# s+is superconductivity with incipient bands: doping dependence and STM signatures

## TL;DR

This paper investigates how doping affects superconductivity in a four-band model of iron-based superconductors, revealing a transition to an $s+is$ state with broken time-reversal symmetry and proposing STM signatures of this state.

## Contribution

It introduces a detailed analysis of doping-dependent superconductivity including incipient bands and BCS-BEC crossover effects, highlighting the emergence of an $s+is$ state.

## Key findings

- Transition from 0 to π phase difference with doping
- Broadening of the $s+is$ region due to chemical potential renormalization
- Predicted STM signatures of the $s+is$ state

## Abstract

Motivated by the recent observations of small Fermi energies and comparatively large superconducting gaps, present also on bands not crossing the Fermi energy (incipient bands) in iron-based superconductors, we analyse the doping evolution of superconductivity in a four-band model across the Lifshitz transition including BCS-BEC crossover effects on the shallow bands. Similar to the BCS case we find that with hole doping the phase difference between superconducting order parameters of the hole bands changes from $0$ to $\pi$ through an intermediate $s+is$ state breaking time-reversal symmetry. The transition however occurs in the region where electron bands are incipient and chemical potential renormalization in the superconducting state leads to a significant broadening of the $s+is$ region. We further present the qualitative features of the $s+is$ state that can be observed in scanning tunnelling microscopy (STM) experiments also taking incipient bands into account.