Microscopic gauge-invariant theory of the $c$-axis infrared response of bilayer cuprate superconductors and the origin of the superconductivity-induced absorption bands

We report on results of our theoretical study of the $c$-axis infrared conductivity of bilayer high-$T_c$ cuprate superconductors using a microscopic model involving the bilayer-split (bonding and antibonding) bands. An emphasis is on the gauge invariance of the theory, which turns out to be essential for the physical understanding of the electrodynamics of these compounds. The description of the optical response involves local (intrabilayer and interbilayer) current densities and local conductivities. The local conductivities are obtained using a microscopic theory, where the quasiparticles of the two bands are coupled to spin fluctuations. The coupling leads to superconductivity and is described at the level of generalized Eliashberg theory. Also addressed is the simpler case of quasiparticles coupled by a separable and nonretarded interaction. The gauge invariance of the theory is achieved by including a suitable class of vertex corrections. The resulting response of the model is studied in detail and an interpretation of two superconductivity-induced peaks in the experimental data of the real part of the $c$-axis conductivity is proposed. The peak around $400{\text{cm}}^{-1}$ is attributed to a collective mode of the intrabilayer regions, which is an analog of the Bogolyubov-Anderson mode playing a crucial role in the theory of the longitudinal response of superconductors. For small values of the bilayer splitting, its nature is similar to that of the transverse plasmon of the phenomenological Josephson superlattice model. The peak around $1000{\text{cm}}^{-1}$ is interpreted as a pair-breaking feature that is related to the electronic coupling through the spacing layers separating the bilayers.

Figure 1

(a) Crystal structure of Y-123. (b) Multilayer model, where intrabilayer and interbilayer current densities $j_{\text{bl}}$ and $j_{\text{int}}$ lead to a charge redistribution between the ${\text{CuO}}_2$ planes, which modifies the local fields $E_{\text{bl}}$ and $E_{\text{int}}$.

Figure 2

(a) Diagrammatic representation of the self-energies of the bonding (B) and antibonding (A) bands. The propagators of the electronic quasiparticles and spin fluctuations are represented by the straight and the wiggly lines, respectively. (b) Simple bubble approximation to current-current correlator (20). Only the part given by Eq. (21) is shown. The black dots are the current vertices corresponding to $j_{\text{bl}}^p$. (c) Current-current correlator with a renormalized current vertex [Eq. (23)]. The diagrams corresponding to the case of $t_{\perp {\mathit{k}}_{\parallel }}^{\prime }=0$ with no intraband contributions are shown. (d) Diagrammatic representation of the Bethe-Salpeter Eq. (24). (e) Diagrammatic representation of the equation determining the current-current correlator including plane-charging effects. The dashed lines correspond to the interplane Coulomb interaction.

Figure 3

(a) Local dielectric function ${\epsilon }_{\text{bl}/\text{bl}}$ in the simplest BCS case with $t_{\perp {\mathit{k}}_{\parallel }}^{\prime }=0$, $t_{\perp \text{max}}=45\text{meV}$, and ${\Delta }_{\text{max}}=30\text{meV}$, vertex corrections are not included (NV). The thin (thick) lines correspond to the normal (superconducting) state, $T=100\text{K}$ $\left(T=20\text{K}\right)$. The solid (dashed) lines represent the imaginary (real) part. (b) The real part of the corresponding total $c$-axis conductivity obtained using Eq. (2). The thin (thick) line corresponds to the normal (superconducting) state. [(c) and (d)] The same as in (a) and (b) but with the VC included (for the superconducting state only).

Figure 4

(a) Schematic representation of the current-density, density, and phase pattern associated with the Bogolyubov-Anderson mode of a single-layer superconductor with $\mathit{q}\parallel c$, $\left|\mathit{q}\right|=\pi /d$. (b) The same for the collective mode of the bilayer system discussed in the text.

Figure 5

Dependence of the real part of the local conductivity ${\sigma }_{\text{bl}/\text{bl}}\left(T=20\text{K}\right)$ on the intrabilayer hopping amplitude $t_{\perp \text{max}}$ calculated with the vertex corrections neglected (a) and included (b). In (a), the absorption peak stops at $2{\Delta }_{\text{max}}=60\text{meV}$ when decreasing $t_{\perp \text{max}}$. In (b), the energy of the peak is proportional to $t_{\perp \text{max}}$.

Figure 6

(a) Real parts of the local conductivities ${\sigma }_{\text{bl}/\text{bl}}$, ${\sigma }_{\text{bl}/\text{int}}$, and ${\sigma }_{\text{int}/\text{int}}$ calculated considering the BCS interaction between the charged quasiparticles for $t_{\perp \text{max}}=45\text{meV}$, $t_{\perp \text{max}}^{\prime }=0.2t_{\perp \text{max}}$, ${\Delta }_{\text{max}}=30\text{meV}$, and $T=20\text{K}$. The spectra of $\text{Re}{\sigma }_{\text{bl}/\text{int}}$ and $\text{Re}{\sigma }_{\text{int}/\text{int}}$ are four times magnified. The dashed (solid) lines correspond to the NV approximation (to the approach with the VC included). (b) Real part of the total $c$-axis conductivity calculated using Eq. (4) for various values of the ratio $t_{\perp \text{max}}^{\prime }/t_{\perp \text{max}}$. The figure demonstrates the effect of the interbilayer hopping.

Figure 7

Scheme illustrating the differences between the local conductivities ${\sigma }_{\text{bl}/\text{bl}}$ and ${\sigma }_{\text{int}/\text{int}}$. The Wannier-type orbitals of the planes are denoted by ${\Psi }_1$ and ${\Psi }_2$, the bonding and antibonding orbitals of the individual bilayers by ${\Psi }_A$ and ${\Psi }_B$, respectively. The current-density operators ${\widehat{j}}_{\text{bl}}^p$ and ${\widehat{j}}_{\text{int}}^p$ can be associated with transitions marked by the solid and the dashed arrows, respectively.

Figure 8

Dependence of (a) $\text{Re}{\sigma }_{\text{bl}/\text{bl}}$ and (b) $\text{Re}{\sigma }_{\text{int}/\text{int}}$ on the hopping amplitude $t_{\perp \text{max}}$. A constant value of the ratio $t_{\perp \text{max}}^{\prime }/t_{\perp \text{max}}$ of 0.2 has been used. If the vertex corrections are included, the B. A.-like mode characterized by a linear $t_{\perp }$ dependence appears in the bl/bl component and, for small values of $t_{\perp \text{max}}$, also in the int/int component.

Figure 9

Optical weight $I_o=2{\hslash }^2/\pi e^2{\int }_{0^{+}}^{\infty }\text{Re}{\sigma }_{\text{bl}}\left(\omega \right)\text{d}\omega$ of the local conductivity ${\sigma }_{\text{bl}}$ divided by $\left|K_{\text{bl}}\right|$ as a function of temperature for various values of the interplane hopping parameter $t_{\perp }$. Here $K_{\text{bl}}$ is the effective $c$-axis kinetic energy defined by Eq. (19). The results were obtained using Eq. (21) within the spin-fermion model with no interbilayer hopping $\left(t_{\perp \text{max}}^{\prime }=0\right)$ and the resonant mode in the odd-interaction channel only, the vertex corrections have been neglected. With the inclusion of the vertex corrections, $I_o/\left|K_{\text{bl}}\right|$ is always equal to 1, both for the normal and the superconducting states.

Figure 10

Effect of the vertex corrections on the conductivities. Real parts of [(a)–(d)] the local conductivity ${\sigma }_{\text{bl}/\text{bl}}$ and of [(e)-(f)] the total $c$-axis conductivity calculated within the spin-fermion model. The spacing layers have been taken to be insulating, i.e., $t_{\perp \text{max}}^{\prime }=0$ and the neutron resonance has been included in the odd-interaction channel only. Results for several values of the hopping parameter $t_{\perp \text{max}}$ are presented. The spectra in (a) and (c) [(b), (d), (e), and (f)] have been obtained for the NV (VC) case. The thin line in (f) represents the estimated energy dependence of the spectral weight of the peak labeled as $T_1$. For $t_{\perp \text{max}}=30\text{meV}$ (45 meV, 60 meV), the estimated value of the spectral weight is $4000{\Omega }^{-1}{\text{cm}}^{-2}$ $\left(4600{\Omega }^{-1}{\text{cm}}^{-2},2300{\Omega }^{-1}{\text{cm}}^{-2}\right)$.

Figure 11

Effect of finite interbilayer hopping $\left(t_{\perp \text{max}}^{\prime }>0\right)$ on the conductivities. Real parts of (a) the local conductivities ${\sigma }_{\text{bl}/\text{bl}}$ and ${\sigma }_{\text{int}/\text{int}}$ and of (b) the total $c$-axis conductivity calculated within the spin-fermion model. The neutron resonance has been distributed equally in both (even and odd) interaction channels. The dashed (solid) lines in (a) represent the spectra obtained with the VC neglected (included). The spectra in (b) have been obtained with the VC included, the dashed (solid) lines correspond to the normal (superconducting) state, results for several values of $t_{\perp \text{max}}^{\prime }$ are shown.

Figure 12

(a) Energies of the B. A. peak and of the $2\Delta$ peak in the local conductivity ${\sigma }_{\text{bl}/\text{bl}}$, and of the structures $T_1$ and $T_2$ in the total $c$-axis conductivity (for examples, see Figs. 10, 11) as a function of the intrabilayer hopping parameter $t_{\perp \text{max}}$. A constant value of the ratio $t_{\perp \text{max}}^{\prime }/t_{\perp \text{max}}$ of 0.2 has been used. (b) Real part of the total conductivity for values of $t_{\perp \text{max}}$, where both structures $T_1$ and $T_2$ are visible. As $t_{\perp \text{max}}$ decreases, the peak $T_1$ emerges from the background at $t_{\perp \text{max}}\approx 100\text{meV}$, and it quickly becomes the dominant feature. Eventually, it covers the $T_2$ peak at $t_{\perp \text{max}}\approx 50\text{meV}$.

Figure 13

(a) Doping dependence of the difference $\text{Re}{\sigma }_c\left(T⪡T_c\right)-\text{Re}{\sigma }_c\left(T\approx T_c\right)$ for Y-123. The abbreviations UD, OPT, and OD stand for underdoped, optimum doped, and overdoped. The values of $p$ are 0.093, 0.116, 0.124, 0.155, and 0.194. Also shown are the data for ${\text{Y}}_{0.86}{\text{Ca}}_{0.14}\text{−}123$ with $T_c=84\text{K}$ and $p=0.176$. Details concerning the samples and the experiment are given in Refs. 20, 39. (b) The spectra of the difference for the sequence $R\text{−}123$ ($R=Y$, Nd, and La) with $p\approx 0.12$. In this sequence, the distance between the ${\text{CuO}}_2$ planes within a bilayer increases. The inset shows the original spectra for Nd-123. Adapted from Fig. 2 of Ref. 20.

Figure 14

Illustrates the potential of the model to provide the order of the spectral structures consistent with experimental data on underdoped Y-123. (a) Real part of the local conductivity ${\sigma }_{\text{bl}/\text{bl}}$ consisting of the result of the BCS approach with $\Delta =50\text{meV}$ and a broad Lorentzian described in the text. (b) Real part of the total $c$-axis conductivity obtained using the local conductivity ${\sigma }_{\text{bl}/\text{bl}}$ shown in (a) and the other three local conductivities resulting from the BCS approach. A constant value of the ratio $t_{\perp \text{max}}^{\prime }/t_{\perp \text{max}}$ of 0.2 has been used.

Figure 15

Representative examples of the spectra of $\text{Re}{\sigma }_c$ compared with the conductivities of a related single-layer system, and with the spectra of the same quantity obtained using the approximation proposed by Shah and Millis. (a) Real part of the total $c$-axis conductivity calculated using the spin-fermion model with $t_{\perp \text{max}}=250\text{meV}$, $t_{\perp \text{max}}^{\prime }=75\text{meV}$, and with $t_{\perp \text{max}}=30\text{meV}$, $t_{\perp \text{max}}^{\prime }=0\text{meV}$. The solid and the dashed lines correspond to the superconducting and the normal state, respectively. The spectra for $t_{\perp \text{max}}=250\text{meV}$ and $t_{\perp \text{max}}^{\prime }=75\text{meV}$ on an extended scale are shown in the inset. (b) Real parts of the in-plane conductivity and of the $c$-axis conductivity for a single-layer superconductor with the plane spacing of $d_{\text{bl}}+d_{\text{int}}$ and with the $c$-axis hopping parameter of the form of Eq. (7) and the maximum of 75 meV. (c) The same as in (a) but using the approach proposed by Shah and Millis, where the nondiagonal components of the conductivity ${\sigma }_{\text{bl}/\text{int}}$ and ${\sigma }_{\text{int}/\text{bl}}$ are neglected, and the diagonal component ${\sigma }_{\text{bl}/\text{bl}}$ $\left({\sigma }_{\text{int}/\text{int}}\right)$ is approximated by the conductivity of the single-layer superconductor with the hopping parameter equal to $t_{\perp }$ $\left(t_{\perp }^{\prime }\right)$. Only the superconducting state spectra are shown.