Frustrated proximity effects between s and s$_{\pm }$ superconductors

The nature of the superconducting order parameter (OP) in iron pnictides and chalcogenides is a hotly debated issue. It was theoretically proposed that the OP has opposite signs on the hole and the electron bands, i.e., it belongs to the unconventional class of s$_{\pm }$ (or extended s)-wave. There are, however, very few experiments that can directly distinguish this state from the ordinary s-wave OP. One way to address this problem is to study the proximity effects in a sandwich composed of conventional and iron pnictide superconductors (SC). If the pnictides indeed have the s$_{\pm }$ OP this system is intrinsically frustrated. In the case of strong frustration, a time-reversal symmetry-breaking (TRSB) SC state emerges, in which the OP phases in different bands are tilted at an angle, different from $\pi $, and controlled by the coupling strength. Observation of such state in the iron-based SC materials would give definite evidence for the s$_{\pm }$ OP. We present a microscopic, fully self-consistent approach to this problem, based on Usadel equations. We have studied the conditions for existence of the TRSB state and its experimental signatures.