Out-of-plane Edelstein effects: Electric-field induced magnetization in $p$-wave magnets

In-plane magnetization is induced by the Edelstein effect in the Rashba spin-orbit interaction system. However, out-of-plane magnetization is more useful for switching a ferromagnetic memory. We study analytically and numerically electric-field induced magnetization in $p$-wave magnets with the aid of the Rashba interaction based on a simple two-band model. The out-of-plane magnetization is induced when the N\'{e}el vector of the $p$-wave magnet is along the $z$ direction. We also show that no magnetization is induced in the absence of the Rashba interaction. The electric-field induced magnetization will be useful for future switching technology of ferromagnetic memories based on the $p$-wave magnet.