Effects of nonmagnetic disorder on the energy of Yu-Shiba-Rusinov states

We study the sensitivity of Yu-Shiba-Rusinov states, bound states that form around magnetic scatterers in superconductors, to the presence of nonmagnetic disorder in both two and three dimensional systems. We formulate a scattering approach to this problem and reduce the effects of disorder to two contributions: disorder-induced normal reflection and a random phase of the amplitude for Andreev reflection. We find that both of these are small even for moderate amounts of disorder. In the dirty limit in which the disorder-induced mean free path is smaller than the superconducting coherence length, the variance of the energy of the Yu-Shiba-Rusinov state remains small in the ratio of the Fermi wavelength and the mean free path. This effect is more pronounced in three dimensions, where only impurities within a few Fermi wavelengths of the magnetic scatterer contribute. In two dimensions the energy variance is larger by a logarithmic factor because impurities contribute up to a distance of the order of the superconducting coherence length.

Figure 1

Sketch of the model used in this work and the relevant length scales. A magnetic impurity with a size comparable to the Fermi wavelength ${\lambda }_F$ is placed inside a superconductor. The coherence length ${\xi }_0$ corresponds to the size of Cooper pairs (orange) that form the superconducting ground state in the absence of disorder. Adding nonmagnetic disorder (gray circles) introduces a mean free path $\ell$.

Figure 2

Sketch of the numerical method in two dimensions. The disordered superconductor is cut into thin circular slices (blue), with infinitesimally thin clean, metallic shells (white) in between. The scattering matrices of the slices are calculated in the narrow limit and combined into the full scattering matrix $S\left(\varepsilon \right)$, which describes the reflection of spherical symmetric waves (black) that are propagating from the magnetic impurity into the bulk.

Figure 3

YSR energy shift (solid lines) versus disorder cutoff length $r_N$ in two dimensions (left panel) and three dimensions (right panel), obtained from the numerical scattering approach. We choose $\xi /\ell =10$ with $k_{\mathrm{F}}\ell =100$. When including the cutoff $r_N$, our perturbative approach (dashed lines) fits well with the numerical results. The shaded region shows the numerical standard error.

Figure 4

YSR energy variance versus $k_{\mathrm{F}}{\xi }_{\varepsilon }$ in two dimensions. The data points are for ${\xi }_{\varepsilon }/\ell =0.5$ (triangles), 1 (circles), and 2 (squares). The scattering phase shifts ${\varphi }_{\uparrow }=-{\varphi }_{\downarrow }$ are chosen such that YSR energy in the absence of disorder is $\varepsilon /\mathrm{\Delta }=0$ (left) and $\varepsilon /\mathrm{\Delta }=0.37$ (right). The solid lines show the lowest-order perturbation theory result from Eq. (17). Error bars are of the size of the markers.

Figure 5

YSR energy variance in three dimensions, as a function of $k_{\mathrm{F}}{\xi }_{\varepsilon }$ at fixed ratios ${\xi }_{\varepsilon }/\ell$. The parameters are the same as in Fig. 4. The dashed and solid curves give the perturbative result (41) and its leading term, respectively.

Figure 6

Comparison of the two contributions $\delta r$ and $\delta \eta$ to the superconductor scattering matrix $S\left(\varepsilon \right)$, in two (left) and three (right) dimensions. See Eq. (26). We choose a ratio ${\xi }_0/\ell =1$ and energy $\varepsilon /\mathrm{\Delta }=0.15$. The contribution from normal reflection dominates in two as well as in three dimensions.

Figure 7

Energy variance over mean-free path at fixed $k_{\mathrm{F}}$ and ${\xi }_0$. The markers show the numerical results in two dimensions (top) and three dimensions (bottom), with $k_{\mathrm{F}}{\xi }_0=50$ (blue triangles) and $k_{\mathrm{F}}{\xi }_0=100$ (green circles). The lines correspond to the perturbative results derived in the main text. The scattering phase shifts ${\varphi }_{\uparrow }=-{\varphi }_{\downarrow }$ of the magnetic impurity are chosen such that, in the absence of disorder, a YSR state forms at $\varepsilon /\mathrm{\Delta }=0$.

Figure 8

Convergence of the energy variance with disorder cutoff $r_N$ in the dirty limit and in two dimensions. The parameters are $k_{\mathrm{F}}{\xi }_0=300$ and ${\xi }_0/\ell =10$, with the same magnetic impurity parameters as in Fig. 7.