Superconducting gap renormalization around two magnetic impurities: From Shiba to Andreev bound states

We study the renormalization of the gap of an $s$-wave superconductor in the presence of two magnetic impurities. For weakly bound Shiba states, we analytically calculate the part of the gap renormalization that is sensitive to the relative orientation of the two impurity spins. For impurities with a strong exchange coupling to the conduction electrons, we solve the gap equation self-consistently by numerics and find that the subgap Shiba state turns into a supragap Andreev state when the local gap parameter changes sign under the impurities.

Figure 1

The energy spectrum (black, green, and brown dots) and the size of the superconducting gap ${\mathrm{\Delta }}_1$ directly under the impurity (red dots) for a square lattice of size $17a\times 13a$ with (a) a single impurity and (b) two identical impurities (${\overline{J}}_1={\overline{J}}_2$) as a function of exchange ${\overline{J}}_1/t$ found numerically. ${\mathrm{\Delta }}_1$ gets suppressed with increasing ${\overline{J}}_1$ and an effective $\text{S}-{\text{S}}^{\prime }$ junction builds up. After the phase transition at the critical value ${\overline{J}}_{1c}$, ${\mathrm{\Delta }}_1$ changes its sign giving rise to a $\pi$ junction. Subgap Shiba states (green dots) evolve into supragap Andreev states (brown dots) at values of ${\overline{J}}_{\mathrm{SA}}$ where the bound-state energy coincides with ${\mathrm{\Delta }}_1$. (b) For two impurities, separated by a distance $x_{12}/a=4$, the hybridization between the two bound states lifts their degeneracy. There are thus two phase transitions, one at ${\overline{J}}_{1c}$ and one at ${\overline{J}}_{2c}$, where these states become occupied, accompanied by jumps in bound-state energies and, hence, in ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$. There are also two distinct Shiba to Andreev state transitions. The gap between the impurities (blue dots) is suppressed for small ${\overline{J}}_1$, when the two bound states have finite overlap, but is restored at larger ${\overline{J}}_1$; see also Fig. 4. For corresponding wave function plots, see Fig. 2. The chosen parameters are ${\mathrm{\Delta }}_0/t=0.4$ and $\mu /t=0.9$.

Figure 2

The wave functions ${\left|\psi \right|}^2$ of the bound states in the case of a single impurity [(a), (c)] and of two identical impurities [(b), (d)], $J_1=J_2=J$, found in the tight-binding model on a lattice of size $17a\times 17a$ by self-consistent numerical diagonalization. For two impurities, separated by a distance $x_{12}/a=4$, the hybridization between the two bound states lifts their degeneracy. Before the phase transition with $J/t=2$ [(a), (b)], the Shiba bound state is well localized directly under the corresponding impurity. The Andreev bound state occurring after the phase transition at stronger values of the exchange coupling, $J/t=4$ [(c), (d)], is delocalized over the entire region where the superconducting gap is suppressed, with a maximum wave function amplitude between the impurities; see (d). The chosen parameters are ${\mathrm{\Delta }}_0/t=0.4$ and $\mu /t=0.9$.

Figure 3

Sketch of the renormalization of the superconducting gap, $\delta \mathrm{\Delta }\left(\mathbf{r}\right)=\mathrm{\Delta }\left(\mathbf{r}\right)-{\mathrm{\Delta }}_0$, in the presence of two spin impurities localized at positions ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ along the $\widehat{x}$ axis connecting the two impurities. The sketch shows the envelopes of three different contributions to the gap renormalization (see text).

Figure 4

The cross section of the spatial profile of the gap $\mathrm{\Delta }$ for two impurities located on the $x$ axis for fixed ${\overline{J}}_2/t=2.3$ and varying ${\overline{J}}_1:{\overline{J}}_1/t=1$ (yellow dotted line), ${\overline{J}}_1/t=2$ (green dotted dashed line), ${\overline{J}}_1/t=2.3$ (red dashed line), ${\overline{J}}_1/t=4$ (blue solid line). The gap under the first impurity is increasingly suppressed with increasing exchange coupling: After the phase transition ${\overline{J}}_1>{\overline{J}}_c$, $\pi$ junctions (negative gap value) under the impurity arise accompanied by restoration of the gap between the two impurities. The gap under the second impurity is suppressed the most when ${\overline{J}}_1\approx {\overline{J}}_2$ since then the two bound states hybridize strongly. For wave function plots, see Fig. 2. All parameters are the same as in Fig. 1.