Effect of quantum tunneling on spin Hall magnetoresistance

We start with the quantum tunneling formalism that calculates the interface spin current in the NM/FMI bilayer, which later serves as the boundary condition for the spin diffusion equation that determines SMR. The quantum tunneling formalism describes the NM/FMI bilayer shown in figure 1(a) by the Hamiltonian

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{H}_{N}} & = & \frac{{{p}^{2}}}{2m}-\mu _{x}^{\sigma}\;\;\;\left(-{{l}_{N}}\leqslant x<0\right), \end{array}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{H}_{FI}} & = & \frac{{{p}^{2}}}{2m}+{{V}_{0}}+ \Gamma \boldsymbol{S}\centerdot \boldsymbol{\sigma }\;\;\;\left(0\leqslant x\leqslant {{l}_{FI}}\right), \end{array}\end{eqnarray} \tag{ 2 }$$

where $\mu _{x}^{\sigma}=\pm \boldsymbol{\mu }_{{x}}\centerdot \widehat{\boldsymbol{z}}/2$ is the spin voltage of $\sigma =\left\{\uparrow,\downarrow \right\}$ produced by an in-plane charge current $J_{y}^{c}\mathbf{\hat{y}}$, ${{V}_{0}}-{{\epsilon}_{\text{F}}}$ is the insulating gap with ${{\epsilon}_{\text{F}}}$ the Fermi energy, and $\boldsymbol{S}=$$S(\sin \theta \cos \varphi,\sin \theta \sin \varphi,\cos \theta )$ is the magnetization. We choose $\Gamma <0$ such that the magnetization has the tendency to align with the conduction electron spin $\boldsymbol{\sigma }$. The wave function near the interface is

$$\begin{eqnarray}&&{{\psi}_{N}}=\left(A{{\text{e}}^{\text{i}{{k}_{0\uparrow}}x}}+B{{\text{e}}^{-\text{i}{{k}_{0\uparrow}}x}}\right)\left(\begin{array}{c} 1 \\ 0 \end{array}\right)+C{{\text{e}}^{-\text{i}{{k}_{0\downarrow}}}}\left(\begin{array}{c} 0 \\ 1 \end{array}\right),\quad \quad \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{\psi}_{FI}}=\left(D{{\text{e}}^{{{q}_{+}}x}}+E{{\text{e}}^{-{{q}_{+}}x}}\right)\left(\begin{array}{c} {{\text{e}}^{-\text{i}\varphi /2}}\cos \frac{\theta}{2} \\ {{\text{e}}^{\text{i}\varphi /2}}\sin \frac{\theta}{2} \end{array}\right)\\ &&+\,\left(F{{\text{e}}^{{{q}_{-}}x}}+G{{\text{e}}^{-{{q}_{-}}x}}\right)\left(\begin{array}{c} -{{\text{e}}^{-\text{i}\varphi /2}}\sin \frac{\theta}{2} \\ {{\text{e}}^{\text{i}\varphi /2}}\cos \frac{\theta}{2} \end{array}\right),\end{eqnarray} \tag{ 4 }$$

where ${{k}_{0\sigma}}=\sqrt{2m\left({{\epsilon}_{\text{F}}}+\mu _{0}^{\sigma}\right)}/\hbar$ and ${{q}_{\pm }}=\sqrt{2m\left({{V}_{0}}\pm \Gamma S-{{\epsilon}_{\text{F}}}\right)}/\hbar$. The amplitudes $B\sim E$ are solved in terms of the incident amplitude A by matching wave functions and their first derivative at the interface. The x  <  −l_N and x  >  l_FI regions are assumed to be vacuum or insulating oxides that correspond to infinite potentials such that the wave functions vanish there for simplicity. We identify the incident flux with $|A{{|}^{2}}={{N}_{F}}|\boldsymbol{\mu }_{{0}}|/{{a}^{3}}$ where N_F is the density of states per a³ with $a=2\pi /{{k}_{F}}=h/\sqrt{2m{{\epsilon}_{\text{F}}}}$ the Fermi wave length.

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The spin current inside the FMI at position x is calculated from the evanescent wave function

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{j}_{{x}}=\frac{\hbar}{4\text{i}m}\left[\psi _{FI}^{\ast}\boldsymbol{\sigma }\left({{\partial}_{x}}{{\psi}_{FI}}\right)-\left({{\partial}_{x}}\psi _{FI}^{\ast}\right)\boldsymbol{\sigma }{{\psi}_{FI}}\right]. \end{array}\end{eqnarray} \tag{ 5 }$$

Angular momentum conservation [8, 28] dictates that the interface spin current to be equal to the STT exerts on the magnetization

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{j}_{{0}}-\boldsymbol{j}_{{{{l}_{FI}}}}=\boldsymbol{j}_{{0}}=\frac{\boldsymbol{\tau }}{{{a}^{2}}}=\frac{ \Gamma S{{N}_{F}}}{\hbar}\left[{{G}_{r}}\widehat{\boldsymbol{S}}\times \left(\widehat{\boldsymbol{S}}\times \boldsymbol{\mu }_{{0}}\right)+{{G}_{i}}\widehat{\boldsymbol{S}}\times \boldsymbol{\mu }_{{0}}\right], \end{array}\end{eqnarray} \tag{ 6 }$$

which defines the field-like G_i and damping-like G_r spin mixing conductance that in turn can be calculated from the interface spin current [28]

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{ \Gamma S{{N}_{F}}}{\hbar}{{G}_{r}} & = & \frac{2j_{0}^{x}\cos \varphi}{|\boldsymbol{\mu }_{{0}}|\sin 2\theta}+\frac{2j_{0}^{y}\sin \varphi}{|\boldsymbol{\mu }_{{0}}|\sin 2\theta}=-\frac{j_{0}^{z}}{|\boldsymbol{\mu }_{{0}}|{{\sin}^{2}}\theta}, \\ \frac{ \Gamma S{{N}_{F}}}{\hbar}{{G}_{i}} & = & \frac{j_{0}^{x}\sin \varphi}{|\boldsymbol{\mu }_{{0}}|\sin \theta}-\frac{j_{0}^{y}\cos \varphi}{|\boldsymbol{\mu }_{{0}}|\sin \theta}. \end{array}\end{eqnarray} \tag{ 7 }$$

A straight forward calculation yields

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{G}_{r,i}} & = & \frac{-4}{{{a}^{3}}|{{\gamma}_{\theta}}{{|}^{2}}}\left(\frac{{{q}_{+}}\coth {{q}_{+}}{{l}_{FI}}-{{q}_{-}}\coth {{q}_{-}}{{l}_{FI}}}{q_{+}^{2}-q_{-}^{2}}\right) \\ {} & {} & \times \,\left(\text{Im},\text{Re}\right)\;\left(n_{\downarrow +}^{\ast}{{n}_{\downarrow -}}\right), \end{array}\end{eqnarray} \tag{ 8 }$$

where ${{\sigma}_{x,y}}$ is x, y component of Pauli matrix, and

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{n}_{\sigma \pm }} & = & \frac{{{k}_{0\sigma}}}{\left({{k}_{0\sigma}}+\text{i}{{q}_{\pm }}\coth {{q}_{\pm }}{{l}_{FI}}\right)}, \\ {{\gamma}_{\theta}} & = & \frac{{{n}_{\downarrow +}}}{{{n}_{\uparrow +}}}{{\cos}^{2}}\frac{\theta}{2}+\frac{{{n}_{\downarrow -}}}{{{n}_{\uparrow -}}}{{\sin}^{2}}\frac{\theta}{2}. \end{array}\end{eqnarray} \tag{ 9 }$$

Equation (8) describes the spin mixing conductance in STT, as well as that in spin pumping since the Onsager relation [34] is satisfied in this approach [28]. Both G_r and G_i have very weak dependence (at most few percent) on the angle of magnetization θ through ${{\gamma}_{\theta}}$, which may be considered as higher order contributions [28]. In the numerical calculation below we set $\theta =0.3\pi$ without loss of generality.

Numerical results of the spin mixing conductance G_r,i are shown in figure 1, plotted as a function the FMI thickness l_FI and at different strength of the interface s  −  d coupling $\Gamma S/{{\epsilon}_{\text{F}}}$. Both G_r and G_i increase with l_FI initially and then saturate to a constant as expected, since they originate from the quantum tunneling of conduction electrons that only penetrate into the FMI over a very short distance. At a FMI thickness small compared to Fermi wave length ${{l}_{FI}}\ll a$, we found that ${{G}_{r}}\propto l_{FI}^{6}$ and ${{G}_{i}}\propto l_{FI}^{3}$, therefore the damping-like to field-like ratio is $|{{G}_{r}}/{{G}_{i}}|\ll 1$. In most of the parameter space, the torque is dominated by field-like component $|{{G}_{r}}/{{G}_{i}}|<1$ throughout the whole range of l_FI. Only when the magnitude of s  −  d coupling is large compared to the insulating gap $\left({{V}_{0}}-{{\epsilon}_{\text{F}}}\right)/{{\epsilon}_{\text{F}}}$ is the torque dominated by the damping-like component $|{{G}_{r}}/{{G}_{i}}|>1$, consistent with that found previously [28] and also in accordance with the result from first principle calculation [26]. The magnitude of G_r,i generally increases with the s  −  d coupling, yet more dramatically for G_r. Note that G_r and G_i do not depend on the NM thickness in this quantum tunneling approach.

We adopt the spin diffusion approach of Chen et al [23] to address the effect of the interface spin current in equation (6) on SMR, which is briefly summarized below. The spin diffusion approach is based on the following assumptions for the spin transport in the NM: (1) The spin current in NM consists of two parts, one from the spatial gradient of spin voltage and the other the bare spin current caused directly by SHE,

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{j}_{{x}}=-\frac{{{\sigma}_{c}}}{4{{e}^{2}}}{{\partial}_{x}}\boldsymbol{\mu }_{{x}}+\frac{{{\theta}_{\text{SH}}}{{\sigma}_{c}}{{E}_{y}}}{2e}\widehat{\boldsymbol{z}}, \end{array}\end{eqnarray} \tag{ 10 }$$

where ${{\theta}_{\text{SH}}}$ is spin Hall angle, ${{\sigma}_{c}}$ is the conductivity of NM, E_y is applied external electric in y direction, and  −e is electron charge. (2) The spin voltage obeys the spin diffusion equation ${{\nabla}^{2}}\boldsymbol{\mu }_{{x}}=\boldsymbol{\mu }_{{x}}/{{\lambda}^{2}}$, where λ is the spin diffusion length. (3) Spin current vanishes at the edge of NM (x  =  −l_N), which serves as one boundary condition. (4) The spin current at the NM/FMI interface is described by equation (6), which serves as another boundary condition. The self-consistent solution satisfying (1)–(4) is [23]

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{\boldsymbol{j}_{{x}}\centerdot \widehat{\boldsymbol{x}}}{{{j}_{\text{SH}}}} & = & {{\beta}_{x}}\sin \theta \left[\cos \theta \cos \varphi \text{Re}\left(\tilde{G}\right)+\sin \varphi \text{Im}\left(\tilde{G}\right)\right], \\ \frac{\boldsymbol{j}_{{x}}\centerdot \widehat{\boldsymbol{y}}}{{{j}_{\text{SH}}}} & = & {{\beta}_{x}}\sin \theta \left[\cos \theta \sin \varphi \text{Re}\left(\tilde{G}\right)-\cos \varphi \text{Im}\left(\tilde{G}\right)\right], \\ \frac{\boldsymbol{j}_{{x}}\centerdot \widehat{\boldsymbol{z}}}{{{j}_{\text{SH}}}} & = & 1-\frac{\cosh \left(\frac{2x+{{l}_{N}}}{2\lambda}\right)}{\cosh \left(\frac{{{l}_{N}}}{2\lambda}\right)}-{{\beta}_{x}}{{\sin}^{2}}\theta \text{Re}\left(\tilde{G}\right), \end{array}\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\beta}_{x}} & = & \frac{\sinh \left(\frac{x+{{l}_{N}}}{\lambda}\right)}{\sinh \left(\frac{{{l}_{N}}}{\lambda}\right)}\tanh \left(\frac{{{l}_{N}}}{2\lambda}\right), \\ \tilde{G} & = & \frac{\alpha {{G}_{c}}}{1-\alpha {{G}_{c}}\coth \left(\frac{{{l}_{N}}}{\lambda}\right)}, \\ \alpha & = & \frac{4 \Gamma S{{N}_{F}}{{e}^{2}}\lambda}{\hbar {{\sigma}_{c}}}, \end{array}\end{eqnarray} \tag{ 12 }$$

and ${{j}_{\text{SH}}}={{\theta}_{\text{SH}}}{{\sigma}_{c}}{{E}_{y}}/2e$ is the bare spin current. Here $\alpha <0$ is a negative parameter (because we assume the interface s  −  d coupling $\Gamma <0$) that bridges our tunneling formalism to the spin diffusion equation, and ${{G}_{c}}={{G}_{r}}+\text{i}{{G}_{i}}$ is the complex spin mixing conductance.

Through ISHE, the spin currents in equation (11) is converted back to a charge current in the longitudinal (along $\widehat{\boldsymbol{y}}$) and transverse (along $\widehat{\boldsymbol{z}}$) direction

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Delta j_{\text{long}}^{c}(x) & = & -2e{{\theta}_{\text{SH}}}\left(\boldsymbol{j}_{{x}}-\frac{{{\theta}_{\text{SH}}}{{\sigma}_{c}}{{E}_{y}}}{2e}\widehat{\boldsymbol{z}}\right)\centerdot \widehat{\boldsymbol{z}}, \end{array}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Delta j_{\text{trans}}^{c}(x) & = & 2e{{\theta}_{\text{SH}}}\left(\boldsymbol{j}_{{x}}-\frac{{{\theta}_{\text{SH}}}{{\sigma}_{c}}{{E}_{y}}}{2e}\widehat{\boldsymbol{z}}\right)\centerdot \widehat{\boldsymbol{y}}. \end{array}\end{eqnarray} \tag{ 14 }$$

The conductivity averaged over the NM layer then follows

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\sigma}_{\text{long}}} & = & \sigma +\frac{1}{{{l}_{N}}{{E}_{y}}}{\int}_{-{{l}_{N}}}^{0}\text{d}x\; \Delta j_{\text{long}}^{c}(x), \end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\sigma}_{\text{trans}}} & = & \frac{1}{{{l}_{N}}{{E}_{y}}}{\int}_{-{{l}_{N}}}^{0}\text{d}x\; \Delta j_{\text{trans}}^{c}(x). \end{array}\end{eqnarray} \tag{ 16 }$$

Using $\theta _{\text{SH}}^{2}\sim 0.01\ll 1$, the longitudinal and transverse component of SMR read

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\rho}_{\text{long}}} & = & \sigma _{\text{long}}^{-1}\approx \rho + \Delta {{\rho}_{0}}+{{\sin}^{2}}\theta \; \Delta {{\rho}_{1}}, \\ {{\rho}_{\text{trans}}} & = & -{{\sigma}_{\text{trans}}}/\sigma _{\text{long}}^{2} \\ {} & \approx & \cos \theta \sin \theta \sin \varphi \; \Delta {{\rho}_{1}}-\sin \theta \cos \varphi \; \Delta {{\rho}_{2}}, \end{array}\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Delta {{\rho}_{0}}/\rho & = & -\theta _{\text{SH}}^{2}\frac{2\lambda}{{{l}_{N}}}\tanh \left(\frac{{{l}_{N}}}{2\lambda}\right), \\ \Delta {{\rho}_{1}}/\rho & = & -\theta _{\text{SH}}^{2}\frac{\lambda}{{{l}_{N}}}{{\tanh}^{2}}\left(\frac{{{l}_{N}}}{2\lambda}\right)\text{Re}\left(\tilde{G}\right), \\ \Delta {{\rho}_{2}}/\rho & = & \theta _{\text{SH}}^{2}\frac{\lambda}{{{l}_{N}}}{{\tanh}^{2}}\left(\frac{{{l}_{N}}}{2\lambda}\right)\text{Im}\left(\tilde{G}\right). \end{array}\end{eqnarray} \tag{ 18 }$$

Clearly the FMI thickness l_FI affects $\Delta {{\rho}_{1}}$ and $\Delta {{\rho}_{2}}$ only through the $\tilde{G}=\tilde{G}\left({{l}_{FI}}\right)$ in equation (12).

To perform numerical calculation of equation (18), we make the following assumption on the parameter α in equation (12) that connects the quantum tunneling formalism with the spin diffusion equation. Firstly, α contains the density of state per a³ at the Fermi surface, which is assumed to be the inverse of Fermi energy ${{N}_{F}}=1/{{\epsilon}_{\text{F}}}$. The combined parameter $\Gamma S{{N}_{F}}= \Gamma S/{{\epsilon}_{\text{F}}}$ therefore represents the strength of s  −  d coupling. Other parameters that influence α are the spin diffusion length assumed to be $\lambda \approx 10$ nm, the conductivity of the NM film taken to be ${{\sigma}_{c}}\approx 5\times {{10}^{6}}{{ \Omega }^{-1}}$ m⁻¹, and Fermi wave length assumed to be roughly equal to the lattice constant $a\approx 0.4$ nm, all of which are the typical values for commonly used materials such as Pt. These lead to the dimensionless parameter $\alpha {{G}_{c}}\approx 10\times \left(\Gamma S/{{\epsilon}_{\text{F}}}\right)\times \left({{G}_{c}}/\left({{e}^{2}}/\hbar {{a}^{2}}\right)\right)$ in equation (12) being expressed in terms of the relative strength of s  −  d coupling and the spin mixing conductance divided by its unit. In what follows, we examine the effect of FMI thickness, NM thickness, insulating gap, and interface s  −  d coupling on SMR. On the contrary, the spin Hall angle, spin diffusion length, and conductivity are treated as constants, although in reality they may also depend on the layer thickness or on each other in such thin films [35].

The numerical result of SMR is shown in figure 2, plotted as a function of the FMI thickness l_FI and NM thickness l_N at several values of insulating gap $\left({{V}_{0}}-{{\epsilon}_{\text{F}}}\right)/{{\epsilon}_{\text{F}}}$ and s  −  d coupling $\Gamma S/{{\epsilon}_{\text{F}}}$. As a function of the FMI thickness l_FI, both $\Delta {{\rho}_{1}}/\rho$ and $\Delta {{\rho}_{2}}/\rho$ initially increase and then saturate at around ${{l}_{FI}}/a\sim 2$, which is expected since conduction electrons only tunnel into the FMI over a short depth, so the interface spin current saturates once the FMI is thicker than this tunneling depth. The insulating gap $\left({{V}_{0}}-{{\epsilon}_{\text{F}}}\right)/{{\epsilon}_{\text{F}}}$ obviously affects the tunneling depth, and is particularly influential on the magnitude of $\Delta {{\rho}_{1}}/\rho$, as can be seen by comparing plots with different $\left({{V}_{0}}-{{\epsilon}_{\text{F}}}\right)/{{\epsilon}_{\text{F}}}$ in figure 2. The magnitude of $\Delta {{\rho}_{1}}/\rho$ also generally increases with the s  −  d coupling $\Gamma S/{{\epsilon}_{\text{F}}}$, while $\Delta {{\rho}_{2}}/\rho$ at large $\Gamma S/{{\epsilon}_{\text{F}}}$ displays a nonmonotonic dependence on the FMI thickness. On the other hand, as a function of NM thickness l_N, both SMR components increase and peak at around ${{l}_{N}}/\lambda \sim 1$ and then decrease monotonically for large l_N. This can be understood because both $\Delta {{\rho}_{1}}$ and $\Delta {{\rho}_{2}}$ are interface effects that become less significant compared to bulk resistivity ρ when NM thickness increases, and the spin voltage is known to be maximal when the NM thickness is comparable to the spin diffusion length [25] ${{l}_{N}}/\lambda \sim 1$.

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We proceed to address the SMR in the NM/FMM bilayer, with the assumption that the FMM film itself has negligible SHE and is much thinner than its spin diffusion length ${{l}_{\text{FM}}}\ll \lambda$, such that quantum tunneling is the dominant mechanism for spin transport in the FMM, while the spin diffusion inside the FMM can be ignored. The calculation of the spin current at the NM/FMM interface starts with the model schematically shown in figure 3(a). The NM and FMM occupy $-{{l}_{N}}\leqslant x<0$ and $0\leqslant x\leqslant {{l}_{\text{FM}}}$, respectively. The NM region is described by equations (1) and (3), while the FMM layer is described by ${{H}_{\text{FM}}}={{p}^{2}}/2m+ \Gamma \mathbf{S}\centerdot \boldsymbol{\sigma }$ and the wave function

$$\begin{eqnarray}&&{{\psi}_{\text{FM}}}=\left(D{{\text{e}}^{\text{i}{{k}_{+}}x}}+{{F}^{-i{{k}_{+}}x}}\right)\left(\begin{array}{c} {{\text{e}}^{-\text{i}\varphi /2}}\cos \frac{\theta}{2} \\ {{\text{e}}^{\text{i}\varphi /2}}\sin \frac{\theta}{2} \end{array}\right)\\ &&+\,\left(E{{\text{e}}^{\text{i}{{k}_{-}}x}}+G{{\text{e}}^{-\text{i}{{k}_{-}}x}}\right)\left(\begin{array}{c} -{{\text{e}}^{-\text{i}\varphi /2}}\sin \frac{\theta}{2} \\ {{\text{e}}^{\text{i}\varphi /2}}\cos \frac{\theta}{2} \end{array}\right),\quad \quad \end{eqnarray} \tag{ 19 }$$

where ${{k}_{\pm }}=\sqrt{2m\left({{\epsilon}_{\text{F}}}\mp \Gamma S\right)}/\hbar$. The wave functions outside of the bilayer in $x>{{l}_{\text{FM}}}$ and x  <  −l_N are assumed to vanish for simplicity. The coefficients $A\sim I$ are again determined by matching wave functions and their first derivative at the interface. The interface spin current and the spin mixing conductance are calculated from equations (5) to (7), with replacing ${{\psi}_{FI}}$ to ${{\psi}_{\text{FM}}}$ and l_FI to l_FM, resulting in

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{G}_{r,i}} & = & \frac{1}{{{a}^{3}}|\gamma _{\theta}^{\prime}{{|}^{2}}}\left(\text{Im},\text{Re}\right)\left[Z_{\downarrow -+}^{\ast}{{Z}_{\downarrow ++}} \right. \\ {} & {} & \left. \times \,\left({{u}_{+-}}-{{u}_{++}}-{{u}_{--}}+{{u}_{-+}}\right)\right], \end{array}\end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{u}_{\alpha \beta}} & = & \text{i}\frac{{{\text{e}}^{\text{i}\left(\alpha {{k}_{+}}+\beta {{k}_{-}}\right){{l}_{\text{FM}}}}}}{\alpha {{k}_{+}}+\beta {{k}_{-}}},\quad {{W}_{\sigma \alpha \beta}}=\frac{{{k}_{0\sigma}}+\beta {{k}_{\alpha}}}{2{{k}_{0\sigma}}}, \\ {{Z}_{\sigma \alpha \beta}} & = & {{W}_{\sigma \alpha \beta}}{{\text{e}}^{-\text{i}{{k}_{\alpha}}{{l}_{\text{FM}}}}}-{{W}_{\sigma \alpha \bar{\beta}}}{{\text{e}}^{\text{i}{{k}_{\alpha}}{{l}_{\text{FM}}}}}, \\ \gamma _{\theta}^{\prime} & = & {{Z}_{\uparrow ++}}{{Z}_{\downarrow -+}}{{\cos}^{2}}\frac{\theta}{2}+{{Z}_{\downarrow ++}}{{Z}_{\uparrow -+}}{{\sin}^{2}}\frac{\theta}{2}, \end{array}\end{eqnarray} \tag{ 21 }$$

with $\bar{\beta}=-\beta$. Apart from a change in magnitude, the pattern of the spin mixing conductance as a function of s  −  d coupling and the FMM thickness shown in figure 3 is almost indistinguishable from that reported in figure 2 of [28], which shows clear signals of quantum interference with respect to both s  −  d coupling and FMM thickness. This similarity is expected, since the only difference between the formalism here and in [28] is the insulating gap ${{V}_{0}}-{{\epsilon}_{\text{F}}}$ of the substrate or vacuum in the $x>{{l}_{\text{FM}}}$ region in figure 3(a), which is assumed to be infinite here for simplicity but finite in [28]. The insulating gap is spin degenerate and essentially does not influence the spin transport.

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To get SMR, we use equations (17) and (18) while taking the ${{G}_{c}}={{G}_{r}}+\text{i}{{G}_{i}}$ obtained from equation (20). The results for the $\Delta {{\rho}_{1}}/\rho$ and $\Delta {{\rho}_{2}}/\rho$ components of SMR as functions of FMM thickness l_FM and NM thickness l_N are shown in figure 4, for several values of s  −  d coupling $\Gamma S/{{\epsilon}_{\text{F}}}$. As a function of NM thickness, both components reach a maximal at around the spin diffusion length ${{l}_{N}}/\lambda \sim 1$ and then decrease monotonically, similar to that reported in figure 2 for NM/FMI bilayer and is due to the spin diffusion effect explained in section 2.2. On the other hand, as a function of FMM thickness, both components show clear modulations with an average periodicity that decreases with increasing s  −  d coupling, a trend similar to that of G_r and G_i shown in figure 3 and is attributed to the quantum interference of spin transport. Intuitively, a larger s  −  d coupling renders a faster precession of conduction electron spin when it travels inside the FMM, hence more modulations appear for a given FMM thickness. The $\Delta {{\rho}_{1}}$ component of SMR is found to be generally one order of magnitude smaller than the $\Delta {{\rho}_{2}}$ component.

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We remark that an oscillating dependence on FMM layer thickness due to quantum effects has been reported for other spin transport related quantities, such as magnetoresistance [36], anisotropic magnetoresistance (AMR) [37], and interlayer exchange coupling [38]. Recent SMR experiments in NiFe/Pt also show hints for an nonmonotonic dependence on the NiFe thickness [39], although in reality other effects not taken into account by our minimal model, such as surface roughness, may also be the origin. We also mention that the longitudinal resistance ${{\rho}_{\text{long}}}$ in equation (17) is invariant under inversion of the magnetization $\mathbf{S}\to -\mathbf{S}$, in accordance with the first harmonic resistance ${{R}_{\omega}}$ measured in Ta/Co and Pt /Co bilayers, where ω is the frequency of the applied a.c. current [14]. On the other hand, a unidirectional spin Hall magnetoresistance (USMR) in the second harmonic ${{R}_{2\omega}}$ that is odd under inversion of magnetization has also been observed [14], and has been related to a spin-dependent interface resistance due to SHE induced spin accumulation. This USMR is anticipated to be outside of the scope of our minimal model, and may require a certain Boltzmann equation type of approach to capture.

Notice that since the FMM itself is conducting and sandwiched between two different materials, an in-plane charge current is expected to induce a spin accumulation in the FMM due to Rashba spin–orbit coupling (SOC) [40–42], which may subsequently modify the boundary conditions discussed below equation (10) that we used to solve the spin diffusion equation. Given that the quantification of the relative weight between the spin accumulation induced by Rashba SOC and that induced by SHE requires a separate treatment, we ignore this SOC-induced spin accumulation and examine solely the influence of SHE. Despite this simplification, the calculated SMR shown in figure 4 has a magnitude similar to that reported experimentally, meaning that the SHE alone already contributes significantly to the SMR. Thus we also anticipate that the predicted quantum interference effect should still be manifest even if the Rashba SOC is included.

The experimental detection of the oscillation of SMR with respect to FMM thickness l_FM shown in figure 4 would be a direct proof of our approach. In a typical NM/FMM set up, however, the resistance of both NM and FMM contribute to the total resistance measured in experiments, therefore it is important to investigate whether there is a situation in which the predicted oscillation of SMR can manifest. To explore this possibility, we use a three-resistor model to characterize the total longitudinal resistance [14], which contains the resistor that represents the NM layer (N), the FMM layer (F), and the interface layer (I) connected in parallel, each denoted by ${{R}_{i}}=R_{i}^{0}+\delta {{R}_{i}}$ with $i=\left\{N,I,F\right\}$. Here $R_{i}^{0}$ is the contribution to the longitudinal resistance in layer i that does not depend on the angle of the magnetization, and $\delta {{R}_{i}}$ is the part that depends on the angle which is generally much smaller $\delta {{R}_{i}}\ll R_{i}^{0}$. Expanding the total longitudinal resistance to leading order in $\delta {{R}_{i}}$ yields

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{R}_{\text{tot}}} & \approx & R_{\text{tot}}^{0}+{{\left(\frac{R_{I}^{0}R_{F}^{0}}{B}\right)}^{2}}\delta {{R}_{N}}+{{\left(\frac{R_{N}^{0}R_{F}^{0}}{B}\right)}^{2}}\delta {{R}_{I}}+{{\left(\frac{R_{N}^{0}R_{I}^{0}}{B}\right)}^{2}}\delta {{R}_{F}}, \\ R_{\text{tot}}^{0} & = & \frac{R_{N}^{0}R_{I}^{0}R_{F}^{0}}{B}, \\ B & = & R_{N}^{0}R_{I}^{0}+R_{I}^{0}R_{F}^{0}+R_{N}^{0}R_{F}^{0}. \end{array}\end{eqnarray} \tag{ 22 }$$

Each resistance is assumed to satisfy the usual relation to the sample size $\left\{R_{i}^{0},\delta {{R}_{i}}\right\}=\left\{\rho _{i}^{0},\delta {{\rho}_{i}}\right\}\times L/{{l}_{i}}\;h$, where L and h are the length and the width of the sample, respectively, $\rho _{i}^{0}$ and $\delta {{\rho}_{i}}$ are the corresponding resistivity, and l_i is the thickness of layer i. The thickness of the interface l_I is assumed to be intrinsically constant, in contrast to l_N and l_F that can be varied experimentally [14]. The percentage change of the total resistance due to the angle of the magnetization is

$$\begin{eqnarray}\begin{array}{*{35}{rl}} \frac{{{R}_{\text{tot}}}-R_{\text{tot}}^{0}}{R_{\text{tot}}^{0}} & \approx & \left(\frac{\rho _{I}^{0}\rho _{F}^{0}}{{{l}_{I}}{{l}_{F}}C}\right)\frac{\delta {{\rho}_{N}}}{\rho _{N}^{0}}+\left(\frac{\rho _{N}^{0}\rho _{F}^{0}}{{{l}_{N}}{{l}_{F}}C}\right)\frac{\delta {{\rho}_{I}}}{\rho _{I}^{0}}+\left(\frac{\rho _{N}^{0}\rho _{I}^{0}}{{{l}_{N}}{{l}_{I}}C}\right)\frac{\delta {{\rho}_{F}}}{\rho _{F}^{0}}, \\ C & = & \frac{\rho _{N}^{0}\rho _{I}^{0}}{{{l}_{N}}{{l}_{I}}}+\frac{\rho _{N}^{0}\rho _{F}^{0}}{{{l}_{N}}{{l}_{F}}}+\frac{\rho _{F}^{0}\rho _{I}^{0}}{{{l}_{F}}{{l}_{I}}}. \end{array}\end{eqnarray} \tag{ 23 }$$

Note that the $\rho _{i}^{0}\rho _{j}^{0}/{{l}_{i}}{{l}_{j}}C$ factors are monotonic functions of the layer thickness $\left\{{{l}_{N}},{{l}_{I}},{{l}_{F}}\right\}$, and are independent from the angle of the magnetization.

The contribution to the angular dependent part of R_F comes from the AMR which takes the form [43, 44] $\delta {{\rho}_{F}}\propto {{\left(\,\,\boldsymbol{j}^{{c}}\centerdot \mathbf{\hat{m}}\right)}^{2}}\propto {{\left({{m}^{y}}\right)}^{2}}$ since the in-plane charge current $\boldsymbol{j}^{{c}}$ runs along $\mathbf{\hat{y}}$ as shown in figure 3(a), and we denote $\mathbf{\hat{m}}=\mathbf{S}/S=(\sin \theta \cos \varphi,\sin \theta \sin \varphi,\cos \theta )$ as the unit vector along the direction of the magnetization. In addition, Zhang et al [44] showed that the interface resistance has a quadratic dependence on both m^y and m^z, a result of surface spin–orbit scattering. On the other hand, the SMR in the NM has the angular dependence [23] described by equation (17). These considerations lead to the parametrization of resistivity by

$$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {} & \rho _{F}^{0}+\delta {{\rho}_{F}}=\rho _{F}^{0}+ \Delta \rho _{F}^{b}{{\left({{m}^{y}}\right)}^{2}}, \\ {} & {} & \rho _{I}^{0}+\delta {{\rho}_{I}}=\rho _{I}^{0}+ \Delta \rho _{I,y}^{s}{{\left({{m}^{y}}\right)}^{2}}+ \Delta \rho _{I,z}^{s}{{\left({{m}^{z}}\right)}^{2}}, \\ {} & {} & \rho _{N}^{0}+\delta {{\rho}_{N}}=\left(\rho + \Delta {{\rho}_{0}}\right)+ \Delta {{\rho}_{1}}\left[{{\left({{m}^{x}}\right)}^{2}}+{{\left({{m}^{y}}\right)}^{2}}\right]. \end{array}\end{eqnarray} \tag{ 24 }$$

Combinig this with equation (23) motivates us to propose the following experiment that should isolate the effect of longitudinal SMR represented by $\delta {{\rho}_{N}}$. From equation (24), we see that $\delta {{\rho}_{F}}$ and $\delta {{\rho}_{I}}$ vanish if the magnetization does not have an in-plane component, i.e. ${{m}^{y}}={{m}^{z}}=0$, while $\delta {{\rho}_{N}}$ remains finite as long as the out-of-plane component is nonzero ${{m}^{x}}\ne 0$. Thus we propose to fix the magnetization of the FMM film to be out-of-plane ${{m}^{x}}\ne 0$, in which case the percentage change of total longitudinal resistance as a function of FMM thickness takes the form

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{{{R}_{\text{tot}}}-R_{\text{tot}}^{0}}{R_{\text{tot}}^{0}}\approx \frac{{{l}_{1}}}{{{l}_{F}}+{{l}_{1}}+{{l}_{2}}}\times \frac{ \Delta {{\rho}_{1}}}{\rho + \Delta {{\rho}_{0}}}{{\left({{m}^{x}}\right)}^{2}}, & {} & \\ \left(\text{for}\;\;{{m}^{y}}={{m}^{z}}=0\right) & {} & \end{array}\end{eqnarray} \tag{ 25 }$$

where ρ, $\Delta {{\rho}_{0}}$, and $\Delta {{\rho}_{1}}$ are those in equations (17) and (18), ${{l}_{1}}={{l}_{N}}\rho _{F}^{0}\,/\rho _{N}^{0}$ and ${{l}_{2}}={{l}_{I}}\rho _{F}^{0}\,/\rho _{I}^{0}$ are two length scales that can be treated as fitting parameters in experiments. Equation (25) indicates that, for the case of only out-of-plane magnetization, the percentage change of magnetoresistance decays with the FMM thickness l_F due to the ${{l}_{1}}/\left({{l}_{F}}+{{l}_{1}}+{{l}_{2}}\right)$ factor, but also oscillates with l_F due to the $\Delta {{\rho}_{1}}/\left(\rho + \Delta {{\rho}_{0}}\right)\approx \Delta {{\rho}_{1}}/\rho$ factor as quantified in equation (18) and shown in figure 4. Thus varying FMM thickness while keeping its magnetization out-of-plane may be a proper set up to observe the predicted oscillation of longitudinal SMR, provided the FMM thickness remains thinner than its spin relaxation length ${{l}_{F}}\ll \lambda$.

Note that the convention of labeling coordinate in SMR or STT experiments is that the charge current flows by definition along $\hat{x}$ and the direction normal to the film is along $\hat{z}$. Therefore the coordinate in our tunneling formalism (x, y, z) corresponds to (z, x, y) in the experimental convention. Finally, we remark that the surface roughness and surface absorption in a realistic NM/FMM bilayer may cause difficulty in experimentally observing the predicted quantum size effect, whose precise effects are however beyond the scope of our minimal model.