Fermiology, charge transfer energy, and robust paramagnons in high-$T_c$ cuprate superconductors

Abstract

Copper-oxide high-temperature (high-$T_c$) superconductors host robust paramagnon excitations whose propagation energies are insensitive to hole concentration and correlate with maximal measured superconducting transition temperatures. Given variation of electronic structure across (and within) cuprate families, elucidation of the relationship between microscopic parameters relevant to high-$T_c$ superconductivity and paramagnon dynamics remains a key challenge to theory. Employing canonical Hubbard- and $t$-$J$-$U$ models of a $\mathrm{CuO_2}$ plane, we relate robust paramagnon energies to high-$T_c$ fermiology (via the ratio $r \equiv t^\prime/|t|$ of next-nearest- to nearest-neighbor hopping integrals) and charge transfer energy, $Δ_\mathrm{CT}$. It is shown that variation of $r$ and $Δ_\mathrm{CT}$ between materials has an opposite effect on paramagnon energy, rationalizing comparable bandwidth of magnetic excitations across multiple cuprates. Utilizing empirical values of $r$ and $Δ_\mathrm{CT}$ as input to theory, we address magnetic dynamics in Bi-family of cuprates with up to three $\mathrm{CuO_2}$ planes, and demonstrate quantitative (within $6\,\%$ margin) agreement of calculated paramagnon energies with experiment. Our work offers a route toward quantitative control of robust paramagnon physics in strongly-correlated electron systems.

Figures (8)

• Figure 1: Illustration of the negative correlation between range parameter, $r \sim t^\prime/|t|$, and charge-transfer energy in high-$T_c$ cuprates, composed based on data of Ref. Weber2012 (tabulated in Weber2012a). Symbols represent first-principle calculation results for multiple cuprates and shaded orange region is guide to the eye. The named copper oxides, La$_2$CuO$_4$ (LSCO) and Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10}$ (Bi2223), are positioned on the opposite ends of $r$ vs. charge-transfer energy phase diagram. Here IP and OP refer to inner- and outer planes, respectively. Those two compounds have the same paramagnon bandwidth within error bars, $311(4)\,\mathrm{meV}$ (LSCO) and $324(15)\,\mathrm{meV}$ (Bi2223) Wang2022, pointing toward an intricate relationship between electronic structure and robust paramagnon dynamics.
• Figure 2: Comparison between the two-parameter (2P) and three-parameter (3P) tight-binding models of high-$T_c$ cuprates for representative density $n_e = 0.85$. Solid circles in panel (a) show the established correspondence between range parameters $r_\mathrm{2P}$ and $r_\mathrm{3P}$. Solid line represents linear fit of those data for $r_\mathrm{3P} \in [0.1, 0.3]$. Red stars are experimental ARPES data for multiple cuprates (superscript "a") Lee2006, whereas vertical lines mark $t^\prime/|t|$ estimated using a more general 3P-model-like parameterization (superscript "b") Markiewicz2005. The high-$T_c$ regime, enclosing available experimental data, is indicated by grey area. The comparison between band structures [panels (b)-(c)] and Fermi surfaces [panels (d)-(f)] of both models for values of $r_\mathrm{2P}$ and $r_\mathrm{3P}$ related via mapping of panel (a) is also shown (employed parameters are detailed inside the figure). In-plane wave vector components in panels (d)-(f) are given in the units of $2\pi/a$ with $a$ being square lattice spacing.
• Figure 3: VWF+$1/\mathcal{N}_f$ imaginary part of the dynamical spin susceptibility at the $X$ Brillouin-zone point, obtained for hole-doped 3P model at representative density $n_e = 0.84$. Blue symbols are simulated data, whereas red line represents total model fit in the energy interval $w \in [0, \omega_\mathrm{th}]$. Shaded regions detail the decomposition of the total model signal into paramagnon (blue) and incoherent particle-hole (green) components. The particle-hole continuum excitation threshold, $\omega_\mathrm{th}$, is marked by a vertical dashed line. Panels (a)-(d) correspond to the Hubbard model ($U = 7 |t|$, $J = 0$, and Fermi-surface parameter $r_\mathrm{3P} = 0.1$-$0.4$). Panels (e)-(h) show analogous results for the $t$-$J$-$U$ model ($U = 15 |t|$, $J = \frac{32}{105} |t| \approx 0.305 |t|$, $r_\mathrm{3P} = 0.1$-$0.4$). The parameters are listed inside the figure.
• Figure 4: VWF+$1/\mathcal{N}_f$ paramagnon characteristics for the 3P (a)-(d) and 2P (f)-(h) Hubbard model, plotted as a function of electronic density and for range parameters $r_\mathrm{2P/3P}$ spanning the range relevant to high-$T_c$ (their values are detailed above the panels). Dotted line segments indicate dynamical instability of the paramagnetic state. The displayed quantities include: paramagnon propagation energy [(a) and (e)], bare energy [(b) and (f)], and damping [(c) and (g)]. Moreover, panels (d) and (e) show and ratio $I(|t|)/I(\infty)$, where $I(\omega) \equiv \int_0^\omega d\nu \chi^{\prime\prime}_s(\nu, X)$ denotes integrated intensity (cf. the text). The simulations have been carried out for $k_B T = 0.35 |t|$, $t = -0.35\,\mathrm{eV}$, and $U = 7|t|$, resulting in effective antiferromagnetic exchange $J_\mathrm{eff} = 200\,\mathrm{meV}$.
• Figure 5: Dependence of the 3P (a) and 2P (b) Hubbard model $X$-point paramagnon propagation energy, $\omega_p$, on inverse charge-transfer gap, $\Delta_\mathrm{CT}^{-1}$. Model parameters are: $n_e = 0.85$, $t = - 0.35\,\mathrm{eV}$, $U = \Delta_\mathrm{CT}$, $J = 0$, and $k_BT = 0.35 |t|$. Range parameters $r_\mathrm{3P}$ ($r_\mathrm{2P}$) are set to experimental values Lee2006 for Bi-cuprate series, and are detailed inside the panels. Symbols represent VWF+$1/\mathcal{N}_f$ simulation result, and lines show the corresponding linear fits. Red stars are experimental $X$-point propagation energies $\omega^\mathrm{exp}_p \equiv 2 J^\mathrm{exp}$, with effective exchange $J^\mathrm{exp}$ extracted from RIXS data for Bi-family of cuprates Wang2022. The 3P-model result [panel (a)] agrees quantitatively with experiment, whereas 2P approximation [panel (b)] fails to account for measured paramagnon energies.