Dynamic phase diagram of dc-pumped magnon condensates

We study the effects of nonlinear dynamics and damping by phonons on a system of interacting electronically pumped magnons in a ferromagnet. The nonlinear effects are crucial for constructing the dynamic phase diagram, which describes how “swasing” and Bose-Einstein condensation emerge out of the quasiequilibrated thermal cloud of magnons. We analyze the system in the presence of magnon damping and interactions, demonstrating the continuous onset of stable condensates as well as hysteretic transitions.

Figure 1

Schematic of the proposed heterostructure. On the top, the normal metal N, with electron temperature $T^{\prime }$, provides spin torque through spin accumulation ${\mu }^{\prime }={\mu }^{\prime }\mathbf{z}$ at the interface with the ferromagnet (F). The F is assumed to be sufficiently thin such that its magnon temperature $T$ is uniform throughout. Collective spin density $\mathbf{s}$ in the F precesses with frequency $\omega$, which is controlled by the applied field $\mathbf{H}$, both pointing in the $z$ direction. Electron-magnon interaction at the $\mathrm{N}\mid \mathrm{F}$ interface is parametrized by the spin-mixing conductance $g^{\uparrow \downarrow }$. The normal metal $\tilde{\mathrm{N}}$ is a poor spin sink, which can, nevertheless, drain energy from magnons and phonons in the ferromagnet.

Figure 2

Graphical representation for obtaining solutions (36) to the equation $\hslash {\dot{n}}_c=i$ with $i$ given by Eq. (28). Here, $\sigma <0$ and ${ı}_x<0$ (corresponding to region ${\mathrm{IV}}_1$, as described in the text), resulting in two fixed points: unstable at $n_c^{-}$ and stable at $n_x^{+}$.

Figure 3

Phase diagram for the solutions of Eqs. (27) and (28) for $n_c$ in the abstract $(\sigma ,{ı}_x)$ space. $O$ stands for the unperturbed (i.e., thermal-equilibrium) point, while $P$ is the critical point for a driven system. The solid lines, ${ı}_x=0$ and ${ı}_x=-{\sigma }^2/4\zeta$, trace out phase transitions between distinct dynamic states: second-order transition between the NP and BEC (I/II boundary) and hysteretic first-order transitions at the ${\mathrm{IV}}_2$/${\mathrm{IV}}_1$ and ${\mathrm{IV}}_1$/III boundaries, where the normalized condensate density, $n_c/s$, jumps by $-\sigma /2\zeta$ and $-\sigma /\zeta$ relative to 0, respectively. The condensate associated with these first-order transitions is interpreted to be “swasing” [12].

Figure 4

Physical phase diagram in the presence of anisotropy $K=\hslash \Omega$ at $k_{B}T=k_B{T}^{\prime \prime }={10}^2\hslash \Omega$, $s{(A/\hslash \Omega )}^{3/2}={10}^4$, and $\alpha /{\alpha }^{\prime }=1$ (black curves), calculated using the linearized current ${ı}_x$ in Eq. (28) [see discussion preceding Eq. (37)]. The white curves show the idealized $\alpha /{\alpha }^{\prime }=0$ case. The analytically evaluated diagram shown here is essentially indistinguishable from the numerical diagram (not shown) produced by the exact expression for $i$ in Eq. (28). The phase-transition lines and crossovers that delineate different dynamic regimes can be inferred from Fig. 3.

Figure 5

Effects of intrinsic damping $\alpha /{\alpha }^{\prime }$ (starting at 1 and decreasing to 0 in increments of $0.2$) and nonlinearity $K/\hslash \Omega$ (going from 1 to 0 in increments of $0.2$), while keeping $\hslash \Omega$ fixed, on the phase-diagram structure, using Eqs. (37) and (38). Decreasing Gilbert damping $\alpha$ [which lowers the swasing threshold (35)] increases the size of the condensate regions, while decreasing anisotropy $K$ increases the size of the hysteretic region [as is evident from Eq. (38)].

Figure 6

Phase diagram with a floating magnon temperature $T$ and density determined by the conditions $i=0$ and $j=0$. Here, $\alpha /{\alpha }^{\prime }=1$, $K=\hslash \Omega$, $s{(A/\hslash \Omega )}^{3/2}={10}^4$, and $k_B{T}^{\prime \prime }={10}^2\hslash \Omega$, similarly to the other plots.