Anisotropy and spin-fluctuation effects on the spectral properties of Shiba impurities

We theoretically consider a quantum magnetic impurity coupled to a superconductor and obtain the local density of states at the position of the impurity taking into account the effect of spin fluctuations and single-ion magnetic anisotropy. We particularly focus on the spectrum of subgap Yu-Shiba-Rusinov (YSR or Shiba) states induced by a quantum impurity with easy- or hard-axis uniaxial anisotropy. Although this is a relevant experimental situation in, e.g., magnetic adatoms on the surface of clean metals, it is customary that theoretical descriptions assume a classical-spin approximation which is not able to account for single-ion anisotropy and other quantum effects. Here, quantum fluctuations of the spin are taken into account in the equations of motion of the electronic Green's function in the weak-coupling limit and considerably modify the energy of the Shiba states compared to the classical-spin approximation. Our results point towards the importance of incorporating quantum fluctuations and anisotropy effects for the correct interpretation of scanning tunneling microscopy (STM) experiments.

Figure 1

(a) Diagram of the experimental setup. (b) Level spectrum for easy-axis anisotropy ($D>0$). This figure shows the spin fluctuations from the ground state of the impurity (states $m=\pm S$) to the first excited state ($m=\pm (S-1)$) at zero temperature. Here we defined $\overline{\mathrm{\Delta }}\equiv D(2S-1)$, see text.

Figure 2

YSR energy $E_{\text{YSR}}$ as a function of the spin impurity $S$ for $\alpha =0.1$ (black points), $\alpha =0.47$ (blue squares), and $\alpha =0.65$ (red triangles). Continuous lines are a guide to the eye. The classical Shiba energy independent of $S$ are showed as orange dashed horizontal lines for the corresponding values of $\alpha$. The arrow marks the putative quantum phase transition predicted by perturbative approach.

Figure 3

YSR subgap-state energy $E_{\text{YSR}}$ as a function of the anisotropy parameter $D$ for (a) $S$ half-integer and (b) $S$ integer. Here has been used the value $\alpha =0.5$.

Figure 4

Local density of states (LDOS) at the site of the impurity for (a) $D=0.1\mathrm{\Delta }$, (b) $D=0.3\mathrm{\Delta }$, (c) $D=0.5\mathrm{\Delta }$, and d) $D=0.7\mathrm{\Delta }$, at temperature $T=0.05\mathrm{\Delta }$. $\alpha =0.5$ and $S=1$. The arrows indicate the center of the YSR resonance. Inset: full spectral density for $D=0.1\mathrm{\Delta }$.

Figure 5

YSR subgap-state energy $E_{\text{YSR}}$ as a function of the temperature $T$, computed for $S=1$ and $\alpha =0.5$.