Strain-induced intervortex interaction and vortex lattices in tetragonal superconductors

In superconductors with strong coupling between superconductivity and elasticity manifested in a strong dependence of transition temperature on pressure, there is an additional contribution to intervortex interactions due to the strain field generated by vortices. When vortex lines are along the $c$ axis of a tetragonal crystal, a square vortex lattice (VL) is favored at low vortex densities, because the vortex-induced strains contribution to the intervortex interactions is long range. At intermediate magnetic fields, the triangular lattice is stabilized. The triangular lattice evolves to the square lattice upon increasing magnetic field, and eventually the system locks to the square structure. We argue, however, that as magnetic field approaches the upper critical field $H_{c2}$ the elastic intervortex interactions disappear faster than the standard London interactions, so that VL should return to the triangular structure. Our results are compared to VLs observed in the heavy fermion superconductor ${\mathrm{CeCoIn}}_5$.

Figure 1

The equilibrium apex angle $\beta$ of the rhombic unit cell for vortex lines along $c$. Vortices form a square lattice both at low and high fields, and the VL is triangular at intermediate fields. For comparison, the results for two different $G_e$'s are displayed. The field at the transition from the triangular to square VL at higher field increases when $G_e$ is reduced.

Figure 2

The apex angle $\beta$ of the rhombic unit cell for $\mathit{B}\perp \mathit{c}$. Insets are the sketches of the corresponding Bragg peaks of maximum intensity in the momentum space at low and high fields. Here ${\eta }_X=0, {\eta }_Y=0.002, \overline{\gamma }=2, \gamma =2$, and $\overline{\kappa }=30$.

Figure 3

The apex angle $\beta$ of the rhombic unit cell when $\mathit{B}\perp \mathit{c}$. Inset is a sketch of the corresponding Bragg peaks of maximum intensity in the momentum space. Here ${\eta }_X=0.001, {\eta }_Y=0, \overline{\gamma }=2, \gamma =2$, and $\overline{\kappa }=30$.