**3.2. Theory of spin fluctuation effect**

This theory was mainly reported by Maekawa, S. and Tachiki, M. [7] in 1970s, with the discovery of field-induced superconductors EuxSn1-xMo6S8-ySey. These types of materials have rare-earth 4f-ions and paired conduction electrons from the 4d-Mo-ions. The rare-earth 4f-ions have large fluctuating magnetic moments, while the conduction electrons from the 4d-Mo-ions have strong electron–electron interactions and they form Cooper pairs.

Without externally applied magnetic field (*H* = 0), the fluctuating magnetic field from the rare-earth 4f-ion moments at the 4d-Mo conduction electrons are so strong that it weakens the BCS coupling of the Mo-electrons. Thus there is no superconductivity without externally applied magnetic field.

However, when external magnetic field is applied (*H* ≠ 0) it suppresses the spin uctuation, causing an increase of the BCS coupling among the conduction electrons. Thus, superconductivity appears in the presence of an applied magnetic field. This scenario can be seen from a typical *H-T* phase diagram shown in Fig. 2.

**Figure 2.** A possible *H-T* phase diagram for the spin fluctuation effect. The thin-red line represents a case without local spin fluctuation moments as a reference.
