Vortex state in a Fulde-Ferrell-Larkin-Ovchinnikov superconductor based on quasiclassical theory

We investigate the vortex state with Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) modulations suggested for a high-field phase of $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$. On the basis of quasiclassical Eilenberger theory, we calculate the three-dimensional structure of pair potentials, internal magnetic fields, paramagnetic moments, and electronic states for $s$-wave and $d$-wave pairings comparatively. The $\pi$-phase shift of the pair potential at the FFLO nodal plane or at the vortex core induces sharp peak states in the local density of states and enhances the local paramagnetic moment. We also discuss the NMR spectrum and neutron scattering as methods to detect the FFLO structure.

Figure 1

Configurations of the vortex lines and the FFLO nodal planes are schematically presented in the $x\text{−}z$ plane including vortex lines (a) and in the $x\text{−}y$ plane (b). The vortex distance is $a$ in the $x$ direction, and the distance between the FFLO nodal planes is $L∕2$. The hatched region indicates the unit cell. In (a), along the trajectories presented by “$0\to \pi$,” the pair potential changes the sign (+ → −) across the vortex line or across the FFLO nodal plane, due to the $\pi$-phase shift of the pair potential. Along the trajectory presented by “$0\to 2\pi$,” the sign of the the pair potential does not change (+ → +) across the intersection point of the vortex line and the FFLO nodal plane, since the phase shift is $2\pi$. In (b), ● indicates the vortex center. ${\mathbf{u}}_1-{\mathbf{u}}_2$ and ${\mathbf{u}}_2$ are unit vectors of the vortex lattice.

Figure 2

(Color online) Spatial structure of the FFLO vortex state in the $x\text{−}z$ plane at $\overline{B}=0.2B_0$, $T=0.1T_c$, and $L=50R_0$. (a) Amplitude of the pair potential $\mid \Delta \left(\mathbf{r}\right)\mid$. (b) Paramagnetic moment $M_{\mathrm{para}}\left(\mathbf{r}\right)$. (c) Internal magnetic field $B_z\left(\mathbf{r}\right)$. The upper panels show the spatial variation within a unit cell [hatched region in Fig. 1a] for $s$-wave pairing. The middle panels present the profiles along the path UNCVU shown in Fig. 1a for $s$-wave pairing. The lower panels are the same as the middle panels, but for $d$-wave pairing. The solid (dashed) lines are for the case when the magnetic field is applied to the antinodal (nodal) direction.

Figure 3

(Color online) Spectrum of the LDOS for up-spin electrons $N_{\uparrow }(E,\mathbf{r})∕N_0$ (solid lines) and for down-spin electrons $N_{\downarrow }(E,\mathbf{r})∕N_0$ (dashed lines) in $s$-wave pairing. $\overline{B}=0.2B_0$, $T=0.1T_c$, and $L=50R_0$. From top panel to bottom, we present the LDOS at positions $U$, $V$, $N$, and $C$. Their locations are shown in Fig. 1a.

Figure 4

(Color online) The same as Fig. 3, but in $d$-wave pairing. (a) $\overline{\mathbf{B}}$ is applied to the antinodal direction. (b) $\overline{\mathbf{B}}$ is applied to the nodal direction.

Figure 5

(Color online) FFLO vortex structure at $\overline{B}=0.15B_0$, $T=0.1T_c$, and $L=50R_0$ in $s$-wave pairing. (a) Spectrum of the LDOS for up-spin electrons $N_{\uparrow }(E,\mathbf{r})∕N_0$ (solid lines) and for down-spin electrons $N_{\downarrow }(E,\mathbf{r})∕N_0$ (dashed lines) at position $C$. (b) Paramagnetic moment $M_{\mathrm{para}}\left(\mathbf{r}\right)$. (c) Internal field $B_z\left(\mathbf{r}\right)$. The upper panels of (b) and (c) show the spatial variation within a unit cell. The lower panels present the profile along the path UNCVU.

Figure 6

(Color) Spectral evolution of the LDOS for up-spin electrons along the path CVUNC shown in Fig. 1a. Density plot (a) and stereographic view (b) of $N_{\uparrow }(E,\mathbf{r})∕N_0$ are presented in $d$-wave pairing, when the magnetic field $\overline{B}=0.2B_0$ is applied to the antinodal direction. $T=0.1T_c$ and $L=50R_0$.

Figure 7

(Color online) (a) Distribution function of the paramagnetic moment. We show $P\left(M\right)$ as a function of $M_{\mathrm{para}}$. (b) Distribution function of the internal magnetic field. We show $P\left(B\right)$ as a function of $B_z$. The solid (dashed) lines are for the vortex state with (without) the FFLO modulation. $\overline{B}=0.2B_0$, $T=0.1T_c$, and $L=50R_0$. The top panels are for $s$-wave pairing. The middle (bottom) panels are for $d$-wave pairing when the magnetic field is applied to the antinodal (nodal) directions. The heights of $P\left(M\right)$ and $P\left(B\right)$ are scaled so that $\int P\left(M\right)dM=\int P\left(B\right)dB=1$.

Figure 8

(Color online) Structure of paramagnetic moment (a) and internal magnetic field (b) when $L∕R_0=70$ (solid lines) and 30 (dashed lines) in $d$-wave pairing when the magnetic field is applied to the antinodal direction. $\overline{B}=0.2B_0$ and $T∕T_c=0.1$. The upper panels show the distribution function $P\left(M\right)$ or $P\left(B\right)$. The lower panels show the profile of $M_{\mathrm{para}}\left(\mathbf{r}\right)$ or $B_z\left(\mathbf{r}\right)$ along the path UNCVU.

Figure 9

(Color online) Structure of paramagnetic moment (a) and internal magnetic field (b) at $T∕T_c=0.1$ (solid lines) and 0.2 (dashed lines) in $d$-wave pairing when the magnetic field is applied to the antinodal direction. $\overline{B}=0.2B_0$ and $L=70R_0$. The upper panels show the distribution function $P\left(M\right)$ or $P\left(B\right)$. The lower panels show the profile of $M_{\mathrm{para}}\left(\mathbf{r}\right)$ or $B_z\left(\mathbf{r}\right)$ along the path UNCVU.

Figure 10

Factor ${\mid F_{h,k,l}\mid }^2$ for the intensity of the neutron scattering, calculated from the internal magnetic field $B_z\left(\mathbf{r}\right)$ (a) or from the paramagnetic moment $M_{\mathrm{para}}\left(\mathbf{r}\right)$ (b) in $d$-wave pairing when the magnetic field is applied to the antinodal direction. ${\mid F_{h,k,l}\mid }^2$ is plotted as a function of $l$ for $(h,k)=(1,0)$, (1,1), and (2,0). In (b), we also show ${\mid F_{h,k,l}\mid }^2$ for $(h,k)=(0,0)$. $\overline{B}=0.2B_0$, $T=0.1T_c$, and $L=70R_0$. Since the period of the FFLO modulation is $L∕2$ for $B_z\left(\mathbf{r}\right)$ and $M_{\mathrm{para}}\left(\mathbf{r}\right)$ along the $z$ direction, the diffraction peaks appear at spots with even $l$.