It is known that measurement of electrical current noise through a device provides useful information about electron transport. For example, nonequilibrium current noise under a high voltage bias, so-called shot noise, can be used for determining the effective charge of a quasiparticle. As expected from fruitful physics of the current noise, fluctuation of the pure spin current, that is, spin-current noise has a potential to provide important information on spin transport in a spintronics device.

We theoretically investigate the fluctuation of a pure spin current induced by the spin Seebeck effect and spin pumping in a normal-metal–(NM-)-ferromagnetic-insulator(FI) bilayer system. Starting with a simple FI-NM interface model, we derive general expressions of the spin current and the spin-current noise at the interface within second-order perturbation of the exchange coupling at the interface, and estimate them for a yttrium-iron-garnet–platinum interface. We show that the spin-current noise can be used to determine the effective spin carried by a magnon. In addition, we show that it provides information on the effective spin of a magnon, heating at the interface under spin pumping, and spin Hall angle of the NM. We also discuss spin transport at the interface between a FI and a superconductor.