Vortex states in superconductors with strong Pauli-paramagnetic effect

Using quasiclassical theory, we analyze the vortex structure of strong-paramagnetic superconductors. There, induced paramagnetic moments are accumulated exclusively around the vortex core. We quantitatively evaluate the significant paramagnetic effect in the $H$ dependence of various quantities, such as low temperature specific heat, Knight shift, magnetization, and the flux line lattice (FLL) form factor. The anomalous $H$ dependence of the FLL form factor observed by the small angle neutron scattering in $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$ is attributable to the large paramagnetic contribution.

Figure 1

(Color online) The magnetic field dependence of the paramagnetic susceptibility $\chi \left(H\right)$ (solid lines) and the zero-energy DOS $\gamma \left(H\right)$ (dashed lines) at $T=0.1T_{\mathrm{c}}$ for various paramagnetic parameters $\mu =0.02$, 0.86, 1.7, and 2.6 in the $d$-wave (a) and $s$-wave (b) pairing cases. The insets show same data as a function of $H∕H_{\mathrm{c}2}$.

Figure 2

(Color online) Spatial structure of the pair potential (a), paramagnetic moment (b), and internal magnetic field (c) at $T=0.1T_{\mathrm{c}}$ and $\overline{B}=0.1B_0$, where $a=11.2R_0$, in $d$-wave pairing. The left panels show $\mid \Delta \left(\mathbf{r}\right)\mid$, $M_{\mathrm{para}}\left(\mathbf{r}\right)$, and $B\left(\mathbf{r}\right)$ within a unit cell of the square vortex lattice at $\mu =1.7$. The right panels show the profiles along the trajectory $r$ from the vortex center to the midpoints between nearest neighbor vortices. $\mu =0.02$, 0.86, 1.7, and 2.6.

Figure 3

(Color online) Local density of states at $r∕R_0=0$ (a), 0.8 (b), and 1.6 (c) from the vortex center towards the nearest neighbor vortex direction in $d$-wave pairing. Solid lines show $N_{\uparrow }(\mathbf{r},E)∕N_0$ for up-spin electrons, and dashed lines show $N_{\downarrow }(\mathbf{r},E)∕N_0$ at $\overline{B}=0.1B_0$. $\mu =1.7$ and $T=0.1T_{\mathrm{c}}$. (d) Spatial-averaged DOS at $\overline{B}∕B_0=0.1$ and 0.01 in $d$-wave pairing. Solid lines show $N_{\uparrow }\left(E\right)∕N_0$ for up-spin electrons, and dashed lines show $N_{\downarrow }\left(E\right)∕N_0$.

Figure 4

(Color online) Field dependence of FLL form factor $F_{1,0}$ for $\mu =0.02$, 0.86, 1.7, and 2.6 at $T=0.1T_{\mathrm{c}}$ in $d$-wave pairing. (a) ${\mid F_{1,0}\mid }^2$ is plotted as a function of $H$. The vertical axis is in logarithmic scale. (b) We plot ln ${\mid F_{1,0}\mid }^2$ as a function of $H∕H_{\mathrm{c}2}$, where $F_{1,0}$ is scaled so that $H_{\mathrm{c}2}$ for each $\mu$ corresponds to $5\mathrm{T}$. Open circles are experimental data in $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$ observed by SANS experiment (Ref. 34).

Figure 5

(a) Magnetic field dependence of magnetization $M_{\text{total}}$ for $\mu =1.7$ at $T∕T_{\mathrm{c}}=0.1$, 0.3, 0.5, 0.7, 0.9, and 1.0 (normal state) in $d$-wave pairing. (b) $M_{\text{total}}$ as a function of $T^2$ at $\overline{B}=0.01,0.02,0.03,\dots ,0.21$. (c) $H$ dependence of factor $\beta \left(H\right)$ at $\mu =0.02$ and 1.7.