Thermalization of hot electrons via interfacial electron-magnon interaction

Recent work on layered structures of superconductors (S) or normal metals (N) in contact with ferromagnetic insulators (FI) has shown how the properties of the previous can be strongly affected by the magnetic proximity effect due to the static FI magnetization. Here we show that such structures can also exhibit a new electron thermalization mechanism due to the coupling of electrons with the dynamic magnetization, i.e., magnons in FI. We here study the heat flow between the two systems and find that in thin films the heat conductance due to the interfacial electron-magnon collisions can dominate over the well-known electron-phonon coupling below a certain characteristic temperature that can be straightforwardly reached with present-day experiments. We also study the role of the magnon band gap and the induced spin-splitting field induced in S on the resulting heat conductance and show that heat balance experiments can reveal information about such quantities in a way quite different from typical magnon spectroscopy experiments.

Figure 1

(a) Schematic of a hybrid bilayer of a normal metal (N) or a superconductor (S) of thickness $d_N$ or $d_S$ in a good contact with a ferromagnetic insulator (FI) of thickness $d_{\mathrm{FI}}$. (b) $P_{\gamma }$ denotes the incident radiation power, which increases the electronic temperature by an amount $\mathrm{\Delta }T$ from some initial temperature $T$ dictated by the temperature of the baths. For a low and constant power $P_{\gamma }$, the magnitude of $\mathrm{\Delta }T=P_{\gamma }/G_{\mathrm{th}}^{\mathrm{tot}}$ is dictated by the total heat conductance, $G_{\mathrm{th}}^{\mathrm{tot}}$, to the heat bath. This heat conductance is typically due to the coupling of the electrons (e) to the phonons (ph), but at a low enough temperature also magnons (m) in the FI film start to be relevant and eventually may become the dominant heat conduction mechanism. Here $h$ denotes the spin-splitting field possibly induced to N or S via the magnetic proximity effect. ${\dot{Q}}_{\mathrm{q}\text{−}\mathrm{ph}}$ and ${\dot{Q}}_{\mathrm{q}\text{−}\mathrm{m}}$ stand for the rates of heat flow due to electron-phonon interaction and electron-magnon collisions, respectively, and $G_{\mathrm{q}\text{−}\mathrm{ph}}$ and $G_{\mathrm{q}\text{−}\mathrm{m}}$ are the corresponding heat conductances.

Figure 2

Electron-magnon $G_{\mathrm{q}\text{−}\mathrm{m}}$ [Eqs. (8)–(10), curves (a)–(c) and $\left({\mathrm{a}}^{\prime }\right),\left({\mathrm{b}}^{\prime }\right)]$ and electron-phonon heat conductance $G_{\mathrm{q}\text{−}\mathrm{ph}}$ [Eq. (13), curves (d)–(f)] vs temperature $T$ for the thin film normal metal-ferromagnetic insulator hybrid structure. In (a) and (b) the chosen magnon band gap is ${\omega }_0=0$ and ${\omega }_0=0.8{\omega }_F$, respectively. Curve (c) is the analytical estimate of electron-magnon heat conductance from Eq. (12b), valid at $k_{B}T\ll {\omega }_F$ for the magnon band gap ${\omega }_0=0$. The curves $\left({\mathrm{a}}^{\prime }\right)$ and $\left({\mathrm{b}}^{\prime }\right)$ in the inset represent the corresponding extended plots of (a) and (b), respectively. Curves (d)–(f) show the electron-phonon heat conductance for three different thicknesses of the normal-metal film, for $d_N/d_l=2,d_N/d_l=6$ and $d_N/d_l=10$, respectively.

Figure 3

Temperature $T^{*}$ below which the electron-magnon thermalization dominates the electron-phonon mechanism, as a function of the thickness $d_N$ of the normal metal film. The curves are for two different values of the magnon band gap. The dotted line shows the analytical estimate from Eq. (19). The thickness is scaled by $d_l$ defined in Eq. (20).

Figure 4

Electron-magnon $G_{\mathrm{q}\text{−}\mathrm{m}}$ [Eqs. (8), (9), and (14), curves (a)–(d)] and electron-phonon heat conductance $G_{\mathrm{q}-\mathrm{ph}}$ [Eqs. (16a)–(16c), curves (e)–(f)] vs temperature $T$ for the thin film superconductor-ferromagnetic insulator hybrid structure. The parameters for the curves are (a) ${\omega }_0=0,{\omega }_F={\mathrm{\Delta }}_0$ and (b) ${\omega }_0=0,{\omega }_F=10{\mathrm{\Delta }}_0$ and (c) ${\omega }_0=0.5{\mathrm{\Delta }}_0,{\omega }_F=10{\mathrm{\Delta }}_0$. The curve (d) is the analytical estimate [Eq. (15)] of the electron-magnon heat conductance for the given parameters in (b) curve. The thin film superconductor thicknesses are in curve (e) $d_S/d_{\mathrm{\Delta }}=0.4$ and (f) $d_S/d_{\mathrm{\Delta }}=0.8$. For all the curves we set $h=0$ and $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Delta }}_0$.

Figure 5

Crossover temperature $T^{*}$ below which electron-magnon thermalization becomes dominant, as a function of the thickness $d_S$ of the superconductor. For all curves we set $h=0$ and $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Delta }}_0$. The parameters for the curves are (a) ${\omega }_0=0,{\omega }_F=26{\mathrm{\Delta }}_0$, (b) ${\omega }_0=0.5{\mathrm{\Delta }}_0,{\omega }_F=50{\mathrm{\Delta }}_0$, and (c) ${\omega }_0=0,{\omega }_F=50{\mathrm{\Delta }}_0$.

Figure 6

Electron-magnon heat conductance $G_{\mathrm{q}\text{−}\mathrm{m}}$ vs spin-splitting field $h$ for the thin film superconductor-ferromagnetic insulator hybrid structure. For all curves ${\omega }_0=0.1{\mathrm{\Delta }}_0,{\omega }_F={\mathrm{\Delta }}_0$, and $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Delta }}_0$. The curves (a), (b), and (c) are for $k_{B}T=0.2{\mathrm{\Delta }}_0,k_{B}T=0.3{\mathrm{\Delta }}_0$, and $k_{B}T=0.4{\mathrm{\Delta }}_0$, respectively.

Figure 7

Electron-magnon heat conductance $G_{\mathrm{q}\text{−}\mathrm{m}}$ vs magnon band gap ${\omega }_0$ for the thin film superconductor-ferromagnetic insulator hybrid structure. For all the curves $h=0.1{\mathrm{\Delta }}_0,{\omega }_F={\mathrm{\Delta }}_0$, and $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Delta }}_0$. The curves (a), (b), and (c) are for $k_{B}T=0.2{\mathrm{\Delta }}_0,k_{B}T=0.3{\mathrm{\Delta }}_0$, and $k_{B}T=0.4{\mathrm{\Delta }}_0$, respectively.