zone-center (off-zone-center) phonons, respectively. Note that in the two fits (300 K and 10 K) the same values of the zone-center-phonon frequencies were used. The broadening parameters were fixed at 10 cm⁻¹ for the rung- and leg-oxygen phonons (both zone-center and off-zone-center) and at 30 cm⁻¹ for the chain-oxygen phonon. The values of the following model parameters have also been tuned: $\epsilon_{\infty}$, γ^h , $\gamma^{\mathrm{d}}$, $\xi_{\mathrm{leg}}^{\mathrm{h}}$, $\xi_{\mathrm{leg}}^{-}$.

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The superscripts $\mathrm{h}$ and $\mathrm{d}$ refer to the homogeneous and the charge modulated (dipole) cases, respectively. The corresponding values of the parameters $\mu_{i}^{j}$ were obtained using the relation $\mu_i^{\,j} = -\frac{a_1+a_2}{a_1}\xi_i^j$ resulting from Newton's third law. The values of the parameters are given in table 1, except for the high frequency dielectric constant, $\epsilon_{\infty} = 3$. It can be seen that the models are capable of reproducing the frequency shift of the leg-oxygen mode when going from the high-temperature to the low-temperature phase and the shapes of the additional peaks C* and G*, except for the dip at the high-frequency side of C*. Some discrepancies in the central part of the figure, including the dip, might be due to the simplified way of incorporating the chain mode.

Overall, we conclude that the main features of the phonon anomalies in the a-axis response of the highly Ca-doped SCCO can be understood and rather well reproduced with a relatively simple model that involves charge oscillations along the rungs of the ladders and their coupling to the phonons via modulations of the local electric fields driving the phonons. In particular, the model explains the drastic increase of SW of the features C$^\ast$ and G$^\ast$, that occurs below $T^\ast$: Below this temperature, the holes start to form pairs and develop an ordered state that gives rise to an activation of off-center phonon modes, the leg- (rung-) oxygen one corresponds to C$^\ast$ (G$^\ast$).

The comparison of the a-axis spectra (field polarized along the rungs) of SCCO with the c-axis spectra (field perpendicular to the CuO₂ planes) of underdoped YBCO has confirmed that in both cases the PG has very similar features. The similarities include the spectral shape of the gap, its energy scale (frequency range of the conductivity suppression), and the characteristic blue-shift of SW from the gapped region to a broad band above the gap edge. For the underdoped YBCO, it was previously shown that the PG energy scale increases with decreasing doping, and a linear extrapolation toward zero doping yields a value close to 2J [55, 56, 104]. Furthermore, the PG is weakened by (nonmagnetic) Zn substitution and enhanced by (magnetic) Ni substitution [105, 106]. These findings suggest that the PG is caused by short range antiferromagnetic spin correlations, most likely connected with the formation of spin singlets. The spin singlet scenario provides indeed a transparent qualitative interpretation of the PG effect: a transfer of an electron from one CuO₂ plane to another necessitates the breaking of a singlet which requires an energy proportional to J [107]. The observation that the ladders, with a different crystal structure but with similar spin excitation spectra [39, 108–111], exhibit a corresponding PG provides additional support for the spin-correlation-based explanation of the PG.

The anomalous temperature dependence of several phonon modes in the a-axis response of SCCO has been investigated in detail for samples with x = 8 and 12. As compared to the sample with x=0, the rung-oxygen mode (peak F in figures 2 and 4) is strongly suppressed and hardly visible. The leg-oxygen mode (peak D in figures 2 and 4) has an anomalous spectral shape that changes drastically below the PG temperature $T^{\,\ast}$. Even more spectacular is the temperature dependence of the modes C* and G* that are only very weak features at high temperatures but acquire a large SW concurrently with the opening of the PG below $T^{\,\ast}$. This raises the question of how these phonon anomalies are related to the ones in the c-axis conductivity of underdoped YBCO. Recall that in the latter, several phonon modes, in particular the bond-bending mode at 320 cm⁻¹, exhibit drastic changes that start at the temperature T^ons and have been associated with the onset (or rapid slowing down) of short-ranged SC pairing correlations. Simultaneously, a transverse plasma mode (tPM) develops that involves charge density fluctuations between the CuO₂ planes of the bilayer units due to the pairing correlations.

We have shown that the anomalies of the regular phonon modes D and F can be accounted for in terms of a model that is conceptually analogous to the one used for the planar cuprate YBCO. For SCCO the model involves charge oscillations between the legs of the ladders, instead of those between the closely spaced CuO₂ planes in YBCO. However, in contrast to the case of YBCO, where the additional mode (the tPM) is of electronic origin, in SCCO the additional modes C* and G* have a strong lattice character. This has been verified by our study of the oxygen isotope effect on the optical spectra of both compounds (see figure 6). The lack of an electronic tPM mode in the low-temperature a-axis spectra of the ladders can be qualitatively understood as follows. In YBCO, the singlet pairs form in individual CuO ₂ planes and the tPM mode arises from the interplane hopping of the pairs (which unlike the hopping of individual electrons does not lead to broken singlet pairs). In the ladders of SCCO, however, the pairs form along the rungs rather than the legs of the ladders such that a purely electronic excitation comparable to the tPM in YBCO is not possible.

The C* and G* modes in the a-axis spectra of SCCO are instead explained in terms of a (short-ranged) ordering of the pairs of the doped holes along the legs of the ladders which gives rise to a crystal of hole pairs of the type discussed in previous experimental and theoretical studies [43, 44, 97, 100, 101]. This kind of charge ordering activates the off-center leg-oxygen and rung-oxygen phonons. What is surprising and has required further explanation is the extraordinarily large SW of these modes, in particular, that of the C* mode. Our calculations, based on a phenomenological model of the a-axis charge dynamics of a hole crystal, have demonstrated that the modes can gain such a large amount of SW from the electronic excitations of the hole crystal. Accordingly, the C* and G* modes can be viewed as hybrids with a mixed lattice and electronic character. The success of this simple model in accounting for the pronounced phonon anomalies implies that even the highly doped ladders at $8 \leqslant x \leqslant 12$ (present data and [39, 80]) exhibit a short-range hole ordering that likely involves a crystal of hole pairs. Note that this interpretation is consistent with the absence of the anomalous C* and G* modes in weakly Ca-doped samples with $x \leqslant 5$. The latter exhibit a static and more long-ranged CDW order that gives rise to an insulator-like gap and thus an almost suppression of the charge excitations in the frequency range of the phonons [39, 79]. In return, our observations suggest that the anomalous phonon features C* and G* are sensitive indicators of the slowly fluctuating short-range order of the hole pairs in the highly doped ladders. For example, they can be used to study the evolution of this charge order under high external pressure, in particular, to find out whether it coexists with the SC order that develops above 3 GPa [45–48].

Essential information about the possible presence of SC pairing correlations in SCCO with x = 12 at ambient pressure has been obtained by analyzing the c-axis response (field polarized along the legs of the ladders) of SCCO and comparing it with the a-axis response (field polarized in the CuO₂ planes, perpendicular to the CuO chains) of underdoped YBCO. Figure 14 reveals that the IR-spectra of these two materials are amazingly similar. For both of them, the conductivity spectra exhibit a dip feature which develops below an onset temperature of about 160–180 K. The formation of this dip feature is connected with a shift of the gapped SW toward the low-frequency side, that is characteristic of an increase of coherence of the electronic response. The only qualitative difference between YBCO and SCCO is that the low-frequency conductivity maximum is centered at zero frequency (or below the low-frequency limit of the measurements) in the former and at a well-resolved finite frequency in the latter.

For underdoped YBCO, we have previously discussed the evidence that the dip feature arises from pairing correlations which set in at a temperature much higher than the bulk SC transition temperature T_c [59], similar conclusions have been reported for underdoped Hg-1201 [75]. The SCCO-YBCO analogy suggests that such pairing correlations occur equally well in SCCO below $T^{\,\ast} \sim 180$–200 K. The pair formation in SCCO seems to be very local and essentially take place within the individual two-leg ladders. The development of a macroscopic SC order appears to be hindered by the reduced dimensionality of the system, due to weak coupling between the ladders. Recall that in purely 1D systems a long-range SC order is prohibited by the Hohenberg-Mermin-Wagner theorem [112–115]. Accordingly, in SCCO a bulk SC state can only occur if the coupling between the ladders and thus the effective dimensionality is increased, e.g. by applying external pressure [45].

The above described arguments that rely on the SCCO-YBCO analogy are also consistent with results of calculations based on the t-J model. Doped t-J ladders are known to exhibit short-range pairing correlations [90, 116–119]. The c-axis conductivity spectra of these ladders display a zero-frequency peak with a SW that is larger (for common values of t and J) than that at finite frequencies [120, 121]. The peak is believed to correspond to a collective motion of the system of hole pairs or, in other words, to a collective motion of electrons with short-range pairing correlations [120, 121]. It is thus natural to attribute the low-frequency conductivity maximum in the c-axis response of SCCO to the zero-frequency peak of the t-J ladders that is shifted toward a finite frequency by effects due to the density wave that remain to be further explored.

Irrespective of the SCCO-YBCO analogy and of the insights gained from the comparison with results of the t-J model, the high SW of the low-frequency maximum in the c-axis conductivity of SCCO indicates that the pattern of holes (probably the crystal of hole pairs) is quite mobile along the c-axis, very likely as a result of short-range SC pairing correlations.

So far, we have used the SCCO-YBCO analogy to understand the physics of SCCO. However, the similarities indicate as well that the collective degrees of freedom of a charge ordered and/or (precursor) SC pairing state, that are manifestly present in the case of SCCO, provide an important contribution to the a-axis conductivity of YBCO below T^ons , and to the in-plane response of underdoped high-T_c cuprates in general. This challenges the various attempts to interpret the low-energy charge dynamics of the high-T_c cuprates solely based on Fermi-liquid-based models [53, 73, 122–125]. The SCCO-YBCO analogy is so pronounced that it even points toward a scenario of a spontaneous charge segregation in the CuO₂ planes of the high-T_c cuprates that gives rise to quasi-1D electronic and magnetic structures, such as predicted early on for stripe-like orders [19] and more recently addressed in the context of a pair-density wave order [20]. The hole pairing in the charge stripes may have the same origin as that in the ladders of SCCO [19].

Finally, we address the temperature scales of the PG, the charge ordering and the SC pairing correlations, and their mutual relationships. Notably, in SCCO with x = 12 the PG, the charge order and the precursor SC pairing correlations have the same (or very similar) onset temperatures of $T^{\,\ast} \approx 180$–200 K, that further coincide with the onset temperature of the spin gap as seen with inelastic neutron scattering and NMR [32–36]. To the contrary, in the underdoped YBCO these various orders have clearly different onset (or crossover) temperatures. This can be seen from figure 15 which gives an overview of the doping dependence of the various onset temperatures in YBCO. The values of T^pg , T^ons , and T_c are reproduced from [59] and those of T^CDW from [68, 69]. It is evident that T^pg exceeds T^ons throughout the entire underdoped regime. The CDW transition temperature, deduced from the x-ray diffraction experiments of [68, 69], on the other hand, is close to T^pg in the weakly underdoped regime but then levels off and saturates at about 160 K around p = 0.12, before it declines rapidly toward lower doping and vanishes around p = 0.07–0.08. Note that it may not be accidental that the maximum of T^ons coincides with the putative quantum critical point at which the static CDW order vanishes [68, 69]. Whereas the static CDW order has been shown to compete with macroscopic superconductivity [67, 126], collective charge fluctuations may well contribute to the local SC pairing [14, 15, 127, 128].

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For SCCO, to the contrary, the three ordering phenomena are found to set on simultaneously. Moreover, the common transition temperature of $T^{\,\ast} = 180$–200 K is close to the maximum value of T^ons in strongly underdoped YBCO. This behavior suggests that in this quasi-1D material the underlying correlations and the subsequent orders are somehow more rigidly coupled than in the planar cuprates. For example, in the ladders the unpaired holes may well be more detrimental to the spin correlations than in the planar cuprates such that the spin singlet correlations underlying the PG can only develop in parallel with the hole ordering and pairing. Clearly, more detailed theoretical and experimental work is required to understand this important difference which may provide new insights into the relationship between the various spin and charge orders and the nature of the anomalous normal state properties of the planar cuprates.

The Hamiltonian of a two-leg t-J ladder reads:

$$\begin{align} \mathcal{H}&=-t_\parallel\sum\limits_{j,r,\sigma} \left[ c_{j,r,\sigma}^\dagger c_{j,r+1,\sigma} + \mathrm{H.c.}\right] -t_\perp\sum\limits_{r,\sigma} \left[ c_{u,r,\sigma}^\dagger c_{\ell,r,\sigma} + \mathrm{H.c.}\right]\nonumber\\ &\quad+ J_\parallel \sum\limits_{j,r}\left[ \mathbf{S}_{j,r}\cdot \mathbf{S}_{j,r+1} - \frac{1}{4} n_{j,r}n_{j,r+1}\right]\nonumber\\ &\quad+ J_\perp \sum\limits_{r}\left[ \mathbf{S}_{u,r}\cdot \mathbf{S}_{\ell,r} - \frac{1}{4} n_{u,r}n_{\ell,r}\right], \end{align} \tag{ B.1 }$$

where $c_{j,r,\sigma}$ are the projected electron operators, $n_{j,r} = \sum\limits_\sigma c_{j,r,\sigma}^\dagger c_{j,r,\sigma}$, and $\mathbf{S}_{j,r}$ are the local spin operators. The indices r and $j\in\{u,\ell\}$ run over the rungs and the legs, respectively, σ denotes spin, and the indices $u,\,\ell$ denote the upper and the lower leg, respectively. Periodic boundary conditions have been used in all our calculations. If not mentioned otherwise, the same values of the input parameters as in [121] have been used: $t_\parallel = 0.45$ eV, $t_\perp = 0.36$ eV, $J_\parallel = 0.15$ eV, and $J_\perp = 0.12$ eV ('realistic values'). The ground states, as well as the dynamical properties, have been obtained using the Lanczos technique [132].

Figures B.1(a)–(d) show the hole-hole correlation function $\langle(1-n_{i}) (1-n_{j})\rangle$ for ladders with 2 and 4 holes on 10 rungs, with the position of one hole fixed. Panels (a) and (c) have been obtained using the realistic values of the input parameters, panels (b) and (d) using the same values of the hopping parameters but three times larger values of the superexchange constants. Panels (a) and (b) illustrate the formation of the hole pairs, and panel (d) suggests that the pairs form a density wave. The latter, however, cannot be identified in (c). Signatures of the pair density wave can be seen in panels (e)–(g) displaying the three hole correlation function $\langle(1-n_{i})(1-n_{j})(1-n_{k})\rangle$, with the positions of two holes fixed.

Here we derive the formula for the a-axis interladder optical conductivity based on the assumption of weakly coupled ladders. We consider the system of two independent ladders. The unit cell contains one rung per ladder and the rungs are shifted by half the leg lattice parameter (see figure 5). The ladders are connected by four hopping terms $t_{\perp2},t_{\perp3},t_{\perp5},t_{\perp7}$ introduced in [91]. We first construct the corresponding contributions to the current density operator j and then express the conductivity in terms of $j = j_2 + j_3 + j_5 +j_7$.

The contribution j₂ can be written as:

$$\begin{align} j_2 \propto \mathrm{i} t_{\perp2}\sum\limits_r\left[-c_{ur}^{1\dagger}\left(c_{\ell r}^2+c_{\ell r+1}^2\right) + c_{\ell r}^{2\dagger}\left(c_{u r}^1+c_{u r-1}^1\right)\right].\end{align} \tag{ B.2 }$$

The two ladders are labeled $1,2$; $u,\ell$ denotes the upper and the lower leg, respectively, and r is the unit cell index. The spin index has been omitted for brevity. By transforming the operators into the along-the-leg quasi-momentum representation we obtain:

$$\begin{align} j_2 \propto \mathrm{i} t_{\perp2}\sum\limits_k\left[\left(c_{0 k}^{2\dagger}c_{0 k}^1\!+\!c_{0k}^{2\dagger}c_{\pi k}^1 \!-\! c_{\pi k}^{2\dagger}c_{0 k}^1 \!-\! c_{\pi k}^{2\dagger}c_{\pi k}^1\right)\frac{1+\mathrm{e}^{\mathrm{i} kc_1}}{2}\! -\! \mathrm{H.c.}\right], \end{align} \tag{ B.3 }$$

where $c_{0k} /c_{\pi k} = \frac{1}{\sqrt{2}}(c_{uk}\pm c_{\ell k})$ correspond to even/odd states with respect to reflection about the line connecting the midpoints of the rungs, and c₁ is the lattice parameter along the leg. Similarly we obtain:

$$\begin{align} j_3 &\propto\, \mathrm{i} t_{\perp3}\sum\limits_r\left[-c_{\ell r}^{1\dagger}\left(c_{\ell r}^2+c_{\ell r+1}^2\right) + c_{\ell r}^{2\dagger}\left(c_{\ell r}^1+c_{\ell r-1}^1\right)\right]\nonumber\\ &=\,\mathrm{i} t_{\perp3}\sum\limits_k\left[\left(c_{0 k}^{2\dagger}c_{0 k}^1-c_{0k}^{2\dagger}c_{\pi k}^1 - c_{\pi k}^{2\dagger}c_{0 k}^1 + c_{\pi k}^{2\dagger}c_{\pi k}^1\right)\frac{1+\mathrm{e}^{\mathrm{i} kc_1}}{2} - \mathrm{H.c.}\right], \end{align} \tag{ B.4 }$$

$$\begin{align} j_5 &\propto\, \mathrm{i} t_{\perp5}\sum\limits_r\left[-c_{ur}^{1\dagger}\left(c_{\ell r-1}^2+c_{\ell r+2}^2\right) + c_{\ell r}^{2\dagger}\left(c_{u r+1}^1+c_{u r-2}^1\right)\right]\nonumber\\ &=\,\mathrm{i} t_{\perp5}\sum\limits_k\left[\left(c_{0 k}^{2\dagger}c_{0 k}^1+c_{0k}^{2\dagger}c_{\pi k}^1 - c_{\pi k}^{2\dagger}c_{0 k}^1 - c_{\pi k}^{2\dagger}c_{\pi k}^1\right)\vphantom{\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2}}\right.\nonumber\\ &\quad\left.\times\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2} - \mathrm{H.c.}\right], \end{align} \tag{ B.5 }$$

$$\begin{align} j_7 &\propto\, \mathrm{i} t_{\perp7}\sum\limits_r\left[-c_{\ell r}^{1\dagger}\left(c_{\ell r-1}^2+c_{\ell r+2}^2\right) + c_{\ell r}^{2\dagger}\left(c_{\ell r+1}^1+c_{\ell r-2}^1\right)\right]\nonumber\\ &=\mathrm{i} t_{\perp7}\sum\limits_k\left[\left(c_{0 k}^{2\dagger}c_{0 k}^1-c_{0k}^{2\dagger}c_{\pi k}^1 - c_{\pi k}^{2\dagger}c_{0 k}^1 + c_{\pi k}^{2\dagger}c_{\pi k}^1\right)\vphantom{\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2}}\right.\nonumber\\ & \qquad\qquad\quad\left.\times\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2} - \mathrm{H.c.}\right]. \end{align} \tag{ B.6 }$$

The k dependent factors associated with the four relevant hopping terms in the total current density operator j are thus:

$$\begin{equation} c_{0 k}^{2\dagger}c_{0 k}^1:\quad \mathrm{i} \left[\left(t_{\perp 2}+t_{\perp 3}\right)\frac{1+\mathrm{e}^{\mathrm{i} kc_1}}{2} +\left(t_{\perp 5}+t_{\perp 7}\right)\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2} \right], \end{equation} \tag{ B.7 }$$

$$\begin{equation} c_{0 k}^{2\dagger}c_{\pi k}^1:\quad \mathrm{i}\left[\left(t_{\perp 2}-t_{\perp 3}\right)\frac{1+\mathrm{e}^{\mathrm{i} kc_1}}{2} +\left(t_{\perp 5}-t_{\perp 7}\right)\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2} \right], \end{equation} \tag{ B.8 }$$

$$\begin{equation} c_{\pi k}^{2\dagger}c_{0 k}^1:\quad \mathrm{i}\left[\left(t_{\perp 2}-t_{\perp 3}\right)\frac{1+\mathrm{e}^{\mathrm{i} kc_1}}{2} -\left(t_{\perp 5}+t_{\perp 7}\right)\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2} \right], \end{equation} \tag{ B.9 }$$

$$\begin{equation} c_{\pi k}^{2\dagger}c_{\pi k}^1:\quad \mathrm{i}\left[\left(t_{\perp 2}+t_{\perp 3}\right)\frac{1+\mathrm{e}^{\mathrm{i} kc_1}}{2} -\left(t_{\perp 5}-t_{\perp 7}\right)\frac{\mathrm{e}^{-2\mathrm{i} kc_1}+\mathrm{e}^{\mathrm{i} kc_1}}{2} \right]. \end{equation} \tag{ B.10 }$$

The interladder conductivity of the ground state is given by the standard formula (see, e.g. [132]):

$$\begin{align} \mathrm{Re}\, \sigma(\omega) &\propto \sum\limits_{M\neq GS} \frac{\left|\left< M\left|j\right|GS\right> \right|^2}{E_M-E_{GS}}\left[\delta(\omega-(E_M-E_{GS}))\right.\nonumber\\ &\quad\left. + \delta(\omega+(E_M-E_{GS}))\right], \end{align} \tag{ B.11 }$$

where M runs over eigenstates of the two ladder system and $\left|GS \right. \rangle$ is the ground state. The hopping terms listed in equations (B.7)–(B.10), and the H.c. terms, can be considered independently, as cross-terms do not contribute under our assumption of independent ladders. We can therefore express the conductivity in terms of the single-ladder spectral function:

$$\begin{align} A_Q(\omega) = A_Q^+(\omega)+A_Q^-(\omega), \end{align} \tag{ B.12 }$$

$$\begin{align} A_Q^+(\omega) = \sum\limits_m \left|\left< m\left|c_Q^\dagger\right|gs \right> \right|^2\delta(\omega-(E_m-E_{gs})), \end{align} \tag{ B.13 }$$

$$\begin{align} A_Q^-(\omega) = \sum\limits_m \left|\left< m\left|c_Q\right|gs\right> \right|^2\delta(\omega+(E_m-E_{gs})), \end{align} \tag{ B.14 }$$

where m runs over the eigenstates of a single ladder, $\left|gs \right. \rangle$ is the ground state, and Q stands for $(q,\,k),\, q\in\{0,\pi\}$.

The final formula reads:

$$\begin{align} \mathrm{Re} \,\Delta \sigma(\omega>0) \propto\sum\limits_{q,q^{\prime},k} |\,f(q,q^{\prime},k)|^2\int\mathrm{d}\omega^{\prime} A^+_{q,k}(\omega^{\prime})A_{q^{\prime},k}^-(\omega^{\prime}-\omega), \end{align} \tag{ B.15 }$$

where $f(q,q^{\prime},k)$ is the form factor of equations (B.7)–(B.10). Results shown in figures B.2 and B.3 have been obtained using equations (B.13)–(B.15) for a ladder of 10 rungs and four doped holes. Realistic values of $t_\parallel,\,t_\perp,\,J_\parallel,\,J_\perp$, and $t_{\perp2} = 0.04,\,t_{\perp3} = 0.05,\,t_{\perp5} = 0.04\,,t_{\perp7} = -0.02$ eV [91] have been used. Figure B.2 demonstrates that the quasi-hole and quasi-electron excitations are separated by a fairly large gap. This justifies our assumption of weakly coupled ladders. Figure B.3(a) shows the calculated inter-ladder conductivity and in addition the intra-ladder (rung) conductivity calculated as in [98]. Figure B.3(b) shows the k dependence of the moduli of the matrix elements of equations (B.7)–(B.10) squared.

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