Quasiclassical analysis of vortex lattice states in Rashba noncentrosymmetric superconductors

Vortex lattice states occurring in noncentrosymmetric superconductors with a spin-orbit coupling of Rashba type under a magnetic field parallel to the symmetry plane are examined by assuming the $s$-wave pairing case and in an approach combining the quasiclassical theory with the Landau level expansion of the superconducting order parameter. The resulting field-temperature phase diagrams include not only a discontinuous transition but a continuous crossover between different vortex lattice structures, and, further, a critical end point of a structural transition line is found at an intermediate field and a low temperature in the present approach. It is pointed out that the strange field dependence of the vortex lattice structure is a consequence of that of its anisotropy stemming from the Rashba spin-orbit coupling, and that the critical end point is related to the helical phase modulation peculiar to these materials in the ideal Pauli-limited case. Furthermore, calculation results on the local density of states detectable in STM experiments are also presented.

Figure 1

Fermi surfaces (FS1 and FS2) under an in-plane magnetic field. The gray arrows indicate the direction of the spin fixed by the spin-orbit coupling of Rashba type. Each surface shifts oppositely from $\mathrm{\Gamma }$ by $\pm {\mathit{Q}}_0$. The vector ${\mathit{Q}}_0$ is defined by Eq. (20) and in Appendix pp1.

Figure 2

Definition of the parameters $\nu$ and $\lambda$, which represents the shape of the vortex lattice. Here, ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are principal lattice vectors of the lattice.

Figure 3

Dependence of the $H_{c2}$ curve on the number of LLs incorporated in the calculation in the $\delta N=0$ case, where $n_{\max }$ is the index of the highest LL incorporated.

Figure 4

Field dependence of eigenvalues of the modes obtained by diagonalizing $F_2$ at $T=0.1T_c$ when $\delta N=0$. The modes with the lowest two eigenvalues (the red and blue lines) are nearly degenerate for $H>0.4H_{\mathrm{orb}}^{2\mathrm{D}}$ with each other and cross at $H=0.48H_{\mathrm{orb}}^{2\mathrm{D}}$.

Figure 5

Resulting phase diagram and vortex lattice states appearing when $\delta N=0$. The first-order structural transition (FOST) points are numerically determined, and the line connecting between them is a guide to the eye. Phases I and II are stretched triangular lattice phases. Phase III is a modulated triangular lattice phase. (B) shows that the modulation is along the direction of the shift of the Fermi surfaces. The figures (C), (E), and (F) indicate that the lattice is compressed along the $x$ axis with increasing field. In (A), the spatial modulation in the region surrounded by the broken white circle does not imply the presence of an additional vortex there. Here, ${\mathrm{\Delta }}_0$ is the magnitude of the order parameter at each temperature in the absence of magnetic fields.

Figure 6

Field dependence of $\nu$, which is the lattice spacing along the $x$ axis (see Fig. 2), of the lattice in phase I of Fig. 5. Note that $\nu$ is almost proportional to $1/\sqrt{H}$ for $H>0.3H_{\mathrm{orb}}^{2\mathrm{D}}$ at $T=0.3T_c$.

Figure 7

Dependence of the $H_{c2}$ curve on the number of LLs incorporated in calculation when $\delta N=0.1$. As in the $\delta N=0$ case, we have assumed $n_{\max }=7$.

Figure 8

Field dependence of the eigenvalues of the modes obtained by diagonalizing $F_2$ at $T=0.1T_c$ when $\delta N=0.1$. The mode with the lowest eigenvalue (red line) is dominant at any field.

Figure 9

Phase diagram and vortex lattice states when $\delta N=0.1$. The first-order structural transition (FOST) points are numerically determined, and the line connecting between them is a guide to the eye. Phases I and II are stretched triangular lattice phases, while phase III is a rectangular lattice one. Furthermore, phase IV is a modulated triangular lattice phase. Figure (A) shows that the modulation develops along the direction of the shift of the Fermi surfaces with the field increasing. (D), (E), and (F) can be continuously transformed to one another circumventing the critical end point (CEP).

Figure 10

Field dependence of $\nu$, the lattice spacing parallel to the $x$ axis, of the states in phase II of Fig. 9. The upturn of the width for $0.4H_{\mathrm{orb}}^{2\mathrm{D}}<H<0.5H_{\mathrm{orb}}^{2\mathrm{D}}$ means that the field-induced compression parallel to the $x$ axis is weakened in higher fields. As for the definition of $\nu$, see Fig. 2.

Figure 11

Panels (a) and (b) are the images of the local density of states (LDOS) at the excitation energy $E=0$ of the states (F) and (D), respectively, in Fig. 9. The smearing factor $\eta$ here is set at 0.1 and 0.15, respectively, in units of $2\pi T_c$. Panels (c) and (d) are the graphs of the energy dependence of the LDOS at the vortex center of the same state as in (a) and (b), respectively, with different smearing factors $\eta$. In (c), the spectrum is stable around $\eta /2\pi T_c=0.1$ and has slightly split peaks. In (d), the spectrum is stable around $\eta /2\pi T_c=0.15$ and has widely split peaks.

Figure 12

Two paths of the complex integration $\oint$.

Figure 13

Relations among the three vectors in Eq. (E4). ${\mathrm{FS}}_0$ denotes the Fermi surface of the bare band, while ${\mathrm{FS}}_a$'s ($a=1,2$) express the two Fermi surfaces split by the ASOC of Rashba type. In the figure, ${\mathrm{FS}}_a$ is represented by that of band 2.