# Magneto-Optical Kerr Effect Driven by Spin Accumulation on Cu, Au, and Pt
## 3 Results
Generated spins on FM by \(g_{\rm s}\) and \(G_{\rm s}\) induce diffusive spin currents to NM,  
\[\frac{\partial\mu_{\rm s}}{\partial t}=D\frac{\partial^{2}\mu_{\rm s}}{\partial z ^{2}}-\frac{\mu_{\rm s}}{\tau_{\rm s}}+\left(\frac{g_{\rm s}}{N_{\rm s}}\right) \tag{3}\]  
where \(\mu_{\rm s}\) is the spin chemical potential, \(D\) is the spin diffusion constant, \(\tau_{\rm s}\) is the spin relaxation time, and \(N_{\rm s}\) is the spin density of states (\(N_{\rm s}=N_{\rm F}/2\), where \(N_{\rm F}\) is the electronic density of states at the Fermi level, \(N_{\rm F}=\gamma/(\pi^{2}k_{B}^{2})\), with \(\gamma\) being the electronic heat capacity coefficient). \(D\) is calculated as \(D=\sigma/\left(e^{2}N_{\rm F}\right)\), where \(\sigma\) is the electrical conductivity. \(\tau_{\rm s}\) is calculated as \(\tau_{\rm s}=l_{\rm s}^{2}/D\), where \(l_{\rm s}\) is the spin diffusion length. The \(\mu_{\rm s}\) of each layer are connected at the interface with the interfacial spin conductance \(G/\left(2e^{2}\right)\), where \(G\) is the electrical conductance at the interface. I summarize the \(N_{\rm s}\), \(D\), and \(\tau_{\rm s}\) of each layer in Table 1, and the \(G\) of each interface in Table 2[18; 22; 23; 24; 25; 26; 27; 28; 29; 30]. As spins diffuse from the bottom (the FM/Cu interface) to the surface, the time of the peak of \(\mu_{\rm s}\) shifts through the Cu thickness (Figure 4). The magnitude of \(\mu_{\rm s}\) slightly decreases through the thickness in the Cu and Cu/Au samples, and it decreases more rapidly through the thickness in the Cu/Pt sample because of the short \(\tau_{\rm s}\) of Pt (Figure 4). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Pt** & **[Co/Ni]** & **Cu** & **Au** \\ \hline \(\gamma\) (J m\({}^{-3}\) K\({}^{-2}\)) & 721  & 930  & 97  \\ \(N_{\rm s}\) (eV\({}^{-1}\) nm\({}^{-3}\)) & 92 & 118 & 13 & 9 \\ \(\sigma\) (10\({}^{6}\)\(\Omega^{-1}\) m\({}^{-1}\)) & 6.6 7 [FOOTNOTE:7]Footnote 7 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. & 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. & 8 Obtained from Ref. [25]. \\ \(D\) (nm\({}^{2}\) ps\({}^{-1}\)) & 220 & 80 & 9500 & 9800 \\ \(l_{\rm s}\) (nm) & 4 [FOOTNOTE:4]Footnote 4 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. 8 Obtained from Ref. [25]. & 8 Obtained from Ref. [25]. & 8 Obtained from Ref. [25]. \\ \(\tau_{\rm s}\) (ps) & 0.07 & 0.1 & 17 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bulk parameters for the spin diffusion modeling. \(\gamma\) is the electronic heat capacity coefficient, \(N_{\rm s}\) is the spin density of states, \(\sigma\) is the electrical conductivity, \(D\) is the diffusion constant, \(l_{\rm s}\) is the spin diffusion length, and \(\tau_{\rm s}\) is the spin relaxation time. The spin accumulations on Cu, Au, and Pt contribute to the MOKE in different amounts. I calculate each layer's relative contribution, \(P\), to the MOKE through the thickness using a transfer matrix method with refractive indexes of \(0.3+i5.6\), \(0.2+i4.9\), and \(2.7+i5.9\) for Cu, Au, and Pt, respectively [18] (Figure 5). The \(P\) is a fractional value, and the integration of \(P\) over the entire thickness of the capping (Au or Pt) and Cu layers is one. Using the thickness-dependent information of \(\mu_{\text{s}}\) and \(P\), the Kerr rotation can be calculated as  
\[\Delta\widetilde{\theta}=\left.\frac{\Delta\widetilde{\theta}}{\Delta M} \right|_{1}\mu_{\text{B}}N_{\text{s}1}\int_{0}^{d_{1}}P(z)\mu_{\text{s}}(z)dz+ \left.\frac{\Delta\widetilde{\theta}}{\Delta M}\right|_{2}\mu_{\text{B}}N_{ \text{s}2}\int_{d_{1}}^{d_{1}+d_{2}}P(z)\mu_{\text{s}}(z)dz \tag{4}\]  
where \(d\) is the thickness of each layer and subscript 1 and 2 indicate the capping (Au or Pt) and Cu layer, respectively.