Effect of field-dependent core size on reversible magnetization of high-$\kappa$ superconductors

The field dependence of the vortex core size $\xi \left(B\right)$ is incorporated in the London model, in order to describe reversible magnetization $M(B,T)$ for a number of materials with large Ginzburg-Landau parameter $\kappa$. The dependence $\xi \left(B\right)$ is directly related to deviations in $M\left(\ln B\right)$ from linear behavior prescribed by the standard London model. A simple method to extract $\xi \left(B\right)$ from the magnetization data is proposed. For most materials examined, $\xi \left(B\right)$ so obtained decreases with increasing field and is in qualitative agreement both with behavior extracted from $\mu \mathrm{SR}$ and small-angle neutron-scattering data and with that predicted theoretically.

Figure 1

(Color online) An example of $M\left(H\right)$ for $\mathrm{Y}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ at $T=9\mathrm{K}$. The main plot shows both up- and down-field scans and the irreversibility field. The left inset illustrates how the normal-state paramagnetic contribution is subtracted with the result shown in the right inset.

Figure 2

(Color online) The magnetization $M\left(\ln B\right)$ for $\mathrm{Y}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ at $T=2$, 5, 9, and $12\mathrm{K}$. The upper critical fields $H_{c2}$ are the positions of kinks in $M\left(\ln B\right)$ not shown in the figure. The solid curves are obtained by fitting the data to Eq. (4) with the fitting parameters $M_0$ (shown in the inset of Fig. 3), $\eta$, and $\alpha$.

Figure 3

(Color online) $\xi \left(B\right)$ extracted from the data of Fig. 2 for $\mathrm{Y}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ with the help of Eq. (8). The inset shows $M_0\left(T\right)$ extracted from the fits of Fig. 2.

Figure 4

(Color online) (a) $M\left(B\right)$ for $\mathrm{Lu}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ for $T=7$, 10, and $12\mathrm{K}$. (b) $\xi \left(B\right)$ corresponding to the graphs of the upper panel.

Figure 5

(Color online) (a) $M\left(B\right)$ for $\mathrm{Lu}{\left({\mathrm{Ni}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{B}}_2\mathrm{C}$ with $x=0$, 3, and 6%. (b) $\xi \left(B\right)$ corresponding to the graphs of the upper panel.

Figure 6

(Color online) (a) $M\left(B\right)$ for $\mathrm{Nb}{\mathrm{Se}}_2$. (b) $\xi \left(B\right)$ corresponding to the graphs of the upper panel.

Figure 7

(Color online) (a) $\mathrm{Mg}{\mathrm{B}}_2$, $T=4.6$, 15, and $25\mathrm{K}$. (b) Corresponding $\xi \left(B\right)$.

Figure 8

(Color online) (a) Magnetization of the $\mathrm{Mg}{\mathrm{B}}_2$ crystal in applied field at 45° to the $c$ axis. (b) Corresponding coherence length. Note the difference in the behavior of ${\xi }^{*}\left(b\right)$ from the case of the field along $c$ of Fig. 7.

Figure 9

(Color online) Magnetization of the ${\mathrm{V}}_3\mathrm{Si}$ single crystal at $T=13$, 14, 15, and $16\mathrm{K}$.

Figure 10

(Color online) The field dependence of $\xi$ calculated microscopically for $T∕T_c=0.1$ and for a few scattering parameters ${\xi }_0∕\ell$ where ${\xi }_0$ is zero-$T$ BCS coherence length and $\ell$ is the mean-free path for scattering on nonmagnetic impurities.

Figure 11

(Color online) $\xi \left(b\right)∕{\xi }_{c2}$ vs $b=B∕H_{c2}$ extracted from the flux-flow resistivity data as explained in the text.