We propose a model and derive analytical expressions for conductivity in heterogeneous fully anisotropic conductors with ellipsoid superconducting inclusions. This model and calculations are useful to analyze the observed temperature dependence of conductivity anisotropy in various anisotropic superconductors, where superconductivity onset happens inhomogeneously in the form of isolated superconducting islands. The results are applied to explain the experimental data on resistivity above the transition temperature $T_c$ in the high-temperature superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$ and in the organic superconductor $\beta \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$. The comparison of resistivity data and diamagnetic response in $\beta \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$ allows us to estimate the size of superconducting inclusions as $d\sim 1\mu \text{m}$.

A theoretical model has been proposed to describe the conductivity of a layered anisotropic normal metal containing small superconducting inclusions at an arbitrary eccentricity of spheroidal superconducting islands. The electron transport and magnetic properties of FeSe single crystals have been measured. The results indicate the existence of superconductivity at temperatures much higher than the critical superconducting transition temperature corresponding to vanishing electrical resistance. Within the proposed model, quantitative agreement has been achieved between the volume fraction of superconducting inclusions and its temperature dependence determined from the transport and magnetic measurements.

We study the coexistence of superconductivity (SC) and density-wave state and reconcile various puzzling experimental data in organic superconductors ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{PF}}_6$ and ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{ClO}}_4$. The anisotropic resistance drop above $T_c$ is qualitatively described by nascent isolated SC islands within a bulk analytical model. However, the observed anisotropic SC onset is explained only when the finite size and flat needle shape of samples is considered. Our results pave a way to estimate the volume fraction and the typical size of SC islands far from the sample surface and apply to many inhomogeneous superconductors, including high-$T_c$ cuprate or Fe-based ones.

The $su\left(2\right)$ coherent-state path-integral technique is employed to study lattice electrons strongly coupled to a quantum spin background. In the large-spin limit it is replaced by its classical counterpart that breaks the time-reversal symmetry. The fermions propagating through a classical large-spin texture may then exhibit the topological Hall effect which arises even for a zero scalar spin chirality of the underlying spin background.

We calculated the electron susceptibility of rare-earth tritelluride compounds RTe₃ as a function of temperature, wave vector, and electron-dispersion parameters. Comparison of the results obtained with the available experimental data on the transition temperature and on the wave vector of a charge-density wave in these compounds allowed us to predict the values and evolution of electron-dispersion parameters with the variation of the atomic number of rare-earth elements (R).

In highly anisotropic organic superconductor (TMTSF)${}_2$ClO${}_4$, superconducting (SC) phase coexists with metallic and spin-density wave phases in the form of domains. Using the Maxwell-Garnett approximation (MGA), we calculate the volume ratio and estimate the shape of these embedded SC domains from resistivity data at various temperature and anion disorder, controlled by the cooling rate or annealing time of (TMTSF)${}_2$ClO${}_4$ samples. We found that the variation of cooling rate and of annealing time affect differently the shape of SC domains. In all cases the SC domains have oblate shape, being the shortest along the interlayer z-axis. This contradicts the widely assumed filamentary superconductivity along the z-axis, used to explain the anisotropic superconductivity onset. We show that anisotropic resistivity drop at the SC onset can be described by the analytical MGA theory with anisotropic background resistance, while the anisotropic $T_c$ can be explained by considering a finite size and flat shape of the samples. Due to a flat/needle sample shape, the probability of percolation via SC domains is the highest along the shortest sample dimension (z-axis), and the lowest along the sample length (x-axis). Our theory can be applied to other heterogeneous superconductors, where the size d of SC domains is much larger than the SC coherence length $\xi$, e.g., cuprates, iron-based or organic superconductors. It is also applicable when the spin/charge-density wave domains are embedded inside a metallic background, or vice versa.