The origin of Cooper pairing in high-temperature superconductors, such as the copper-oxide and iron-based system, is still under debate. High transition temperatures together with unconventional pairing states support the picture of an electronic pairing glue in the cuprates, where superconductivity is mediated by collective bosonic excitations of the electron fluid. In other materials, most importantly iron based systems with only hole or only electron pockets, the microscopic origin is hotly debated. Scanning tunneling microscopy (STM) has been shown to be a powerful experimental probe to detect electronic excitations and further allows to deduce some fingerprints of bosonic collective modes. Here, we demonstrate that the inclusion of inelastic tunnel events is crucial for the interpretation of tunneling spectra and allows to directly probe bosonic excitations via STM. We develop a model describing both the elastic tunneling current, which displays the electronic spectral function, and the inelastic current, that contains the information about the bosonic spectrum, in the superconducting state. Adopting this extended tunneling formalism we can naturally reproduce the tunneling spectra of various unconventional superconductors and trace the occurring features back to an opening of a spin gap in the superconducting state. More generally, our approach is a strong argument in favour of a collective mode mediating the pairing state in particular in iron-based systems. In particular, we conclude that the debated pairing mechanism in LiFeSe is also of electronic origin with sign-changing pairing symmetry.
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