Field-dependent nonelectronic contributions to thermal conductivity in a metallic ferromagnet with low Gilbert damping

Heat conduction in metals is typically dominated by electron transport since electrons carry both charge and heat. In magnetic metals magnons, or spin waves, excitations of the magnetic order can be used to transport information. Heat conduction via magnons has been previously shown mostly for insulating magnets with low Gilbert damping and resulting long spin-wave lifetimes where conduction electrons cannot contribute. Here we show that thin films of properly optimized metallic ferromagnetic (FM) alloys show significant nonelectronic contributions to heat conduction, which furthermore depend on the direction of an applied magnetic field. These measurements are enabled by micromachined thermal isolation platforms optimized for thermal conductivity measurements of thin-film systems. Electrical conductivity measurements on exactly the same samples allow application of the Wiedemann-Franz relation, which shows large nonelectronic contributions to thermal conductivity for the cobalt-iron alloy with $25\%$ Co. This composition has been shown to have exceptionally low damping for a metallic FM. The thermal conductivity of a 75-nm-thick film of the $25\%$ Co alloy changes by more than $20\%$ at some temperatures, while a reference sample with $50\%$ cobalt that has much higher damping shows no field-direction dependence. Our measurements indicate that applied magnetic fields alter the magnon lifetimes in these films and that these magnons contribute to thermal conductivity in this metallic magnetic alloy with low Gilbert damping.

Figure 1

(a) Oblique angle scanning electron micrograph of the Si-N platform with $\mathbf{\nabla }$ T, $H_{\parallel }$ and $H_{\perp }$ and sample area (blue shading) shown. The length and width of the sample are 2040 and $88\mu \mathrm{m}$, respectively. (b) $K_{\mathrm{tot}}$ vs $T$ for the Si-N background and three ${\mathrm{Co}}_x{\mathrm{Fe}}_{1-x}$ films. Inset: Thermal model used to determine $K_{\mathrm{tot}}$. (c) $K_{\mathrm{tot}}$ vs $T$ for $H_{\parallel }$ and $H_{\perp }$ for the low-$\alpha$ sample ${\mathrm{Co}}_{25}{\mathrm{Fe}}_{75}\#2$. Inset: Schematic side view of the sample layers. (d) $K_{\mathrm{tot}}$ vs $T$ for $H_{\parallel }$ and $H_{\perp }$ for the reference sample ${\mathrm{Co}}_{50}{\mathrm{Fe}}_{50}$ shows no dependence on field direction. Inset: $\alpha$ (left axis) and density of states (right axis) vs $x$ [42]. Dashed lines indicate the two $x$ studied here.

Figure 2

(a) $L=K_{\mathrm{S}}R/T$ vs $T$ for three ${\mathrm{Co}}_x{\mathrm{Fe}}_{1-x}$ films compared with $L_o$. The two low-$\alpha$ films have dramatically increased $L$, indicating nonelectronic contributions to $k_{\mathrm{film}}$. (b) $k_{\mathbf{film}}$ vs $T$ for these samples, with estimated $k_{\mathrm{el}}$ also shown. (c) The nonelectronic contribution $k_{\mathrm{ex}}=k_{\mathrm{film}}-k_{\mathrm{el}}$ vs $T$ (left axis) for both $x=25$ films show peaks near 200 K.

Figure 3

(a) $R$ vs $H$ for $H_{\perp }$ and $H_{\parallel }$ for ${\mathrm{Co}}_{25}{\mathrm{Fe}}_{75}\#2$ at 200 K. (b) AMR ratio vs $T$ for ${\mathrm{Co}}_{25}{\mathrm{Fe}}_{75}\#2$ and ${\mathrm{Co}}_{50}{\mathrm{Fe}}_{50}$. (c) $k_{\mathrm{film}}$ vs $T$ for ${\mathrm{Co}}_{25}{\mathrm{Fe}}_{75}\#2$ in $H=0, H_{\perp }$ and $H_{\parallel }$ compared with $k_{\mathrm{el}}$. (d) $k_{\mathrm{film}}$ vs $T$ for ${\mathrm{Co}}_{50}{\mathrm{Fe}}_{50}$ shows no change with field. (e) The resulting anisotropic thermal conductivity ratio for ${\mathrm{Co}}_{25}{\mathrm{Fe}}_{75}\#2$ is $>10\times$ larger than the AMR at all $T$ and exceeds $25\%$ at low $T$.

Figure 4

(a) Modeled spin-wave dispersion curves, $f_{\mathrm{sw}}\mathrm{vs}q$ for low-$q$ waves generated when $\vec{H}\perp \vec{q}$ (blue line), dominated by magnetostatic surface waves and for waves generated when $\vec{H}\parallel \vec{q}$ (red lines), which arise from a combination of backward volume and exchange spin waves. (b) Magnon group velocity $v_{\mathrm{g}}$ determined from the dispersion relations. Note the discontinuity in the $\vec{H}\parallel \vec{q}$ case arises from our choice of the simplest possible model and likely results in the underestimation of $v_{\mathrm{g}}$ in this range of $q$, which could lead to an overestimate of $\ell$. (c) Integrating the curve shown here produced by the product $C{v}_{\mathrm{g}}\ell$ vs $\vec{q}$ gives $3k$, which shows that a significant contribution from these low-$q$ magnons comes only when $\vec{H}\parallel \vec{q}$. This curve used an assumed constant $\ell =100\mu \mathrm{m}$. Inset: Estimated BVSW mean free path for the data of Fig. 3.