Many-body theory of spin-current driven instabilities in magnetic insulators

We consider a magnetic insulator in contact with a normal metal. We derive a self-consistent Keldysh effective action for the magnon gas that contains the effects of magnon-magnon interactions and contact with the metal to lowest order. Self-consistent expressions for the dispersion relation, temperature, and chemical potential for magnons are derived. Based on this effective action, we study instabilities of the magnon gas that arise due to spin current flowing across the interface between the normal metal and the magnetic insulator. We find that the stability phase diagram is modified by an interference between magnon-magnon interactions and interfacial magnon-electron coupling. These effects persist at low temperatures and for thin magnetic insulators.

Figure 1

Schematic illustration of spin-transfer induced spin-wave excitations of the insulating magnetic film in contact with a normal metal ($F|M$). A spin accumulation $\mathrm{\Delta }\mu$ is induced in the metal via the spin Hall effect. The magnons are assumed to be in quasiequilibrium with a chemical potential ${\mu }_m$. In turn, the spin accumulation exerts a torque on the magnetization in $F$. A temperature gradient $T_m-T_e$, applied longitudinally also contributes to the net spin-current flow through the interface.

Figure 2

Feynman diagrams contributing to the magnon dynamics due to electron-magnon interactions at the $F|M$ interface. (a) Electron bubble diagram representing the annihilation and creation of magnons via spin-flip processes. (b) Diagram for the enhanced spin polarization. The momentum carried by the magnon is denoted by $\mathbf{p}$.

Figure 3

(a) Feynman diagram for the dressed Green's function of magnons due to the coupling with the $M$. Bubble diagram represents the self-energy given in Eq. (13). (b) Diagrammatic representation for the interacting self-energy and (c) the $T$-matrix ladder approximation, where the wavy lines denote the contact interaction $K_z$. Note that the self-energy and $T$ matrix are determined by the dressed and free propagators, respectively.

Figure 4

Chemical potential (top panel) and temperature of magnons (bottom panel) in the steady state as a function of spin accumulation. The plots are displayed for different values of the temperature of the electrons.

Figure 5

Phase diagram for the stability of a driven magnon system as a function of the thermal imbalance $\eta$ and effective chemical potential $\delta \mu$ in units of the thermal energy $k_B{T}_e$. Each curve delimits both phases, being the right-hand side of the graph where the condition Eq. (18) is fulfilled and thus corresponding to the unstable phase of magnons. In panel (a) we have $e_0=0.1,e_1=0.5$, and $e_2=0.1-0.9$. Meanwhile, in panel (b) $e_0=0.1,e_2=0.1$, and $e_1=0.1-0.9$.