Magnon transport through microwave pumping

We present a microscopic theory of magnon transport in ferromagnetic insulators (FIs). Using magnon injection through microwave pumping, we propose a way to generate magnon dc currents and show how to enhance their amplitudes in hybrid ferromagnetic insulating junctions. To this end, focusing on a single FI, we first revisit microwave pumping at finite (room) temperature from the microscopic viewpoint of magnon injection. Next, we apply it to two kinds of hybrid ferromagnetic insulating junctions. The first is the junction between a quasiequilibrium magnon condensate and magnons being pumped by microwave, while the second is the junction between such pumped magnons and noncondensed magnons. We show that quasiequilibrium magnon condensates generate ac and dc magnon currents, while noncondensed magnons produce essentially a dc magnon current. The ferromagnetic resonance (FMR) drastically increases the density of the pumped magnons and enhances such magnon currents. Lastly, using microwave pumping in a single FI, we discuss the possibility that a magnon current through an Aharonov-Casher phase flows persistently even at finite temperature. We show that such a magnon current arises even at finite temperature in the presence of magnon-magnon interactions. Due to FMR, its amplitude becomes much larger than the condensed magnon current.

Figure 1

A schema of the time evolution of the quasiequilibrium magnon-BEC density due to microwave pumping: (i) microwave pumping, (ii) thermalization (or relaxation) process, and (iii) quasiequilibrium magnon-BEC state. This schematic picture is based on the experiment by Serga et al. of Ref. [5]. The applied microwave drives the system into a nonequilibrium steady state and continues to excite the zero mode of magnons (i.e., pumped magnons). After the microwave is switched off, the system experiences a thermalization process for a short time, and immediately reach a quasiequilibrium state where the pumped magnons can form a quasiequilibrium magnon BEC. The thermalization period is [5] about $50\mathrm{ns}$, while the decay time of quasiequilibrium magnon BECs amounts [5] to $400\mathrm{ns}$ and they can survive for about $1\mu \mathrm{s}$.

Figure 2

Schematic representations of a C-P junction between two FIs, one prepared as a BEC (left FI) and the other (right FI) with pumped magnons. (a) Both magnetic fields are applied along the $z$ axis, ${\mathbf{B}}_{\mathrm{L}\left(\mathrm{R}\right)}=B_{\mathrm{L}\left(\mathrm{R}\right)}{\mathbf{e}}_z$. The circles in the right FI are the magnons pumped by microwaves (frequency $\mathrm{\Omega }\sim 100$ MHz) and the cloud of circles in the BEC represents the quasiequilibrium magnon condensate. Both phases correspond to a macroscopic coherent precession in terms of spin variables ${\mathbf{S}}_{{\mathrm{\Gamma }}_{\mathrm{L}}}$ and ${\mathbf{S}}_{{\mathrm{\Gamma }}_{\mathrm{R}}}$. A distinct difference, however, is that the frequency of the magnon BEC is controlled by the applied magnetic field $B_{\mathrm{L}}\left(\sim \mathrm{mT}\right)$, while that of the pumped magnons is controlled by the frequency $\mathrm{\Omega }$ of the applied microwave. The relation between $g{\mu }_{\mathrm{B}}{B}_{\mathrm{L}}/\hslash$ and $\mathrm{\Omega }$ plays the key role in the ac-dc conversion of magnon currents. Tuning $\hslash \mathrm{\Omega }=g{\mu }_{\mathrm{B}}{B}_{\mathrm{R}}$, a FMR occurs in the right FI and consequently the number of pumped magnons drastically increases. (b) Close-up of the C-P junction. Only the boundary spins ${\mathbf{S}}_{{\mathrm{\Gamma }}_{\mathrm{L}}}$ in the left FI and ${\mathbf{S}}_{{\mathrm{\Gamma }}_{\mathrm{R}}}$ in the right FI are relevant to magnon transport. We assume a large spin $S\gg 1$. The two FIs are separated by an interface and thereby weakly exchange coupled with strength $J_{\text{ex}}$. For typical values $J_{\text{ex}}\sim 0.1\mu \mathrm{eV}$ this gives rise to an interface of width ∼10 Å.

Figure 3

A plot of the number density of pumped magnons ${\langle n_{\mathrm{R}}\rangle }_{\mathrm{P}}$ (in ${\mathrm{cm}}^{-3}$) as a function of the magnetic field $B_{\mathrm{R}}$ (in mT) obtained by numerically solving Eq. (12). As an example, for $\mathrm{\Omega }=400$ MHz, ${\mathrm{\Gamma }}_0=0.25\mu \mathrm{T}$, ${\tau }_{\mathrm{p}}=100\mathrm{ns}$, $\alpha \approx 1$ Å, $g=2$, and $S=16$, the number density of pumped magnons amounts to ${\langle n_{\mathrm{R}}\rangle }_{\mathrm{P}}=3.2\times {10}^{21}{\mathrm{cm}}^{-3}$ at FMR ($\hslash \mathrm{\Omega }=g{\mu }_{\mathrm{B}}{B}_{\mathrm{R}}$, i.e., $B_{\mathrm{R}}=2$ mT).

Figure 4

Plots of the rescaled magnon number $N_{\mathrm{L}}\equiv n_{\mathrm{L}}{\alpha }^3$ as a function of the rescaled time $\mathcal{T}\equiv (J_{\mathrm{ex}}S/\hslash )t$ obtained by numerically solving Eq. (28) for the values $g=2$, $S=16$, $J_{\mathrm{ex}}\approx 0.1\mu \mathrm{eV}$, ${\mathrm{\Gamma }}_0=0.125\mu \mathrm{T}$, ${\tau }_{\mathrm{p}}=100\mathrm{ns}$, $\alpha \approx 1$ Å, ${\vartheta }_{\mathrm{L}}\left(0\right)-{\vartheta }_{\mathrm{ac}}=\pi$, $N_{\mathrm{L}}(\mathcal{T}=0)={10}^{-5}$, and $\mathrm{\Omega }=400$ MHz. The rescaled time $\mathcal{T}=1$ corresponds to $t=5\mathrm{ns}$ and $N_{\mathrm{L}}=1$ corresponds to $n_{\mathrm{L}}={10}^{24}{\mathrm{cm}}^{-3}$. $d{N}_{\mathrm{L}}/\left(d\mathcal{T}\right)$ represents the magnon current. (a) $\hslash \mathrm{\Omega }=g{\mu }_{\mathrm{B}}{B}_{\mathrm{L}}$ (i.e., $B_{\mathrm{L}}=2$ mT). Since the sign of the function $d{N}_{\mathrm{L}}/\left(d\mathcal{T}\right)$ does not change all the time, it can be regarded as a dc magnon current. (b) $\hslash \mathrm{\Omega }\ne g{\mu }_{\mathrm{B}}{B}_{\mathrm{L}}:B_{\mathrm{L}}=2.05$ mT. This gives the ac magnon current whose period becomes of the order of ${10}^{-1}\mu \mathrm{s}$.

Figure 5

Schematic representation of the NC-P junction, consisting of a left FI with noncondensed magnons (NC) and a weakly exchange-coupled right FI with pumped magnons (P). The microwave at frequency $\mathrm{\Omega }$ is applied only to the right FI. The distinction to the C-P junction in Fig. 2 is that the magnons in the left FI are not condensed (i.e., $\langle a_{\mathrm{L}}\rangle =0$). Due to the exchange interaction ${\mathcal{H}}_{\mathrm{ex}}$ [Eq. (4)], the applied microwave to the right FI can indirectly affect the noncondensed magnons of the left FI and thereby generate an indirect FMR there.