## 3 Results
### Acoustic Propagation at Interface
Using the relationship \(f=\frac{v_{\mathrm{s}}}{2d_{\mathrm{tot}}}\), where \(d_{\mathrm{tot}}\) is the thickness of \(\mathrm{Co_{50}B_{50}}\)+Pt and \(v_{\mathrm{s}}\) is the effective speed of sound of \(\mathrm{Co_{50}B_{50}}\)+Pt, we determined \(v_{\mathrm{s}}\) to be 4900 m s\({}^{-1}\). Then, by combining \(f\) and \(\tau\), we obtained a damping parameter of \(\alpha=1/2\pi f\tau\) (Figure 5b). \(\alpha\) describes the amplitude attenuation during one period of oscillation; it ranges from zero for no attenuation to \(1/2\pi\) for full attenuation after one period. The attenuation of acoustic waves in a medium is caused by the wave transmission at interfaces and bulk attenuation inside the medium. Considering the energy conservation during the wave propagation at interfaces, the wave transmission is related to the wave reflection. Then, \(\alpha\) can be expressed as  
\[2\pi\alpha=\{1-|r_{\mathrm{int1}}|\}+\{1-|r_{\mathrm{int2}}|\}+\{1-\exp(-2 \beta_{\mathrm{bulk}}d_{\mathrm{tot}})\}, \tag{3}\]  
where \(r_{\mathrm{int1}}\) is the reflection coefficient at the bottom interface between the substrate and transducer, \(r_{\mathrm{int2}}\) is the reflection coefficient at the top interface between the transducer and air, and \(\beta_{\mathrm{bulk}}\) is the bulk attenuation factor in the transducer layer (we note that \(r\) can be negative when \(Z_{1}<Z_{2}\) in Equation (2), so that the reflected wave can change its sign. Since \(\alpha\) describes the attenuation of the absolute magnitude of the acoustic wave, \(\alpha\) should be related to \(\left|r\right|\) but not \(r\)). \(r_{\mathrm{int2}}\) was found to be close to one, considering the large mismatch in the \(Z\) values of the transducer and air. The \(\beta_{\text{bulk}}\) value of metals depends on the frequency [36; 37], and there are no experimental reports regarding ferromagnetic metals at a high frequency of 200 GHz. When \(\beta_{\text{bulk}}\) is negligible and \(\alpha\) is dominated by \(r_{\text{int1}}\), an \(\alpha\) of 0.034 \(\pm\) 0.011 leads to an \(\mid\)\(r_{\text{int1}}\)\(\mid\) of 0.79 \(\pm\) 0.07 at the Co\({}_{50}\)B\({}_{50}\)/sap interface. When \(\beta_{\text{bulk}}\) is as high as \(\approx\)10\({}^{7}\) m\({}^{-1}\)[38], the bulk attenuation can explain most of \(\alpha\), and \(\mid\)\(r_{\text{int1}}\)\(\mid\) will be close to one. Such a large \(\mid\)\(r_{\text{int1}}\)\(\mid\) of >0.79 cannot be explained by the \(Z\) of the transducer and substrate alone; thus, a soft layer should exist at the interface. Then, \(r_{\text{int1}}\) is expressed with the acoustic properties of the soft layer [1]:  
\[r_{\text{int1}}=\sqrt{\frac{Z_{3}^{2}\big{(}Z_{1}-Z_{2}\big{)}^{2}+\big{(}Z_{1 }^{2}-Z_{3}^{2}\big{)}\big{(}Z_{2}^{2}-Z_{3}^{2}\big{)}\ \sin^{2}\phi}{Z_{3}^{2}\big{(}Z_{1}+Z_{2}\big{)}^{2}+\big{(}Z_{1}^{2}-Z_{3}^{2 }\big{)}\big{(}Z_{2}^{2}-Z_{3}^{2}\big{)}\ \sin^{2}\phi}}}, \tag{4}\]  
where \(\phi=2\pi fd_{3}/v_{3}\); here, \(d_{3}\) is the thickness of the soft layer and \(v_{3}\) is the speed of sound of the soft layer. Assuming the same \(Z\) of 4 \(\times\) 10\({}^{7}\) kg m\({}^{-2}\) s\({}^{-1}\) for CoB and sapphire, and a typical \(Z_{3}\) of 1.7 \(\pm\) 0.5 \(\times\) 10\({}^{6}\) kg m\({}^{-2}\) s\({}^{-1}\) and \(v_{3}\) of 1.5 \(\pm\) 0.2 km s\({}^{-1}\) for soft materials [34; 35], an \(\mid\)\(r_{\text{int1}}\)\(\mid\) of >0.79 \(\pm\) 0.07 leads to a \(d_{3}\) of >2 \(\pm\) 1 A at an\(f\) of 200 GHz.