Fig. 5

Dynamic SSE model. a–d, Model schematic of the F|N interface. To illustrate the action of the exchange torque exerted by N on F, it is sufficient to consider the “up” (\(\odot\), see panels a, b) and “down” case (⊗, see panels c, d) of an N-cell spin fluctuation s^N perpendicular to the YIG magnetization M. a At time t′, an N electron entering the interaction region induces a fluctuation \({\mathbf{s}}^{\mathrm N}\left( {t\prime } \right)\) of the total N-cell spin, thereby exerting the effective magnetic field \(J_{{\mathrm{sd}}}{\mathbf{s}}^{\mathrm N}\left( {t\prime } \right)\) on the adjacent F-cell spin (torque #1). b Consequently, at a slightly later time t, the F-cell spin has changed by \({\mathrm{\Delta }}{\mathbf{s}}^{\mathrm F}\left( t \right)\) proportional to \(J_{{\mathrm{sd}}}{\mathbf{s}}^{\mathrm N}\left( {t\prime } \right)\). c The opposite fluctuation \(- {\mathbf{s}}^{\mathrm N}\left( {t\prime } \right)\) at time t′ induces d the change \(- {\mathrm{\Delta }}{\mathbf{s}}^{\mathrm F}\left( t \right)\), resulting in zero change in the F-cell spin, \(\langle{\mathrm{\Delta }}{\mathbf{s}}^{\mathrm F}\left( t \right)\rangle = 0\). However, as seen in panels (b) and (d), a second interaction at \(t\, > \,t\prime\) with the N-cell field (torque #2) leads to the same rectified torque \(J_{{\mathrm{sd}}}{\mathbf{s}}^{\mathrm N}\left( t \right) \times {\mathrm{\Delta }}{\mathbf{s}}^{\mathrm F}\left( t \right)\) for both +s^N and −s^N and, thus, a net spin current between F and N. e Calculated time-domain spin susceptibility of the F cell (transverse \(\chi _ \bot ^{\mathrm{F}}\left( t \right)\) of YIG) and the N cell (isotropic \(\chi ^{\mathrm{N}}\left( t \right)\) of Pt). f Calculated dynamics of the SSE response functions \(\kappa ^{\mathrm{N}}\left( t \right)\) and \(\kappa ^{\mathrm{F}}\left( t \right)\) which quantify, respectively, the spin current induced by a \(\delta \left( t \right)\)-like temperature change of the N (Pt) and F (YIG) layer. The area under both curves equals the DC SSE constant \({\cal K}\). g Evolution of the generalized electronic temperature of Pt as obtained by simulations based on the Boltzmann equation. Excitation conditions are similar to those used in the experiment. For direct comparison, the measured spin current j_s(t) (see Fig. 4b) and calculated SSE response function \(\kappa ^{\mathrm{N}}\left( t \right)\) (see panel f) are also shown