Strong pinning and vortex energy distributions in single-crystalline Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$

The interrelation between heterogeneity and flux pinning is studied in Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ single crystals with widely varying Co content $x$. Magnetic Bitter decoration of the superconducting vortex ensemble in crystals with $x=0.075$ and $x=0.1$ reveals highly disordered vortex structures. The width of the Meissner belt observed at the edges of the crystals, and above the surface steps formed by cleaving, as well as the width of the intervortex distance distribution, indicate that the observed vortex ensemble is established at a temperature just below the critical temperature $T_c$. The vortex interaction energy and pinning force distributions extracted from the images strongly suggest that the vortex lattice disorder is attributable to strong pinning due to spatial fluctuations of $T_c$ and of the superfluid density. Correlating the results with the critical current density yields a typical length scale of the relevant disorder of 40–60 nm.

Figure 1

(a) Photograph of Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$ crystal 2. (b) Scanning electron micrograph of the decorated sample 2.1 cut from the larger crystal 2. (c) Magneto-optical images of Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$ crystal 2 at $T=15$ K and indicated values of the applied magnetic field. (d) Differential magneto-optical (DMO) images in the vicinity of $T_c$ for ${\mu }_0\Delta H_a=0.1$ mT. (e) DMO image of Ba(Fe${}_{0.9}$Co${}_{0.1}$)${}_2$As${}_2$ crystal 1 at full screening ($T=17.5$ K) and (f) at midtransition ($T=19.25$ K). The arrows indicate regions of paramagnetic transmissivity at the superconducting transition. (g) Local transmissivity ${\mathcal{T}}_H$ measured on the three regions of Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$ crystal 2. (h) ${\mathcal{T}}_H$ measured on the three regions of Ba(Fe${}_{0.9}$Co${}_{0.1}$)${}_2$As${}_2$ crystal 1 indicated in (e). Scale bars correspond to a length of 100 $\mu$m unless indicated otherwise.

Figure 2

(a) Transition temperature $T_c$ versus Co doping level. The error bars denote the local spread of $T_c$ values within a given crystal. For each doping level, 1, 2, and 3 denote different crystals. For $x=0.075$ and $0.1$, the numbering denotes that of the decorated crystals. (b) Co doping level dependence of the critical current density $j_c$ measured by MOI at $B=30$ mT and a reduced temperature of $T/T_c=0.47$.

Figure 3

Bitter decoration images of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ single crystals (a) 2.1 with $x=0.075$ and (b) 1 with $x=0.1$. The graph above panel (a) shows a profilometer measurement upon crossing the step that traverses the image from top to bottom; full vertical scale is 2.5 $\mu$m. The width $u$ of the Meissner belt behind the step is also indicated in (a).

Figure 4

Delaunay triangulation of the vortex structure of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ single crystals (a) 2.1 with $x=0.075$ and (b) 2 with $x=0.1$. The blue dots represent vortices with 6 nearest neighbors while red ones represent vortices with a different coordination number. The insets show the respective Fourier transforms of the vortex positions.

Figure 5

Representation of vortex lines near a surface step under field-cooled conditions.

Figure 6

Left-hand panels: Normalized color-coded maps of the vortex interaction energy calculated from the images of Fig. 3 in Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ single crystals with (a) $x=0.075$ and (b) $x=0.1$. Right-hand panels: Histograms of the normalized interaction-energy for (c) $x=0.075$ and (d) $x=0.1$.

Figure 7

Color-coded maps of the modulus of the individual vortex pinning force per unit length, calculated from the images in Fig. 3 for Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ single crystals with (a) $x=0.075$ and (b) $x=0.1$. (c) and (d) represent the pinning force distribution for $x=0.075$ and $x=0.1$, respectively.

Figure 8

Critical-current densities in our Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$ crystals. (a) Temperature dependence of the low-field $j_c$ of crystals 1 ($x=0.1$) and 2.1 ($x=0.075$), as obtained from MOI. Error bars represent the dispersion of $j_c$ within a given crystal. (b) Field dependence of $j_c$ for crystal 2.1, obtained from magnetic hysteresis measurements using a SQUID magnetometer. Straight lines indicate fits with Eq. (6); see Sec. 4.