Systematic study of electron-phonon coupling to oxygen modes across the cuprates

The large variations in $T_c$ across the cuprate families is one of the major unsolved puzzles in condensed matter physics and is poorly understood. Although there appears to be a great deal of universality in the cuprates, several orders of magnitude changes in $T_c$ can be achieved through changes in the chemical composition and structure of the unit cell. In this paper we formulate a systematic examination of the variations in electron-phonon coupling to oxygen phonons in the cuprates, incorporating a number of effects arising from several aspects of chemical composition and doping across cuprate families. It is argued that the electron-phonon coupling is a very sensitive probe of the material-dependent variations in chemical structure, affecting the orbital character of the band crossing the Fermi level, the strength of local electric fields arising from structural-induced symmetry breaking, doping-dependent changes in the underlying band structure, and ionicity of the crystal governing the ability of the material to screen $c$-axis perturbations. Using electrostatic Ewald calculations and known experimental structural data, we establish a connection between the material’s maximal $T_c$ at optimal doping and the strength of coupling to $c$-axis modes. We demonstrate that materials with the largest coupling to the out-of-phase bond-buckling $\left(B_{1g}\right)$ oxygen phonon branch also have the largest $T_c$’s. In light of this observation we present model $T_c$ calculations using a two-well model where phonons work in conjunction with a dominant pairing interaction, presumably due to spin fluctuations, indicating how phonons can generate sizeable enhancements to $T_c$ despite the relatively small coupling strengths. Combined, these results can provide a natural framework for understanding the doping and material dependence of $T_c$ across the cuprates.

Figure 1

(Color online) The $T_c$ at optimal doping for many cuprate superconductors plotted as a function of the distance between the apical oxygen site and the mirror plane located at the center of the unit cell. The Hg-(red) circles, Tl-(blue) squares, and Bi-families (green) triangles are shown as well as some LSCO and ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ (LBCO) systems open (black).

Figure 2

(Color online) The five-band model used to derive the form of the el-ph couplings $g\left(\mathbf{k},\mathbf{q}\right)$.

Figure 3

(Color online) Plots of the el-ph coupling constant $g\left({\mathbf{k}}_F,\mathbf{q}\right)$ for fermionic momentum on the Fermi surface as a function of transferred momentum $\left(q_x,q_y=0\right)$ (a1–a4) and $q_x=q_y$ (b1–b4), respectively. a1, b1 (a2, b2) plot coupling to the $A_{1g}$ $\left(B_{1g}\right)$ branches, respectively, a3, b3 plot coupling to the breathing branch and a4, b4, plots coupling to the apical branch. The colors denote angles from the corner of the BZ (shown in the inset of Fig. 4) given by solid black—$0°$, solid red—$-15°$, solid green—$30°$, solid blue—$45°$ (nodal), dashed green—$60°$, dashed red—$75°$, and dashed black—$90°$ (antinodal).

Figure 4

(Color online) Plots of ${\lambda }_{\nu }\left({\mathbf{k}}_F\right)$ for momentum points along the Fermi surface. Inset: the definition of the angle $\theta$ used in Figs. 3, 4. The parameters used are defined in the text.

Figure 5

(Color online) A contour plot of the band structure of optimal doped Bi-2212 obtained from the five-parameter tight-binding model of Ref. 80. The outermost contour (solid black) corresponds to the FS. The remaining contours correspond to ${ϵ}_{\mathbf{k}}-\mu ={\Omega }_{\nu }$ for the four phonon branches considered in this work. Reading from the FS to the $\Gamma$ point, the contours correspond to the $B_{1g}$, $A_{1g}$, breathing and apical branches, respectively.

Figure 6

(Color online) (a) Plots of the real (black) and imaginary ($-1\times$, red) parts of the charge susceptibility ${\Pi }_{1,1}$ and [(b)–(f)] the renormalized el-ph coupling $\overline{g}\left(\mathbf{k},\mathbf{q}\right)$ in arbitrary units for $\mathbf{k}=\left(0,\pi /a\right)$ and $\mathbf{q}=\left(q_x,0\right)$ for different values of $q_z$ as indicated by the legend. For (b)–(f) the black dashed line corresponds to the bare coupling $g\left(\mathbf{k},\mathbf{q}\right)$. A momentum-independent bare coupling is considered in (b), followed by the $A_{1g}$, $B_{1g}$, apical and breathing modes in (c)–(f), respectively. In the case of the breathing modes a small $q_y$ component has been added to visualize the effects of screening at small ${\mathbf{q}}_{2\text{D}}$. The $q_x$ axis in all plots is on a logarithmic scale to highlight the small ${\mathbf{q}}_{2\text{D}}$ region.

Figure 7

(Color online) Plots of the momentum dependent coupling ${\lambda }_{\nu }\left({\mathbf{q}}_{2\text{D}}\right)$, Eq. (25), for (a) $\nu =A_{1g}$, (b) $B_{1g}$, (c) apical, and (d) bond-stretching modes. The coupling are plotted on a logarithmic scale for ${\mathbf{q}}_{2\text{D}}$ to highlight the region of poor screening.

Figure 8

(Color online) Plots of the $d$-wave coupling ${\lambda }_{\nu ,d}\left({\mathbf{q}}_{2\text{D}}\right)$, Eq. (26), for (a) $\nu =A_{1g}$, (b) $B_{1g}$, (c) apical, and (d) bond-stretching modes. The coupling are plotted on a logarithmic scale for ${\mathbf{q}}_{2\text{D}}$ to highlight the region of poor screening.

Figure 9

(Color online) Plots of the screened e-ph $\lambda \left({\mathbf{k}}_F\right)$ for fermions at the Fermi level as a function of $\theta$ for several values of the plasma frequency ${\Omega }_{\text{pl}}$. The value of ${\Omega }_{\text{pl}}$ varies from 0.289 eV (black, solid) to 0.914 eV (purple, dashed). The black arrows indicate the direction of increasing ${\Omega }_{\text{pl}}$. The parameters used to generate this figure are defined in the text.

Figure 10

(Color online) Plots of ${\lambda }_z$ (left) and ${\lambda }_{\varphi }$ (right) as a function of the plasma frequency ${\Omega }_{\text{pl}}$. The parameters taken are defined in the text.

Figure 11

(Color online) The ${\text{HgBa}}_2{\text{Ca}}_{n-1}{\text{Cu}}_n{\text{O}}_{4n+\delta }$ $\left(n=1–5\right)$ unit cell showing the locations of the mirror plane (yellow), low symmetry breaking plane (green), and high symmetry breaking planes (red). The degree of symmetry breaking establishes the strength of the electric field which couples to the $c$-axis phonons.

Figure 12

(Color online) The Madelung potential at the ${\text{CuO}}_2$ planar oxygen site as function of distance from the plane of mirror symmetry: (a) ${\text{HgBa}}_2{\text{Ca}}_{n-1}{\text{Cu}}_n{\text{O}}_{2n+2}$ ($n=1–6$, red circles). (b) ${\text{Tl}}_2{\text{Ba}}_2{\text{Ca}}_{n-1}{\text{Cu}}_n{\text{O}}_{2n+2}$, ($n=1–4$, blue squares). (c) ${\text{Bi}}_2{\text{Sr}}_2{\text{Ca}}_{n-1}{\text{Cu}}_n{\text{O}}_{2n+2}$ ($n=1–3$, green triangles). (d) An overlay of the Hg, Tl, and Bi plots rescaled by the lattice spacing, $a$. The black line is a fitted curve of the form $a\Phi =A+B\text{cos}\left(2\pi z/c\right)$.

Figure 13

(Color online) The Madelung energy of the apical oxygen site as a function of distance from the plane of mirror symmetry: (a) ${\text{HgBa}}_2{\text{Ca}}_{n-1}{\text{Cu}}_n{\text{O}}_{2n+2}$ ($n=1–6$, red circles). (b) ${\text{Tl}}_2{\text{Ba}}_2{\text{Ca}}_{n-1}{\text{Cu}}_n{\text{O}}_{2n+2}$, ($n=1–4$, blue squares). (c) ${\text{Bi}}_2{\text{Sr}}_2{\text{Ca}}_{n-1}{\text{Cu}}_n{\text{O}}_{2n+2}$ ($n=1–3$, green triangles). (d) An overlay of the Hg, Tl, and Bi plots in addition of points obtained for LSCO and LBCO (black open symbols). The dashed and solid black lines are fitted curves of the form $a\Phi =A+B\text{cos}\left(2\pi z/c\right)$. The dashed curve is obtained when all of the data is included while the solid curve is obtained when the La data points are excluded.

Figure 14

(Color online) [(a1)–(a3)] The magnitude of the electric field at the apical oxygen site of the Hg, Tl, and Bi families of cuprates, respectively. The data is plotted as a function of the apex distance from the plane of mirror symmetry located at the center of the ${\text{CuO}}_2$ planes. [(b1)–(b3)] The magnitude of the electric field at the planar oxygen site of the outermost ${\text{CuO}}_2$ plane for the set of same compounds. [(c1)–(c3)] The strength of the electric field scaled by the unit cell lattice parameters for the planar and apical oxygen fields, respectively. The dashed lines represents a fit $a^{2}E\left(z/c\right)/c=A\text{sin}\left(2\pi z/c\right)$, where $\text{A}=7.1$ and 83.9 eV for the planar and apical sites, respectively.

Figure 15

(Color online) (a) The planar oxygen character at the Fermi surface $A_{\text{O}}^2$ as a function of the material’s $T_c$. (b) The corresponding local $E$ field at the planar oxygen site of the outermost ${\text{CuO}}_2$ plane in the parent compound. (c) The value of ${\lambda }_z$ for the $B_{1g}$ modes of the outermost plane for each material.

Figure 16

The double square well model for $\Delta \left(\omega \right)$ and $Z\left(\omega \right)$.

Figure 17

(Color online) The transition temperature $T_c$ as a function of el-ph coupling in the $d$-wave channel ${\lambda }_{1,\varphi }$, calculated in the two-square well model, defined in the text. The dashed line is $T_c$ expected for the phonon alone.

Figure 18

(Color online) The isotope exponent $\alpha$ as a function of el-ph coupling ${\lambda }_{1,\varphi }$ for the same parameter set used in Fig. 17.

Figure 19

(Color online) ${\text{Cu}}_2{\text{O}}_9$ and ${\text{Cu}}_2{\text{O}}_{10}$ clusters used to compute the modifications of the effective Zhang-Rice singlet hopping $t$ and $t^{\prime }$ and magnetic coupling $J$ and $J^{\prime }$.

Figure 20

(Color online) The effect of different modes on various Zhang-Rice parameters for the ${\text{Cu}}_2{\text{O}}_9$ cluster: plots (a) and (b) show the effect of a given mode on the ground-state energy and on the effective ZRS hopping $t$ with $\Delta t=\left|t\left(\delta \right)-t\left(\delta =0\right)\right|$ and $\Delta E=E\left(\delta \right)-E\left(\delta =0\right)$. (a) The effect of the $A_{1g}$ modes. (b) The effect of the $B_{1g}$ mode. (c) The effect of the apical mode on $2t$.

Figure 21

(Color online) Variation in the effective Zhang-Rice hopping $t^{\prime }$ and magnetic exchange $J^{\prime }$ to next-nearest neighbor. The calculation performed for the ${\text{Cu}}_2{\text{O}}_{10}$ cluster for $A_{1g}$, $B_{1g}$, and apex modes.