Calculating the spin memory loss at Cu$|$metal interfaces from first principles

In an axially symmetric layered structure, the current flow in the $z$ direction, perpendicular to the interfaces, is described by the spin diffusion equation [46]

where $l_{\rm sf}$ represents the SDL and $\mu_{s}$ is the spin accumulation defined as the difference between the spin-up and spin-down chemical potentials, $\mu_{s}=\mu_{\uparrow}-\mu_{\downarrow}$. The spin-dependent chemical potential $\mu_{\uparrow(\downarrow)}$ drives the corresponding current density $j_{\uparrow(\downarrow)}$ according to Ohm’s law

where $\rho_{\uparrow(\downarrow)}$ is the spin-dependent bulk resistivity and it is assumed that the two spin channels are weakly coupled. The general solutions of (1) and (2) are

and

respectively, where the coefficients $A$ and $B$ are determined by appropriate boundary conditions. Here $\beta=(\rho_{\downarrow}-\rho_{\uparrow})/(\rho_{\uparrow}+\rho_{\downarrow})$ is the bulk spin asymmetry, $j=j_{\uparrow}+j_{\downarrow}$ denotes the total current density and $\rho$ is the total resistivity given by $\rho^{-1}=\rho_{\uparrow}^{-1}+\rho_{\downarrow}^{-1}$. We define the normalized spin-current density to be $j_{s}=(j_{\uparrow}-j_{\downarrow})/(j_{\uparrow}+j_{\downarrow})$ which can be written

Equation (5) can be used together with the spin current density calculated from first principles to extract $\beta$ and $\l_{\rm sf}$ [45] while $\rho$ is determined independently from the scattering matrix using the Landauer formula [76, 77].

For a NM metal or alloy with an equal number of spin-up and spin-down electrons, $\rho_{\uparrow}=\rho_{\downarrow}=2\rho$, $\beta=0$ and (5) simplifies to

If a fully polarized spin current is injected from an artificial half-metallic left lead ($z\leq 0$) into a diffusive NM$|$NM^′ bilayer so $j_{s}(0)=1$, then the spin current will decay in both nonmagnetic metals and, provided the bilayer is sufficiently thick, will vanish at the right lead, $j_{s}(\infty)=0$. On each side of the interface, (6) can be used to fit the calculated $j_{s}(z)$ to obtain the corresponding SDL $l_{i}\equiv l^{i}_{\rm sf}$ for each metal, $i=$NM, NM^′. Extrapolation of the fitting curves to the interface yields two different values for the spin current at the interface, $j_{s}(z_{\rm I}-\eta)\neq j_{s}(z_{\rm I}+\eta)$ where $\eta$ is a positive infinitesimal. The corresponding discontinuity, shown in Fig. 3, represents the interface SML.

In the absence of appropriate boundary conditions for a NM$|$NM^′ interface, we follow [35, 36, 43, 27] and model the interface (I) as a fictitious bulklike material with a finite thickness $t$, a resistivity $\rho_{\rm I}$ and SDL $l_{\rm I}$. By doing so, the NM$|$NM^′ bilayer becomes a NM$|$I$|$NM^′ trilayer in which the spin current and spin accumulation are everywhere continuous. Taking the limit $t\rightarrow 0$, we recover the conventional interface resistance $AR_{\rm I}$ as well as the SML $\delta$,

where the cross sectional area $A$ should not be confused with the disposable constant in (6). In this way, the discontinuity at the interface is naturally included in the semiclassical spin diffusion theory. The result of imposing continuity on $\mu_{s}$ and $j_{s}$ and then taking the limit $t\to 0$ is

By calculating the bulk parameters $\rho_{\rm NM^{\prime}}$ and $l_{\rm NM^{\prime}}$ and the interface resistance $AR_{\rm I}$ independently, only the single unknown parameter $\delta$ needs to be determined from (8) which can be straightforwardly solved numerically. The calculation of $AR_{\rm I}$ using the Landauer-Büttiker formalism will be presented in Sec. III.2.

Unlike NM metals for which $\beta=0$, the asymmetry between the spin-up and spin-down conducting channels in a FM metal leads to $\beta$ saturating to a finite value well inside the ferromagnet, on a length scale of $l_{\rm sf}$. We therefore construct a symmetric NM$|$FM$|$NM trilayer with the origin $z=0$ at the centre of the FM layer and inject an electric current from a NM lead imposing the boundary conditions $j_{s}(-\infty)=j_{s}(\infty)=0$. The calculated spin current in the FM metal can be fitted using (5) to extract $\beta$, $l_{\rm FM}$ [45] and $j_{s}(z_{\rm I}-\eta)$ [57, 62]. In the NM metal, the fitting using (6) is still applicable resulting in $l_{\rm NM}$ and $j_{s}(z_{\rm I}+\eta)$ where because of the symmetry and without loss of generality, we just consider the rightmost FM$|$NM interface. Just as we did for the NM$|$NM^′ interface, we transform the FM$|$NM interface into an FM$|$I$|$NM trilayer with a finite thickness $t$ of the interface region. The only difference is that the FM$|$NM interface resistance is spin dependent and the parameter $\gamma\equiv(AR_{\downarrow}-AR_{\uparrow})/(AR_{\uparrow}+AR_{\downarrow})$ is introduced to characterize the spin polarization.

Taking the limit $t\to 0$ yields the interface discontinuity $j_{s}(z_{\rm I}-\eta)-j_{s}(z_{\rm I}+\eta)\neq 0$ and the analysis [62] results in two implicit equations

which can be solved to yield the remaining unknown variables $\delta$ and $\gamma$ [57, 62].

The boundary conditions $j_{s}(-\infty)=j_{s}(\infty)=0$ are appropriate for structures with NM materials like Pt, for which $l_{\rm sf}$ is short, embedded between nonmagnetic leads [62]. When $l_{\rm sf}$ is as long as it is for NM=Cu, it is not possible to treat a thickness of the NM metal so large that the spin current can be assumed to have decayed to zero at the interfaces with the leads. Instead, we consider the injection or detection of a spin-polarized current corresponding to the boundary conditions $j_{s}(-\infty)$ or $j_{s}(\infty)$ equal to the current polarization $P$ which can have an arbitrary value in the range $[-1,1]$. For simplicity, a half-metallic ferromagnetic lead is considered in Appendix A, where $j_{s}(-\infty)=\pm 1$ is explicitly illustrated. Within the accuracy of the calculations, both sets of boundary conditions lead to the same interface parameters.

Our starting point is an electronic structure for the layered metallic system of interest, calculated self-consistently within the framework of density functional theory [83, 84]. The electronic wave functions are expanded in terms of tight-binding linearized muffin-tin orbitals [85, *Andersen:85, *Andersen:prb86] in the atomic spheres approximation (ASA) [88]. We use the local density approximation with the exchange-correlation functional parameterized by von Barth and Hedin [89]. Transport is addressed by solving the quantum mechanical scattering problem for a general two-terminal geometry using Ando’s wave-function matching method [90] implemented [54, 91] with the tight-binding muffin-tin orbital basis. The conductance is calculated directly from the scattering matrix for the scattering region embedded between semi-infinite ballistic leads [92]. The full quantum-mechanical wave functions are explicitly determined throughout the scattering region from which we calculate position dependent charge and spin currents [93, 45]. Spin-orbit coupling was neglected in calculating the self-consistent atomic potentials but included in the transport calculations [77].

Temperature-induced lattice and spin disorder, alloy disorder and lattice mismatch can be efficiently modelled in “lateral supercells” that assume a measure of periodicity in the directions transverse to the transport direction [77, 45]. The lattice constant of Pt is $a_{\rm Pt}=3.924\,$Å, that of Pd just 0.8% smaller, $a_{\rm Pd}=3.891\,$Å [94]. Those of Py, Co and Cu are about 10% smaller but quite similar with $a_{\rm Py}=3.541$ Å [77], $a_{\rm Co}=3.539$ Å [95] and $a_{\rm Cu}=3.615$ Å [94]. For the Cu$|$Pt, Cu$|$Pd, Py$|$Pt and Co$|$Pt interfaces studied here, an 8$\times$8 interface unit cell of (111) Cu, Py or Co is chosen to match to a 2$\sqrt{13}\times$2$\sqrt{13}$ interface unit cell of Pt or Pd (neglecting the 0.8% difference) which allows all materials to be kept fcc. The two-dimensional Brillouin zone of the 8$\times$8 supercell of (111) Cu is sampled using $28\times 28$ $k$ points and the same $k$ mesh density is used for all the transport calculations. Py is chosen to have its equilibrium lattice constant. To study Cu$|$Py and Cu$|$Co interfaces, Cu must be compressed slightly to match Py, respectively Co; as shown for Au in [95, 82], this changes the Fermi surface, and hence the transport properties, of the noble metal negligibly. The alloy disorder of bulk Py is modelled by randomly populating lateral supercell sites with Fe and Ni atomic sphere potentials subject to the required stoichiometry [77], where the atomic potentials of the alloy are calculated self-consistently within the coherent potential approximation [96, 97]. Interface disorder arising from interface mixing (see Sec. III.4) is modelled as a number of layers of interface alloy [51, 8, 52, 53, 54, 57, 82].

Temperature-induced disorder is modelled in the adiabatic approximation using a “frozen thermal lattice disorder” scheme with atoms displaced at random from their equilibrium lattice positions with a Gaussian distribution of displacements. The root mean square displacement $\Delta$ characterizing the distribution is chosen so that the resistivity of the NM metal at a finite temperature is reproduced as detailed in [55, *LiuY:prb15]. For example, we need a value of $\Delta=0.021a_{\rm Cu}$ to reproduce the room-temperature resistivity of bulk Cu, $1.8\,\mu\Omega\,\mathrm{cm}$. This value of $\Delta$ may be compared to values obtained by populating phonons at 300 K ($0.027a_{\rm Cu}$) [56], from room temperature molecular dynamics simulations ($0.023a_{\rm Cu}$) [98] or from 160 K X-ray diffraction measurements ($0.018a_{\rm Cu}$) [99]. For a FM metal, not only are the atom positions influenced by temperature but also their magnetic ordering. We use a simple Gaussian model of spin disorder parameterized to reproduce the experimental magnetization of the ferromagnet at a given temperature, combined with lattice disorder so that together they reproduce the experimental resistivity as discussed in [55, *LiuY:prb15, 77]. This spin disorder is equivalent to the Fisher distribution [100] at low temperature when the tilting angle measured from the global quantization axis is small [77]. The numerical convergence has been extensively examined with respect to the maximum angular momentum in the basis, size of the lateral supercell, $k$-mesh sampling of the two-dimensional Brillouin zone, the number of random configurations of lattice and spin disorder, as well as the influence of the three-center integrals in the spin-orbit interaction [77, 45, 62, 82]. Compared to calculations of the conductance, we find that more configurations are needed to reduce the error bars on the local currents to acceptable levels.

In the semiclassical theory commonly used to describe spin transport, five bulk parameters $\rho_{\rm NM}$, $l_{\rm NM}$, $\rho_{\rm FM}$, $l_{\rm FM}$, $\beta$ and three interface parameters $AR_{\rm I}$, $\delta$ and $\gamma$ are required to describe transport through the FM$|$NM interface of a bilayer, while the bulk and interface spin polarizations $\beta$ and $\gamma$ vanish for an interface between two NM metals. The numerical techniques required to determine the resistivity of a NM or FM metal, as well as the SDL of most transition metals and alloys using scattering theory are well documented [55, 43, 56, 77, 45]. For free-electron like metals like Cu, Ag and Au, $l_{\rm sf}$ may be as large as several hundred nanometers to micrometers and a quantitative estimation requiring the length of the scattering region to be longer than 3-4$\times l_{\rm sf}$ becomes computationally very demanding [101]. To estimate $l_{\rm Cu}$ we therefore calculated $\rho_{\rm Cu}$ and $l_{\rm Cu}$ simultaneously at elevated temperatures, as a function of the root-mean-square-displacement $\Delta(T)$, to determine the product $\rho(T)l_{\rm sf}(T)$ that for the Elliott-Yafet mechanism of spin relaxation should be temperature independent and then used the confirmed linear relationship to extrapolate $l_{\rm Cu}$ to the required lower temperature.

$l_{\rm Cu}(\Delta(T))$ is plotted in Fig. 1 as a function of the simultaneously determined conductivity for a number of values of $\Delta$ (solid squares); the approximate linearity indicates that the Elliott-Yafet mechanism [102, 103] is dominant [101]. The product $\rho_{\rm Cu}l_{\rm Cu}=(9.0\pm 1.0)\times 10^{-15}\ \Omega\ {\rm m}^{2}$ is obtained from linear least squares fitting. This linear relationship is extrapolated to the lower temperatures corresponding to the documented conductivity of bulk Cu at 100, 200, 300 and 400 K which are shown as the empty circles in Fig. 1. In particular, corresponding to the room temperature resistivity $\rho_{\rm Cu}=1.8\pm 0.1\ \mu\Omega$ cm, we estimate a value of $l_{\rm Cu}\sim 502$ nm.

We begin by examining the spin memory loss at the Cu$|$Pt interface that commonly appears in nonlocal spin-valve, spin-pumping and spin-Seebeck experiments. We then compare our results when Pt is replaced by the isoelectronic and isostructural but lighter element Pd.

Before addressing the SML, the Cu$|$Pt interface resistance needs to be determined using the standard Landauer-Büttiker formalism [92, 104]. As sketched in Fig. 2(a), we do this by sandwiching a symmetric, diffusive Cu$|$Pt$|$Cu trilayer between ballistic Cu leads in a two-terminal scattering configuration where element specific parameters $\Delta_{i}$ are used to reproduce the room-temperature resistivities of bulk Cu and Pt, respectively. The total calculated area resistance product for a cross-section $A$ of the scattering geometry can be written as

where $G_{\rm Sh}$ denotes the Sharvin conductance of the ballistic Cu leads, $AR_{\rm Cu|Lead}$ is the resistance of the interface between the ballistic Cu lead and diffusive Cu, $2\rho_{\rm Cu}L_{\rm Cu}$ with $L_{\rm Cu}=10\,$nm is the resistance of a length $2L_{\rm Cu}=20\,$nm of diffusive Cu ¹¹1An explicit numerical check showed that $L_{\rm Cu}=10\,$nm is enough to reproduce the linear dependence of the resistance on the Cu length and therefore a longer $L_{\rm Cu}$ would not alter the final value of the SML that we are interested in., $\rho_{\rm Pt}L_{\rm Pt}$ is the resistance of a length $L_{\rm Pt}$ of diffusive Pt and $AR_{\rm Cu|Pt}$ is the sought-after Cu$|$Pt interface resistance. By repeating the calculations without the central Pt layer, we obtain the contribution from the first three terms on the right hand side of (10) separately. This is subtracted from the total resistance in (10) leaving us with $2AR_{\rm Cu|Pt}+\rho_{\rm Pt}L_{\rm Pt}$ that is plotted in Fig. 2(b) as a function of $L_{\rm Pt}$ where each data point is obtained by averaging over seven configurations of random thermal disorder. The slope of the linear least squares fit reproduces the resistivity of bulk Pt at room temperature while the intercept results in the interface resistance $AR_{\rm Cu|Pt}=0.64\pm 0.03\ {\rm f}\Omega\ {\rm m}^{2}$.

$l_{\rm Pt}$ [45, 101], $l_{\rm Cu}$ and $AR_{\rm Cu|Pt}$ have now been calculated, $\rho_{\rm Cu}$ and $\rho_{\rm Pt}$ have their experimental values by construction, leaving just $j_{s}(z_{\rm I}+\eta)$ and $j_{s}(z_{\rm I}-\eta)$ to be determined before $\delta_{\rm Cu|Pt}$ can be evaluated using (8). To do so, we inject a fully polarized spin current from an artificial half-metallic Cu lead [77, 45] into a diffusive, room temperature (RT: 300 K) Cu(10 nm)$|$Pt(30 nm) bilayer. The $z$-dependent spin current density is plotted in Fig. 3, where each data point corresponds to $j_{s}$ averaged over an atomic layer and over twenty random configurations of thermal lattice disorder. On both sides of the interface, the spin current can be fitted very well using (6) as shown by the solid orange and red lines in Fig. 3. As expected from the large value of $l_{\rm Cu}$, the spin current decays extremely slowly in Cu, as shown in the inset. To reduce the uncertainty from fitting, we use the room temperature value $l_{\rm Cu}\sim 502$ nm yielding very good agreement with the calculated data. The spin current entering Pt exhibits an exponential decay which can be characterized by a value of $l_{\rm Pt}=5.3\pm 0.1\,$nm consistent with the value $5.26\pm 0.04$ nm reported previously in [57]. The fitted semiclassical description of the spin current exhibits a substantial discontinuity at the Cu$|$Pt interface corresponding to the interface SML, Fig. 3. Using the values $j_{s}(z_{\rm I}+\eta)=0.997$ and $j_{s}(z_{\rm I}-\eta)=0.512$ obtained by extrapolation, $\rho_{\rm Pt}l_{\rm Pt}=0.6~{}{\rm f}\Omega~{}{\rm m}^{2}$, and $AR_{\rm Cu|Pt}=0.64\pm 0.03~{}{\rm f}\Omega~{}{\rm m}^{2}$, we finally estimate the SML to be $\delta_{\rm Cu|Pt}=0.77\pm 0.04$ for the Cu$|$Pt interface. This value of $\delta$ is very close to and only slightly smaller than the value of 0.81 we reported for the relaxed Au$|$Pt interface and substantially larger than the value 0.62 we found for the compressed Au$|$Pt interface [57] indicating the importance of taking lattice mismatch into account, Table 1.

By repeating the complete procedure for the Cu$|$Pd interface and neglecting the 0.8% smaller lattice constant of Pd, we find $\delta=0.45\pm 0.03$ at 300 K that is smaller than the SML for the Cu$|$Pt interface because of the weaker spin-orbit interaction in Pd. This value for the Cu$|$Pd interface is also consistent with the SML values of 0.43 and 0.63 calculated for the “commensurate” (with Au compressed to match to Pd) and “incommensurate” Au$|$Pd interfaces [57], respectively. We attribute this agreement to the very similar electronic structures of Cu and Au, where the free-electron-like $s$ band dominates transport at the Fermi level; the filled 3$d$ or 5$d$ bands are well below the Fermi energy so the difference in spin-orbit coupling strength between Cu and Au does not play a significant role.

We compare our calculated interface resistances and SML parameters with experimental and theoretical values available in the literature in Table 2. For the Cu$|$Pt interface, our calculated values of $AR_{\rm I}$ and $\delta$ are somewhat smaller than the experimental values. Experimentally, the influence of temperature on the SML of a NM$|$NM^′ interface is found to be weak [106] which is consistent with our theoretical findings [43, 57]. This is because the main scattering mechanism at the interface is the abrupt variation in the atomic potentials experienced by conduction electrons [43]. For the Cu$|$Pd interface, the value we calculate for $\delta$ is in perfect agreement with that calculated by Belashchenko et al. [58] but is larger than the only reported, low-temperature experimental value [81]. To make more progress, structural properties need to be correlated with transport measurements. We identify a lack of structural characterization of interfaces on the atomic level as a major stumbling block to developing more comprehensive understanding. The theoretical description we have presented can be applied to considerably more complex structural models.

We proceed to study the SML at the interfaces between Cu and the FM metals Py and Co. The exchange interaction in FM metals automatically generates a spin polarization of the conduction electrons so that we do not need to artificially inject a spin-polarized current from the half-metallic lead as in the previous section.

Even in the best experimental samples, the interfaces will almost certainly not be as well ordered as those we have considered so far. In the process of growing thin layers, the kinetic energy of the deposited atoms will lead to interface mixing, rendering the interfaces less sharp [81]. We model the mixing of an A$|$B interface by completely mixing one (or two) layers on either side of the interface that then leads to two (or four) layers with the composition A₅₀B₅₀. The lattice and spin disorder is assumed to be unchanged, as are the atomic sphere potentials. The spin currents that result from these calculations for Cu$|$Py$|$Cu structures with intermixed interfaces are shown in Fig. 7(a). They are fitted in Py and Cu using equations (5) and (6), respectively and extrapolated to the Py$|$Cu interface at $z_{\rm I}$ to determine $j_{s}(z_{\rm I}-\eta)$ on the Py side and $j_{s}(z_{\rm I}+\eta)$ on the Cu side, Fig. 7(a), lower panel. The interface resistance $AR_{\rm I}$ is determined in separate calculations and finally $\delta$ and $\gamma$ are extracted by solving equations (9). The results are shown for both Cu$|$Py and Cu$|$Co interfaces in Fig. 7(b). $AR_{\rm I}$ is seen to increase monotonically with increasing thickness of the mixed interface layer because of the strong scattering by alloy disorder. The interface spin asymmetry parameter $\gamma$ is not changed by the intermixing. The majority-spin potentials of Co, and of both Ni and Fe in Py, are perfectly matched to the potential of Cu while the minority-spin potentials all differ. For this reason, both Cu$|$Py and Cu$|$Co interfaces exhibit strong spin filtering and this effect is not significantly weakened by mixing the magnetic and Cu atoms.

The effect of interface mixing on the SML for Cu$|$Py and Cu$|$Co interfaces is complex. First, the stronger interface scattering by the interface alloy enhances the spin flipping, as we found for the nonmagnetic Au$|$Pt interface [57]. For a Cu$|$FM interface, alloying reduces the magnetic order on the FM side and this reduces the SML by analogy with the reduction we found for Pt$|$Co and Pt$|$Py interfaces on increasing the temperature [57]. Competition between the two effects results in the (slightly) nonmonotonic dependence of $\delta$ on the thickness of the interface alloy layer at Cu$|$Py and Cu$|$Co interfaces shown in Fig. 7(b). This nonmonotonic behavior is clearer for Cu$|$Co since Co has a higher degree of magnetic order than Py.

Cu is commonly used in experiments as a spacer material between a heavy metal like Pt with a large spin susceptibility and a FM metal, to prevent magnetism being induced in Pt by the proximity effect. Because of its weak spin-orbit interaction and correspondingly long SDL, a thin layer of Cu is usually assumed to have no effect on a spin current thus making the interpretation of experiments simpler. However, though bulk Cu may have little effect on a spin current, insertion of a Cu layer between a FM metal and Pt replaces the singel FM$|$Pt interface with two different interfaces, namely, FM$|$Cu and Cu$|$Pt interfaces whose effect on a spin current is, at best, poorly known. Here we consider Py and Co as typical FM metals and Pt as a typical heavy metal to investigate the influence of inserting Cu layers on the spin memory loss.

As shown schematically in Fig. 8(a), we calculate the spin current distribution at 300 K for a Pt$|$Cu$|$Py$|$Cu$|$Pt multilayer when a charge current is passed through it. The thickness of the Py (Pt) layers is 26 nm (25 nm) and we consider $N=0$, 1 and 2 atomic layers of Cu. The spin current in the transport direction ($z$) that we calculate for a symmetric Pt$|$Cu($N$)$|$Py$|$Cu($N$)$|$Pt multilayer is shown in Fig. 8(b) for $N=2$. Both the saturated plateau in the center of Py and the vanishing spin current at the interfaces between Pt and the leads confirm that Py and Pt are sufficiently thick and satisfy the boundary condition $j_{s}(\pm\infty)=0$. We fit $j_{s}(z)$ using the spin diffusion equations in Py and Pt and use these fits to extrapolate $j_{s}(z)$ on the Py side to the Py$|$Cu interface to calculate $j_{s}(z_{\rm I}-\eta)$ and on the Pt side to the Cu$|$Pt interfaces to calculate $j_{s}(z_{\rm I}+\eta)$, respectively as illustrated in the exploded plot in Fig. 8(c). The somewhat arbitrary choice of $z_{\rm I}$ within the Cu insert has a negligible effect on the values of $\delta$ and $\gamma$ that we extract because of the thinness of Cu. The SML determined in this way accounts for the spin-flipping at the “compound interface” between Py and Pt.

Using the above scheme, we calculate the interface parameters $AR_{\rm I}$, $\gamma$ and $\delta$ for Py$|$Cu$|$Pt and Co$|$Cu$|$Pt interfaces at room temperature and summarize the results in Fig. 9 and Table 5. The Cu insert increases $AR_{\rm Py|Pt}$ to between 1.06 and $1.11~{}{\rm f}\Omega\,{\rm m}^{2}$, which is very close to the sum of the two interface resistances connected in series, $AR_{\rm Py|Cu}+AR_{\rm Cu|Pt}=1.05\pm 0.04~{}{\rm f}\Omega~{}{\rm m}^{2}$. Because Cu is so thin and conductive, its contribution to $AR_{\rm I}$ can be neglected. The parameter $\gamma$ is small for both the Py$|$Pt and Co$|$Pt interfaces but increases with increasing Cu thickness. This is because the Py$|$Cu and Co$|$Cu interfaces have strong spin filtering effects. The scattering rate for minority-spin electrons is enhanced by the Cu insert leading to an increase in $\gamma$. The dependence of $AR_{\rm I}$ and $\gamma$ on the thickness of Cu is the same for both Py$|$Cu$|$Pt and Co$|$Cu$|$Pt interfaces.

We finally consider the SML with and without the Cu insert. Without it, $\delta_{\rm Py|Pt}=0.76\pm 0.11$ at room temperature. As shown in Table 5 and Fig. 9, inserting Cu increases $\delta_{\rm Py|Pt}$ slightly to $0.86\pm 0.12$ for $N=1$ and to $1.00\pm 0.14$ for $N=2$ compared to $\delta_{\rm Py|Cu}+\delta_{\rm Cu|Pt}=(0.00^{+0.09})+(0.77\pm 0.04)=0.77\pm 0.10$ for the two separate interfaces. For the Co$|$Pt interface, the SML increases from $\delta_{\rm Co|Pt}=0.77\pm 0.13$ to $1.12\pm 0.16$ for $N=1$ and $1.18\pm 0.17$ for $N=2$. The sum of the room temperature SML values for the individual interfaces, $\delta_{\rm Co|Cu}+\delta_{\rm Cu|Pt}=(0.11\pm 0.04)+(0.77\pm 0.04)=0.88\pm 0.06$, is substantially lower than the SML we find for the compound Co$|$Cu$|$Pt interface. Thus, contrary to the expectation that separating the FM and heavy metal should enhance the interface transparency for a spin current [114], we find that it increases the SML.