Tunneling conductance and local density of states in time-reversal symmetry breaking superconductors under the influence of an external magnetic field

We consider different effects that arise when time-reversal symmetry breaking superconductors are subjected to an external magnetic field, thus rendering the superconductor to be in the mixed state. We focus in particular on two time-reversal symmetry breaking order parameters which are believed to be realized in actual materials: $p+i{p}^{\prime }$ wave and $d+is$ or $d+i{d}^{\prime }$ wave. The first-order parameter is relevant for ${\text{Sr}}_2{\text{RuO}}_4$, while the latter order parameters have been suggested to exist near surfaces in some of the high-$T_c$ cuprates. We investigate the interplay between surface states and vortex states in the presence of an external magnetic field and their influence on both the tunneling conductance and the local density of states. Our findings may be helpful to experimentally identify the symmetry of unconventional time-reversal symmetry breaking superconducting states.

Figure 1

(Color online) Plot of the surface states spectrum for (a) chiral $p+ip$-wave, (b) $d+is$-wave, and (c) $d+id$-wave superconductors. Spectrum in zero magnetic field is shown by solid lines. Blue (dash-dotted) and red (dotted) lines correspond to the spectrum transformation due to magnetic field directed along and opposite the $z$ axis correspondingly.

Figure 2

(a) Sketch of QP trajectories forming surface and vortex states and qualitative plot of spectrum transformation due to the interaction of surface and vortex states in the case of a chiral $p$-wave superconductor; vorticity and chirality have (b) equal and (c) opposite values.

Figure 3

Plot of the normalized zero-energy LDOS $N\left(0\right)$ for (a) $p$-wave superconductor with $\chi =1$, (b) $d+is$-wave case with ${\Delta }_s=0.4{\Delta }_0$, and (c) $d+id$-wave case with ${\Delta }_d=0.4{\Delta }_0$. Dashed lines are guides for eyes. The vertical ones denote positions of LDOS peaks and the horizontal ones correspond to the level of normal-metal DOS $N_0$.

Figure 4

(Color online) Plot of the normalized zero-energy LDOS $N\left(0\right)$ in the presence of a vortex near the surface of a chiral $p$-wave superconductor. (a) and (c) correspond to equal vorticity and chirality and (b) and (d) correspond to opposite vorticity and chirality. The distance from vortex to the surface is $a=2\xi$ for (a) and (b) and $a=\xi$ for (c) and (d).

Figure 5

(a) Plot of the normalized zero-energy LDOS at the point on the surface which is closest to the vortex core. Different curves correspond to different vorticities. (b) Plot of the vortex-induced conductance in chiral $p+ip$-wave superconductor for equal and opposite values of vorticity and chirality. The strength of interface barrier is $Z=5$. Large-distance asymptotes for $N$ and $G$ are shown by dash lines.

Figure 6

(Color online) Plot of the normalized zero-energy LDOS profile $N\left(0\right)$ in the presence of vortex near the surface of chiral $d+is$-wave superconductor with ${\Delta }_s=0.2{\Delta }_0$. (a)–(d) correspond to different vortex orientations with respect to the $z$ axis. The distance from vortex to the surface is $a=4\xi$ for (a) and (b) and $a=2\xi$ for (c) and (d).

Figure 7

(Color online) Plot of the normalized zero-energy LDOS profile $N\left(0\right)$ in the presence of vortex near the surface of chiral $d+id$-wave superconductor with ${\Delta }_d=0.2{\Delta }_0$ (a)–(d) correspond to different vortex orientations with respect to the $z$ axis. The distance from vortex to the surface is $a=4\xi$ for (a) and (b) and $a=2\xi$ for (c) and (d).

Figure 8

Plots of the vortex-induced conductance in cases of (a) chiral $d+is$ superconductor for ${\Delta }_s=0.2{\Delta }_0$ and (b) $d+id$ superconductor for ${\Delta }_d=0.2{\Delta }_0$. The strength of interface barrier is $Z=5$. Different curves on each plot correspond to different vortex orientations with respect to the $z$ axis.