Meissner response of a bulk superconductor with an embedded sheet of reduced penetration depth

We calculate the change in susceptibility resulting from a thin sheet with reduced penetration depth embedded perpendicular to the surface of an isotropic superconductor, in a geometry applicable to scanning superconducting quantum interference device microscopy, by numerically solving Maxwell’s and London’s equations using the finite-element method. The predicted stripes in susceptibility agree well in shape with the observations of Kalisky et al. [Phys. Rev. B 81, 184513 (2010)] of enhanced susceptibility above twin planes in the underdoped pnictide superconductor $\text{Ba}{\left({\text{Fe}}_{1-x}{\text{Co}}_x\right)}_2{\text{As}}_2$. By comparing the predicted stripe amplitudes with experiment and using the London relation between penetration depth and superfluid density, we estimate the enhanced Cooper-pair density on the twin planes, and the barrier force for a vortex to cross a twin plane. Fits to the observed temperature dependence of the stripe amplitude suggest that the twin planes have a higher critical temperature than the bulk, although stripes are not observed above the bulk critical temperature.

Figure 1

Scanning SQUID susceptibility image of an underdoped $\left(x=0.051,T_c=18.25\text{K}\right)$ single crystal of $\text{Ba}{\left({\text{Fe}}_{1-x}{\text{Co}}_x\right)}_2{\text{As}}_2$ at $T=17\text{K}$. The arrow indicates the direction along which the stripes were averaged to form the cross section plotted in Fig. 2.

Figure 2

(Color online) (a) Averaged cross section along the stripes direction of the bottom 3/5 of the data of Fig. 1 (symbols, dashed line). Subtracting out the piecewise linear background [dotted line in (a)] results in the symbols and dashed line in (b). The solid line in (b) is modeling as described in the text.

Figure 3

(Color online) Modeling geometry: (a) three-dimensional rendering; (b) top view; (c) side view; and (d) expanded view of the field coil. A superconductor occupying the half space $z<0$ is assumed to have a penetration depth ${\lambda }_b$ except for a sheet of width $w$ with penetration depth ${\lambda }_s$ centered at $x=x_0$. The susceptometer field coil, represented by a torus with major radius $R$ and minor radius $a$ oriented with its plane parallel to the surface of the superconductor and centered at $x=0,y=0,z=z_0$, carries a current $I$. The susceptibility $\chi =\Phi /I$ is calculated by integrating the $z$ component of the resultant magnetic field over a circle of radius $r$ centered at $\left(0,0,z_0\right)$ to obtain the flux $\Phi$. For the modeling in this paper we used $R=8.85\mu \text{m}$, $z_0/R=0.17$, $a/R=0.05$, $r/R=0.25$. The volume of space modeled was of size $12\times 12\times 8R^3$.

Figure 4

(Color online) Calculated normalized supercurrent densities at $z=0$ for $w/R=0.2$ in the [(a) and (c)] $\widehat{x}$ and [(b) and (d)] $\widehat{y}$ directions for the sheet penetration depth ${\lambda }_s/R=0.2$ and [(a) and (b)] bulk penetration depth ${\lambda }_b/R=0.2$ and [(c) and (d)] ${\lambda }_s/R=0.1$, ${\lambda }_b/R=0.2$. The small scale inhomogeneities in the image are due to discretization error.

Figure 5

(Color online) Dependence of the stripe peak height $\Delta H_{z}R/I$ vs (a) $1/{\lambda }_s$ and (b) $w^{1/2}$, where $\Delta H_z$ is the difference between $x_0/R=0$ and $x_0/R=2$ of the $z$ component of the magnetic field at the center of the field coil induced by the field coil current $I$. The symbols are the numerical simulation and the dashed lines are linear fits.

Figure 6

(Color online) Plot of the dimensionless stripe susceptibility peak height ${\lambda }_s\Delta H_z{R}^{1/2}/I{w}^{1/2}$ vs $\left({\lambda }_b-{\lambda }_s\right)/R$ for various values of the reduced sheet width $w/R$ and sheet penetration depth ${\lambda }_s/R$. For large values of ${\lambda }_b-{\lambda }_s$ the susceptibility peak height is nearly independent of ${\lambda }_b$, indicating that the susceptibility peak height is proportional to $w^{1/2}/{\lambda }_s$, and therefore proportional to the square root of the two-dimensional sheet Cooper-pair density $N_s$. The susceptibility peak height approaches zero linearly as ${\lambda }_b\to {\lambda }_s$. The solid line is the empirical function, Eq. (1), used for the modeling of the temperature dependence of the stripe amplitude below.

Figure 7

(Color online) Plot of the temperature dependence of the stripe amplitude $\Delta \chi \left(T\right)$ (symbols). The solid lines are best fits to the data using the empirical formula, Eq. (1), and the two-fluid temperature dependence [Eq. (2), $p=4$] for the penetration depths, and different critical temperatures $T_{c,s}$ and $T_{c,b}$ for the sheet and bulk, respectively, for widths $w/R=0.5,0.2,0.1,0.05,0.02,0.01,0.005,0.002$, corresponding to the sheet width $w$ between $4.4\mu \text{m}$ and 17.6 nm. The best-fit values and uncertainties for ${\lambda }_s$ and $T_{c,s}/T_{c,b}$ are plotted in Fig. 8.

Figure 8

(Color online) Best-fit values and uncertainties (using a doubling of the best-fit value for ${\chi }^2$ as a criterion) for the parameters ${\lambda }_s/R$ and $T_{c,s}/T_{c,b}$ obtained by fitting the data of Fig. 7, for various assumed values for the sheet width $w$, assuming ${\lambda }_b\left(T=0\right)/R=0.04$.

Figure 9

(Color online) Best-fit values, and upper and lower bounds for the enhancement of the two-dimensional sheet Cooper-pair density as a function of sheet width from fits to experimental susceptibility stripe amplitudes.