First avertical temperature gradient generates a temperature differenceT me between magnons in FM and electrons in NM described by themagnon temperature model10–12 and a magnon spin current asso-ciate

The temperature difference T m?e between magnons in a ferromagnet (FM) and electrons in a normal metal (NM) is generated by a vertical temperature gradient. This temperature difference arises due to the different thermal properties of magnons and electrons.

In addition to the temperature difference, there is also a magnon spin current that is associated with a magnon chemical potential gradient ∇μm in the FM. This magnon spin current is described by the magnon drift-diffusion model. The magnon chemical potential gradient refers to the variation in the chemical potential of magnons across the FM material. This gradient drives the flow of magnon spin current, leading to the transport of spin information in the FM.

These two effects, the temperature difference and the magnon spin current, play important roles in the transport and exchange of energy and spin between magnons and electrons in FM/NM heterostructures. Understanding and controlling these effects are crucial for developing novel spintronic devices and exploring the fundamental physics of magnon-electron interactions