Inhomogeneous Superconductivity Onset in FeSe Studied by Transport Properties

Abstract

Heterogeneous superconductivity onset is a common phenomenon in high-Tc superconductors of both the cuprate and iron-based families. It is manifested by a fairly wide transition from the metallic to zero-resistance states. Usually, in these strongly anisotropic materials, superconductivity (SC) first appears as isolated domains. This leads to anisotropic excess conductivity above Tc, and the transport measurements provide valuable information about the SC domain structure deep within the sample. In bulk samples, this anisotropic SC onset gives an approximate average shape of SC grains, while in thin samples, it also indicates the average size of SC grains. In this work, both interlayer and intralayer resistivity were measured as a function of temperature in FeSe samples of various thicknesses. To measure the interlayer resistivity, FeSe mesa structures oriented across the layers were fabricated using FIB. As the sample thickness decreases, a significant increase in superconducting transition temperature Tc is observed: Tc raises from 8 K in bulk material to 12 K in microbridges of thickness ∼40 nm. We applied analytical and numerical calculations to analyze these and earlier data and find the aspect ratio and size of the SC domains in FeSe consistent with our resistivity and diamagnetic response measurements. We propose a simple and fairly accurate method for estimating the aspect ratio of SC domains from Tc anisotropy in samples of various small thicknesses. The relationship between nematic and superconducting domains in FeSe is discussed. We also generalize the analytical formulas for conductivity in heterogeneous anisotropic superconductors to the case of elongated SC domains of two perpendicular orientations with equal volume fractions, corresponding to the nematic domain structure in various Fe-based superconductors.

Keywords: FeSe; anisotropic superconductivity onset; heterogeneous superconductor; high-Tc; iron selenide; superconducting domain; twin boundary.

Figure 1

Photos of the microbridges, used in our experiment, from two different angles. (a) illustrates the 3D shape of our mesa structure, while (b) shows its dimensions.

Figure 2

Illustration of two current channels, corresponding to two terms in Equation (1) for interlayer conductivity ${\sigma }_{zz}$ in a strongly anisotropic metal containing isolated superconducting islands.

Figure 3

Measured temperature dependence of normalized resistance $R_{zz}\left(T\right)/R_{zz}(T=15K)$ in several samples of the same in-plane size $2\times 2$ µm${}^2$ but of different thickness.

Figure 4

(a) Calculated probability p of current percolation along the in-plane x (solid and dashed curves) and the out-of-plane z axes (dotted and dash-dotted curves) via SC domains of spheroid shape with aspect ratio $a_z/a_x=0.62$ as a function of SC volume fraction $\varphi$ for two different domain heights $d=d_z=20$ nm (red curves) and 5 nm (blue curves) in a sample of dimensions $2\times 2\times 0.2$ µm${}^3$; (b) Two dimensional (2D) illustration showing that the current percolation along the sample thickness is easier than along the sample length. Circular SC islands (blue) with diameter d = 0.4 are randomly distributed inside a rectangular sample (yellow) of dimensions $7\times 2$, forming SC channels between contact electrodes (black).

Figure 5

Calculated probability p of current percolation along the in-plane x (red dashed curves) and out-of-plane z (dotted blue curves) axes via the SC domains of spheroid shape with aspect ratio $a_z/a_x=0.12$ as a function of SC volume fraction $\varphi$ in a sample of thickness $L_z=40$ nm (a), $L_z=125$ nm (b), and $L_z=235$ nm (c). The sample area in the conducting x-y plane is taken $2\times 2$ µm${}^2$, as in our experiment. The domain height in our calculations is $d_z=20$ nm. The percolation probability is almost isotropic for the sample thickness $L_z=235$ nm at $a_z/a_x=0.12$.

Figure 6

Probability p of current percolation along the in-plane x (red dashed curves) and the out-of-plane z axes (dotted blue curves) via SC domains of spheroid shape with aspect ratio $a_z/a_x=0.15$ and height $d_z=20$ nm as a function of SC volume fraction $\varphi$ in a sample of x-y area $2\times 2$ µm${}^2$, calculated for four different sample thicknesses $L_z=40$ nm (a), $L_z=125$ nm (b), $L_z=235$ nm (c), and $L_z=300$ nm (d). The percolation probability is isotropic for $L_z=300$ nm.