Peculiarities in the low-temperature specific heat related to nanoscale structural inhomogeneity in fine-crystalline YBa$_2$Cu$_3$O$_{6.93}$ high-$T_c$ superconductors

A comparative study of the low-temperature specific heat for two types of YBa$_2$Cu$_3$O$_{6.93}$ high-$T_c$ superconductor samples is performed within the temperature range of $2$–$10$ K. The samples of the first type are fine-crystalline optimally doped ones with different degrees of nanoscale structural inhomogeneity. The second type includes coarse-crystalline equilibrium samples with different hole doping levels. A similarity in the behavior of different contributions to the specific heat for structurally inhomogeneous and underdoped samples is revealed. The samples of both types exhibit a metal-like contribution linear in temperature to the specific heat $\sim\gamma T$, which is not characteristic of the superconducting phase. It is found that this contribution moderately grows with the decrease in the oxygen content, whereas with the increase in the structural inhomogeneity, such growth of the linear contribution ($\gamma T$) becomes anomalously large. This leads to the conclusion about the coexistence of metallic and superconducting states in the bulk of the samples under study. Such common feature of electron systems could be related to the formation of the pseudogap regime. It is demonstrated that this regime suppresses just the superconducting states, leaving intact the metallic ones.