Scaling of Yu-Shiba-Rusinov energies in the weak-coupling Kondo regime

The competition of the free spin state of a paramagnetic impurity on a superconductor with its screened counterpart is characterized by the energy scale of Kondo screening $k_B{T}_K$ compared to the superconducting pairing energy $\mathrm{\Delta }$. When the experimental temperature suppresses Kondo screening, but preserves superconductivity, i.e., when $\mathrm{\Delta }\mathrm{/}{k}_B>T>T_K$, this description fails. Here, we explore this temperature range in a set of manganese phthalocyanine molecules decorated with ammonia (MnPc-NH${}_3$) on Pb(111). We show that these molecules suffice the required energy conditions by exhibiting weak-coupling Kondo resonances. We correlate the energy of the Yu-Shiba-Rusinov (YSR) bound states inside the superconducting energy gap with the intensity of the Kondo resonance. We reveal that the bound state energy follows the expectations for a classical spin on a superconductor. This finding is important in view of many theoretical predictions using a classical spin model, in particular for the description of Majorana bound states in magnetic nanostructures on superconducting substrates.