# Magnetic Nanowires
## 3 Results
### Spin Caloritronics in Nanowire Networks
In the limit of no-spin relaxation, most of the CPP-GMR data can be understood using a simple two-current series-resistor model, in which the resistance of layers and interfaces simply add and where 'up' and 'down' charge carriers are propagating independently in two spin channels with large spin asymmetries of the electron's scattering [105; 106]. Similarly, significantly different Seebeck coefficients for spin-up and spin-down electrons, \(S_{\uparrow}\) and \(S_{\downarrow}\), are expected because the d-band is exchange-split in these ferromagnets, as suggested from previous works performed on dilute magnetic alloys [107; 108]. Assuming that the layers of the magnetic multilayers are thin compared to the spin-diffusion lengths and according to the usual rule when the currents split to flow along two parallel paths (see Figure 14a), the corresponding thermopowers \(S_{\text{AP}}\) and \(S_{\text{P}}\) in the high- and low-resistance states, respectively, are simply given by [109]:  
\[S_{\text{AP}}=\frac{S_{\uparrow}\rho_{\uparrow}+S_{\downarrow}\rho_{\downarrow}} {\rho_{\uparrow}+\rho_{\downarrow}} \tag{9}\]  
and  
\[S_{\text{P}}=\frac{S_{\uparrow}\rho_{\downarrow}+S_{\downarrow}\rho_{\uparrow }}{\rho_{\uparrow}+\rho_{\downarrow}}, \tag{10}\]  
where separate resistivities \(\rho_{\uparrow}\) and \(\rho_{\downarrow}\) and Seebeck coefficients \(S_{\uparrow}\) and \(S_{\downarrow}\) are defined for majority and minority spin channels. Therefore, the spin-dependent Seebeck coefficients can be expressed as follows:  
\[S_{\uparrow}=\frac{1}{2}\big{[}S_{\text{AP}}\big{(}1-\beta^{-1}\big{)}+S_{ \text{P}}\big{(}1+\beta^{-1}\big{)}\big{]}, \tag{11}\]  
\[S_{\downarrow}=\frac{1}{2}\big{[}S_{\text{AP}}\big{(}1+\beta^{-1}\big{)}+S_{ \text{P}}\big{(}1-\beta^{-1}\big{)}\big{]}, \tag{12}\]  
where \(\beta=(\rho_{\downarrow}-\rho_{\uparrow})/(\rho_{\downarrow}+\rho_{\uparrow})\) denotes the spin asymmetry coefficient for resistivity. From Equations (11) and (12), it can be easily deduced that \(S_{\uparrow}=S_{\text{P}}\) and \(S_{\downarrow}=S_{\text{AP}}\) in the limit of an extremely large MR ratio (\(\beta\to 1\)). Figure 14b shows the temperature evolutions of \(S_{\text{AP}}\), \(S_{\text{P}}\), \(S_{\uparrow}\) and \(S_{\downarrow}\) for CoNi/Cu CNWs using a rough estimate of \(\beta=\text{MR}^{1/2}\). Below RT, the various Seebeck coefficients decrease almost linearly with decreasing temperature, which is indicative of the dominance of diffusion thermopower. Similar results were obtained on Co/Cu CNWs (not shown in Figure 14). From the analysis, the estimated values for \(\Delta S=S_{\uparrow}-S_{\downarrow}\) of \(-8.5\)\(\mu\)V/K and  
Figure 13: **(a)** calculated thermopower for Co/Cu multilayers in the layer parallel (dash-dotted line) and perpendicular (solid line) directions vs thickness ratio \(\lambda=t_{\text{FM}}/t_{\text{Cu}}\) using Equations (7) and (8), and bulk values for \(S_{\text{Co}}\), \(\rho_{\text{Co}}\), \(S_{\text{Cu}}\) and \(\rho_{\text{Cu}}\). The grey dashed line shows the values for \(\lambda=1\). The inset shows a FM/Cu multilayer stack; **(b)** experimental set-up for Seebeck coefficient and the magneto-thermoelectric effect measurements. The color represents the generated temperature profile in the NW networks; **(c,d)** Electrical resistance **(c)** and Seebeck coefficient **(d)** of a Co/Cu NW network (80 nm in diameter and 3% packing density) at room temperature. The curves in **(c,d)** were obtained with the applied field in-plane (IP—in blue) and out-of-plane (OOP—in red) of the NW network film (adapted from [65]). Figure adapted from ref. [65] with permission of the author. \(-10\)\(\mu\)V/K for Co/Cu and CoNi/Cu CNWs, respectively, are much larger than the one of \(-1.8\)\(\mu\)V/K extracted from measurements performed on Co/Cu/Co nanopillar spin valve using a 3D finite-element model [96; 110]. Recently, we have demonstrated the fabrication of nanowire arrays based on diluted NiCr and CoCr alloys by electrodeposition in 3D porous membranes [66].