Torque theory of anisotropic superconductors with no phenomenological parameter in determining vortex core size

The contribution of the vortex core has been taken into account properly in constructing a torque theory for multiband superconductors. We employ the prescription for describing the internal magnetic field in the vortex lattice by Hao et al. and by Yaouanc et al. to derive a torque formula as a natural extension of a preceding London theory. In marked contrast with the preceding model, our formula does not contain a phenomenological parameter $\eta$, which prevents us from obtaining a true upper critical field $H_{\mathrm{c}2}$ by analyzing an experimental torque curve. The parameter $\eta$ was originally introduced to take care of the uncertainty in determining the vortex core size ${\xi }_v$. Furthermore, we reveal that the $\eta$ value is universally scaled by anisotropy $\gamma$, magnetic field $B$, and $H_{\mathrm{c}2}$ due to field dependence of ${\xi }_v$. This may revitalize the single-band Kogan model in combination with a universal function $\eta (\gamma ,B,H_{\mathrm{c}2})$ instead of a constant $\eta$.

Figure 1

The parameter $\eta$ can be scaled as a function of $\gamma /(B/H_{\mathrm{c}2}^{\left|\right|c})$ on a single curve, where $\gamma$ is the anisotropy parameter, $B$ is the applied field, and $H_{\mathrm{c}2}^{\left|\right|c}$ is the upper critical field parallel to the $c$ plane. Note that $\gamma H_{\mathrm{c}2}^{\left|\right|c}$ becomes the upper critical field $H_{\mathrm{c}2}^{\left|\right|\mathit{ab}}$ parallel to the $\mathit{ab}$ plane. The inset shows the torque curve of the London model (see the dashed line) of Eq. (7) for $\gamma =7$ and $B/H_{c2}^{\parallel c}=0.3$, where the phenomenological parameter $\eta$ is chosen so as to give the same peak angle ${\theta }_p$ with our model (see solid line) of Eq. (13).

Figure 2

The qualitative behavior of ${\eta }^{\text{'}}$ as a function of $1/b$ expected from the theory of Hao et al. (Ref. 15). The parameter ${\eta }^{\text{'}}$ is a major term in the uncertainty parameter $\eta ={\eta }^{\text{'}}\exp ({\eta }_c-1)$ as given by Kogan et al. (Ref. 9), ${\xi }_v$ is a vortex core size, and $b$ is the reduced field.

Figure 3

The multiband curves are obtained by Eq. (13) for several different combinations of ${\gamma }_{\lambda }$ and ${\gamma }_{\xi }$ (see text). The magnetic field is fixed as $H_{c2}^{\parallel c}/B=3.38$ (see Ref. 3). The torque $\tau$ of Eq. (13) is given in units of ${\Phi }_{0}BV/64{\pi }^2{\lambda }^2$, $B$ is the applied field, $H_{\mathrm{c}2}^{\left|\right|c}$ is the upper critical field parallel to the $c$ axis, and $p$ and $q$ are lattice indices in reciprocal space.

Figure 4

The torque curves using the London model Eq. (7) for several different combinations of ${\gamma }_{\lambda }$ and ${\gamma }_{\xi }$ in units of ${\Phi }_0\mathit{BV}/64{\pi }^2{\lambda }^2$ (see Figs. 1, 2, and 3 of Ref. 3 for comparisons). The magnetic field is fixed as $4e^2{H}_{c2}^{\parallel c}/B=100$ ($e\simeq 2.71$), $B$ is an applied field, and $H_{\mathrm{c}2}^{\left|\right|c}$ is an upper critical field parallel to the $c$ axis.