## 3 Results
### Heat Transport at Interface  
Next, we investigated the heat transport at the interface. A thin, soft layer at the interface has a critical role in heat flow from the metal transducer to sapphire substrate. We measured the \(\Delta\theta\) of the sap/Co (10 nm)/Pt (2 nm) and sap/Co\({}_{50}\)B\({}_{50}\) (10 nm)/Pt (4 nm) samples up to 2000 ps. Then, we obtained the ratio \(\mid\)\(\Delta\theta_{\text{in}}\)/\(\Delta\theta_{\text{out}}\)\(\mid\), where \(\Delta\theta_{\text{in}}\) is the signal that is in-phase with the pump modulation and \(\Delta\theta_{\text{out}}\) is the signal that is 90\({}^{\circ}\) out-of-phase with the pump modulation. This ratio makes the MOKE data more robust to errors caused by the defocusing of the pump beam and misalignment between the pump and probe beams [31; 32; 33]. The motion of the delay stage caused a slight spatial defocusing and misalignment in the pump spot on the sample, especially at long timescales of nanoseconds. The aforementioned ratio suppressed the errors associated with these issues because both \(\Delta\theta_{\text{in}}\) and \(\Delta\theta_{\text{out}}\) had the same dependencies on defocusing and overlap. \(\Delta\theta\) is caused by \(\Delta M\), and there are two contributions:  
\[\Delta\theta\propto\Delta|M|+|M|\Delta\sin\phi, \tag{5}\]  
Figure 5: Frequency and damping of acoustic oscillation. (**a**) The frequency of the acoustic oscillation of the sap/Co\({}_{50}\)B\({}_{50}\) (5, 7, 10, and 15 nm)/Pt (4 nm) samples. The \(d_{\text{tot}}\) represents the total thickness of Co\({}_{50}\)B\({}_{50}\) and Pt. The red line is the fitting with \(f=\frac{v_{3}}{Z_{\text{int1}}}\), where \(v_{\text{s}}\) is 4900 m s\({}^{-1}\). (**b**) The damping (\(\alpha\)) of acoustic oscillations of the sap/Co\({}_{50}\)B\({}_{50}\) (5, 7, 10, and 15 nm)/Pt (4 nm) samples. The red dotted line indicates an \(\alpha\) of 0.034.  
where \(|M|\) is the magnitude of magnetization and \(\phi\) is the angle of magnetization (Figure 6a). When the magnetic field is not large enough to fully align the magnetization, a pump pulse can trigger an oscillation of \(\phi\). In the case of the Co transducer, the measured \(|\Delta\theta_{\text{in}}/\Delta\theta_{\text{out}}|\) exhibited a large oscillation and a slow decrease (Figure 6b). With the Co\({}_{50}\)B\({}_{50}\) transducer, a magnetic field of 0.5 T was sufficient to fully align the magnetization; there was no magnetization oscillation. The required magnetic field for the full alignment of magnetization was approximately \(\mu_{0}M_{\text{S}}\), where \(\mu_{0}\) is the vacuum permeability and \(M_{\text{S}}\) is the saturation magnetization. With an \(M_{\text{S}}\) of 1.4 \(\times\) 10\({}^{6}\) A m\({}^{-1}\) of Co, \(\mu_{0}M_{\text{S}}\) was 1.8 T. \(M_{\text{S}}\) decreased significantly with B alloying, and the \(\mu_{0}M_{\text{S}}\) of Co\({}_{50}\)B\({}_{50}\) was smaller than 0.5 T [39]. For the thermal analysis of the Co sample, we focused on the \(\Delta|M|\) signal, which was proportional to \(\Delta T\) of Co, when the oscillation signal decayed completely after 500 ps. We analyzed the slow decrease in \(|\Delta\theta_{\text{in}}/\Delta\theta_{\text{out}}|\) using the time-domain solution of the temperature response of the top surface in the substrate/transducer structure [31]. We treated the Co\({}_{50}\)B\({}_{50}\) and Pt layers as a single layer considering the high thermal conductance at the metallic interfaces [40]. To estimate the sensitivity of thermal modeling to a thermodynamic parameter (\(\gamma\)), we defined the sensitivity parameter as the logarithmic derivative of \(|\Delta\theta_{\text{in}}/\Delta\theta_{\text{out}}|\) to changes in \(\gamma\) of the thermal model (Figure 6c):  
\[S_{\gamma}=\frac{\partial[\text{ln}(|\Delta\theta_{\text{in}}/\Delta\theta_{ \text{out}}|)]}{\partial[\text{ln}(\gamma)]}, \tag{6}\]  
where \(\gamma\) could be the thermal conductivity (\(\Lambda\)) of each layer or thermal conductance (\(G\)) at the interface.