Suppression of the superconducting gap near $d$-wave nodes caused by the structural disorder in fine-crystalline YBa$_2$Cu$_3$O$_y$ high-$T_C$ superconductors

The low-temperature ($2\,\mathrm{K}\leqslant T\leqslant10\,\mathrm{K}$) specific heat of the series of fine-crystalline samples of YBa$_2$Cu$_3$O$_y$ high-$T_C$ superconductor optimally doped with oxygen and having different degrees of nanoscale structural inhomogeneity has been studied at the applied magnetic field $H=8\mathrm{T}$. The result are compared to those obtained for the equilibrium polycrystalline samples with different oxygen contents $y$. Information on the quasiparticle excitations in the magnetic field near $d$-wave nodes of the gap function is obtained. The changes introduced by the structural inhomogeneity to the nodal gap slope ($\nu\Delta$) in the $k$-space, which is a key parameter of high-$T_C$ superconductors, have also been studied. It is found that $\nu\Delta$ increases with the degree of structural disorder, but the superconducting transition temperature remains nearly the same ($T_C=(91.5\pm 0.5)\,\mathrm{K}$). It is shown that this is possible if superconductivity is suppressed not only at the nodal point but also near it (owing to the structural disorder). In this case, the density of states in such parts of the Fermi surface is nonzero even at zero temperature, promoting the existence of the linear in temperature contribution to the specific heat ($\sim\gamma(o)T$), which is characteristic of metals.