Quasiparticle tunneling spectra of the high-Tcmercury cuprates: Implications of thed-wave two-dimensional van Hove scenario

Quasiparticle tunneling measurements of the high-temperature superconductors ${\mathrm{HgBa}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ $(\mathrm{Hg}-12(n-1)n,n=1,2,3)$ are considered in the context of $d_{x^2-y^2}$ symmetry of the superconducting order parameter and a two-dimensional (2D) van Hove singularity (vHs) related to saddle points in the electronic band structure. Normal-metal–insulator–superconductor tunneling spectra taken at 4.2 K with a scanning tunneling microscope on Hg-1212 $c$-axis epitaxial films, as well as on Hg-1201 and Hg-1223 polycrystalline samples, show distinct gap characteristics which cannot be easily reconciled with the simple $s$-wave BCS density of states. The data are analyzed with the nodal $d$-wave gap function ${\Delta }_{\mathbf{k}}={\Delta }_0\left(\cos k_x-\cos k_y\right)/2\mathrm{}$ and the 2D tight-binding electronic dispersion ${\xi }_{\mathbf{k}}=-2t(\cos k_x+\cos k_y{)+4t}^{\prime }(\cos k_x\cos k_y)-\mu ,$ using the quasiparticle tunneling formalism for elastic and specular transmission. The analysis indicates a highly directional and energy-dependent spectral weighting, related to the gap anisotropy and band-structure dependence of the tunneling matrix element $|T{|}^2,$ and successfully explains the observed gap spectra. Values for the $d$-wave gap maximum are determined to be ${\Delta }_0\approx 33,$ 50, and 75 meV, respectively, for optimally doped Hg-1201, Hg-1212, and Hg-1223, corresponding to reduced-gap ratios of $2{\Delta }_0{/k}_B{T}_c\approx 7.9,$ 9.5, and 13. These ratios are substantially larger than the BCS weak-coupling limit of 3.54. A comparison with data from other high-$T_c$ cuprates indicates an overall trend of $2{\Delta }_0{/k}_B{T}_c$ rising with $T_c,$ in violation of BCS universality.