Theory of Spin Hall Magnetoresistance from a Microscopic Perspective

TL;DR

This paper develops a microscopic theory for spin Hall magnetoresistance in metals interfaced with magnetic insulators, linking interface parameters to observable resistance effects and predicting complex magnetic field and temperature dependencies.

Contribution

It introduces a microscopic framework expressing spin-mixing conductances via interface parameters and spin correlations, advancing understanding of spin Hall magnetoresistance phenomena.

Findings

Predicts complex magnetic field and temperature dependence of resistance.

Provides a microscopic link between interface properties and magnetoresistance.

Offers explanations for experimental observations in heavy metal/magnetic insulator systems.

Abstract

We present a theory of the spin Hall magnetoresistance of metals in contact with magnetic insulators. We express the spin-mixing conductances, which govern the phenomenology of the effect, in terms of the microscopic parameters of the interface and the spin-spin correlation functions of the local moments on the surface of the magnetic insulator. The magnetic field and temperature dependence of the spin-mixing conductances leads to a rich behaviour of the resistance due to an interplay between the Hanle effect and spin mixing at the interface. Our theory provides a useful tool for understanding the experiments on heavy metals in contact with magnetic insulators of different kinds, and it predicts striking behaviours of magnetoresistance.

Equations

$$V_{sd}=-J_{sd}\sum _i{\mathbf{S}}^i\cdot \mathbf{s}({\mathbf{r}}_i),$$

$$⟨S_{\alpha }^i(t)S_{\gamma }^j⟩={\delta }_{ij}⟨S_{\alpha }^i(t)S_{\gamma }^i⟩+(1-{\delta }_{ij})⟨S_{\alpha }^i⟩⟨S_{\gamma }^j⟩.$$

$${\dot{\mu }}_s^{\alpha }-\frac{1}{e{\nu }_F}{\partial }_i{J}_{s,i}^{\alpha }-{\omega }_L{ϵ}_{\alpha \gamma \kappa }{n}_{\gamma }{\mu }_{\mathrm{s}}^{\kappa }=-{\Gamma }_{\alpha \kappa }{\mu }_s^{\kappa },$$

$${\omega }_L(z)={\omega }_B-\frac{n_{\text{imp}}^{\text{2D}}{J}_{\text{sd}}}{\hslash }⟨{\hat{S}}_{\parallel }⟩{\delta }_b(z),$$

$${\Gamma }_{\alpha \kappa }(z)=\frac{{\delta }_{\alpha \kappa }}{{\tau }_s}+\left[\frac{{\delta }_{\alpha \kappa }}{{\tau }_{\perp }}+\left(\frac{1}{{\tau }_{\Vert }}-\frac{1}{{\tau }_{\perp }}\right)n_{\alpha }{n}_{\kappa }\right]{\delta }_b(z),$$

$$\frac{1}{{\tau }_{\Vert }}$$

$$\frac{1}{{\tau }_{\perp }}$$

$$-\frac{1}{e{\nu }_F}{J_{s,z}^{\upsilon }|}_{z=-b}^{z=0}$$

$$-e{\mathbf{J}}_{s,z}(0)=G_s{\mathbf{\mu }}_s+G_r\mathbf{n}\times (\mathbf{n}\times {\mathbf{\mu }}_s)+G_i\mathbf{n}\times {\mathbf{\mu }}_s,$$

$$G_s$$

$$G_r$$

$$G_i$$

$${\rho }_L$$

$${\rho }_T$$

$$\Delta {\rho }_0$$

$$\Delta {\rho }_1$$

$$\Delta {\rho }_2$$

$$\mathcal{R}(\mathcal{G},\ell )=\frac{2\ell }{d_N}\tanh \left(\frac{d_N}{2\ell }\right)\frac{1-{\rho }_D\mathcal{G}\ell \coth \left(\frac{d_N}{2\ell }\right)}{1-2{\rho }_D\mathcal{G}\ell \coth \left(\frac{d_N}{\ell }\right)}.$$

$$G_r-G_s\ll G_i^2{\lambda }_s{\rho }_D\coth \left(d_N/{\lambda }_s\right),$$

$$G_{i,c}^2=\frac{{\sinh }^2(d_N/2{\lambda }_s)}{2{\lambda }_s^2{\rho }_D^2\cosh (d_N/{\lambda }_s)}\left[\frac{{\lambda }_s}{d_N}\sinh \left(\frac{d_N}{{\lambda }_s}\right)-1\right].$$

$$⟨{\hat{S}}_{\Vert }⟩=-S{\mathcal{B}}_S\left[S(\hslash {\omega }_B-6⟨{\hat{S}}_{\Vert }⟩J_m)/T\right],$$

Full text

Introduction- The spin-orbit coupling (SOC) in metals and semiconductors leads to a conversion between the charge and spin currents, which results in the spin Hall effect (SHE) and its inverse effect Dyakonov and Perel (1971a, b); Hirsch (1999); Sinova et al. (2004); Valenzuela and Tinkham (2006); Kimura et al. (2007); Kato et al. (2004); Sih et al. (2005); Wunderlich et al. (2005); Zhou et al. (2018); Maekawa and Kimura (2017); Sinova et al. (2015); Nakayama et al. (2016). A manifestation of the SHE in a normal metal (NM) is a modulation of the magnetoresistance (MR) with respect to the direction of the applied magnetic field when the metal is in contact with a magnetic insulator (MI) in NM/MI structures Isasa et al. (2016); Althammer et al. (2013); Huang et al. (2012). This effect, called the spin Hall magnetoresistance (SMR), has been observed in several experiments Weiler et al. (2012); Nakayama et al. (2013); Avci et al. (2015); Hahn et al. (2013); Dejene et al. (2015). The origin of the SMR is the spin-dependent scattering at the NM/MI interface which depends on the angle between the polarization of spin Hall current and the magnetization of the MI Nakayama et al. (2013); Chen et al. (2013). The latter can be controlled by magnetic fields.

Although the theory of SMR Nakayama et al. (2013); Chen et al. (2013) is well established and provides a qualitative description of the effect, it does not describe the dependence of the resistivity on the strength of the applied magnetic field $B$, nor on the temperature $T$. The spin mixing conductances, which are at the heart of the SMR effect, have traditionally been regarded as phenomenological parameters in every experiment, because their computation was thought to be a formidable task which could only be carried out by ab initio methods Jia et al. (2011); Carva and Turek (2007); Zhang et al. (2011); Xia et al. (2002); Dolui et al. (2019). Recent experiments Meyer et al. (2014); Vélez et al. (2018); Das et al. (2019) show, however, that the SMR effect depends both on $B$ Vélez et al. (2018) and on $T$ Meyer et al. (2014); Vélez et al. (2018); Das et al. (2019), and that the magnetic state of the MI plays an important role for SMR. Furthermore, the magnetic field alone leads to the Hanle magnetoresistance (HMR) Dyakonov (2007); Vélez et al. (2016), which has an identical angular dependence to SMR Vélez et al. (2016), but is not requiring an MI. Despite the fact that SMR and HMR have different origins, they cannot always be easily separated in experiments, which adds onto the uncertainties of interpreting the experimental data. It is, therefore, desirable to have a theory of SMR which has predictive power about the dependence of the spin mixing conductances on $B$ and $T$ and is able to cover a wide range of magnetic system, from classical to quantum magnets.

In this letter, we present a general theory of the electronic transport in NM/MI structures. We describe the spin-dependent scattering at the NM/MI interface via a microscopic model based on the sd coupling between local moments on the MI surface and itinerant electrons in the NM. The temperature and magnetic-field dependence of the interfacial scattering coefficients is obtained by expressing them in terms of spin-spin correlations functions. The latter are determined by the magnetic behavior of the MI layer. As examples, we study the MR of a metallic film adjacent to either a paramagnet (PM) or a Weiss ferromagnet (FM). At low temperatures, we find a striking non-monotonic behavior of the MR as a function of $B$, which we explain in terms of an interplay between the SMR and HMR effects. Our model provides a tool to reveal, by MR measurements, magnetic properties of NM/MI interfaces.

Model and Method- We consider an NM in contact with an MI, as shown in Figure 1. We assume both layers to be homogeneous in the $(x,y)$ plane and the NM/MI interface to be located at $z=0$. The system Hamiltonian reads $H=H_{\textrm{NM}}+H_{\textrm{MI}}+V_{sd}$, where $H_{\textrm{NM}}$ is the Hamiltonian of a disordered metal with SOC and Zeeman field, $H_{\textrm{MI}}$ is the Heisenberg Hamiltonian in a magnetic field, and $V_{sd}$ describes the coupling at the NM/MI interface. We model the interface by an exchange interaction between local moments and itinerant electrons

[TABLE]

where $\bm{S}^{i}$ is the spin operator of the local moment, $\bm{s}(\bm{r}_{i})$ is the operator of the itinerant spin density at position $\bm{r}_{i}$, and $J_{sd}$ is the sd-coupling constant at the NM/MI interface. We assume that the metal is strongly disordered, such that the mean-free path $l$ is much smaller than both the thin-film thickness $d_{N}$ and the spin-relaxation length $\lambda_{s}$. For such a diffusive motion of the electron in the thin film, the events of interaction with the local moments located on the surface of the MI appear as spikes of short duration, randomly distributed along the semiclassical trajectory of the electron. The precise positions of the spikes on the trajectory is clearly unimportant, because the trajectory is sufficiently random. In this diffusive limit, we may allow ourselves to displace the local moments in a random fashion on the scale of $l$ without any consequence for the disorder-averaged quantities, as long as we are interested in the dependence of those quantities on a larger scale, set by $\lambda_{s}$. Thus, we arrive at considering a fictitious layer of thickness $b$ in which both the itinerant electrons and the local moments coexist, with the latter being randomly distributed but maintaining their spin-spin coupling. We apply the Born-Markov approximation to $H$ in this $b$-layer, with $V_{sd}$ in Eq. (1) as perturbation. Although the thickness $b$ should be kept small ($b\sim l$), we obtain physically meaningful results by sending first $l\to 0$ in the diffusive limit, and only in a second step $b\to 0$, going thus through an intermediate stage of the calculation in which $l\ll b\ll\lambda_{s}$. This order of taking the limits represents a significant simplification in the derivation, because powerful disorder-averaging techniques devised for homogeneously distributed impurities in the metal can be applied here to calculate the spin-relaxation tensor inside the $b$-layer in a local continuum approximation 111It is important to remark that the coupling in Eq. (1) acts more efficiently when the spin $\bm{S}_{i}$ is embedded in the metal as compared to the case when it is at the surface and interacts only with the tail of the electron wave function appearing in $\bm{s}(\bm{r}_{i})$. We should, therefore, reduce $J_{\textit{sd}}$ in Eq. (1) by a factor $n(z>\lambda_{F})/n(z=0)$, where $n(\bm{r})$ is the average charge density in the metal and $\lambda_{F}$ is the Fermi wave length. However, this suppression factor is expected to be on the order of unity in well-coupled systems, for which the local moments at the surface form bonds with the metal. We absorb this suppression factor into $J_{sd}$ hereafter..

To simplify the magnetic problem, we employ the Weiss mean-field theory for $H_{\textrm{MI}}$. In this approximation, the state of the magnetic system is a product state of individual local moments, yielding

[TABLE]

The equilibrium properties of the local moments are determined by the spin expectations $\langle S_{\alpha}\rangle$ and $\langle S^{2}_{\alpha}\rangle$ ($\alpha=x,y,z$), which depend on $T$ and $B$. We do not consider here the feedback effect of the itinerant electrons on the local moments; the latter act merely as a spin bath on the itinerant electrons.

In this approach, we arrive at the usual continuity equation for the non-equilibrium spin bias $\bm{\mu}_{s}$ in the metal (including the $b$-layer) following a standard derivation

[TABLE]

where superscript Greek indices denote spin projections ($\alpha=\{x,y,z\}$) and subscript Latin indices denote current directions ($i=\{x,y,z\}$), $\bm{n}=\bm{B}/B$ is the unit vector of the $B$-field, $e$ is the elementary charge ($e>0$), $\nu_{F}$ is the density of states per spin species at the Fermi level, $\epsilon_{\alpha\beta\gamma}$ is the antisymmetric tensor, and repeated indices are implicitly summed over. The spin current $J_{s,i}^{\alpha}$ has units of electrical current, with $-J_{s,i}^{\alpha}/2e$ giving the amount of spin with polarization $\alpha$ transported in direction $i$ through a unit cross section and per unit of time. Both the Larmor precession frequency, $\omega_{L}$, and the spin relaxation tensor, $\Gamma_{\alpha\kappa}$ are inhomogeneous in space due to the $b$-layer insertion. Specifically, for the geometry in Figure 1, we have

[TABLE]

where $\omega_{B}=\textsl{g}\mu_{B}B/\hbar$, with $\textsl{g}\approx 2$ being the electron g-factor and $\mu_{B}$ the Bohr magneton, $n_{\textit{imp}}^{\textit{2D}}$ is the number of local moments per unit area at the MI/NM interface, $\hat{S}_{\parallel}=\hat{\bm{S}}\cdot\bm{n}$ is the longitudinal spin operator, and $\delta_{b}(z)$ equals to $1/b$ in the $b$-region and zero elsewhere. In the limit $b\to 0$, $\delta_{b}(z)$ tends to the Dirac $\delta$-function. The second term on the right-hand side in Eq. (4) describes the interfacial exchange field. For instance, this field is particularly well pronounced in Al/EuS, leading to a directly measurable splitting of the density of states in the superconducting regime Hao et al. (1990); Strambini et al. (2017).

The spin relaxation tensor in Eq. (3) reads

[TABLE]

where $\tau_{s}$ is the spin relaxation time in the NM. We assume the spin relaxation in the NM to remain isotropic for the experimentally relevant magnetic fields, $B\lesssim 10\;\textrm{T}$. Indeed, the Pauli paramagnetism has a weak effect on the SOC-induced spin relaxation at the Fermi level, because the density of states is almost spin-independent, $\nu^{\uparrow}_{F}\approx\nu^{\downarrow}_{F}\equiv\nu_{F}$, owing to the large Fermi energy of the NM. In Eq. (5), $\tau_{\|}$ and $\tau_{\bot}$ denote, respectively, the longitudinal and transverse spin relaxation times per unit length for the itinerant electron in the $b$-region. In our notations, $T_{1}=b\tau_{\parallel}$ is the relaxation time of the longitudinal spin component ${S}_{\parallel}=\bm{S}\cdot\bm{n}$, and $T_{2}=b\tau_{\perp}$ is the decoherence time of the transverse spin components $\bm{S}_{\perp}=\bm{n}\times\left(\bm{S}\times\bm{n}\right)$. Within the Born-Markov approximation Slichter (2013), we obtain

[TABLE]

where $n_{B}(\omega)=1/(e^{\hbar\omega/k_{B}T}-1)$ is the Bose-Einstein distribution function and $\omega_{m}=\omega_{B}-\langle\hat{S}_{\|}\rangle\sum_{j}J_{ij}/\hbar$, with $J_{ij}$ being the coupling constant of the Heisenberg ferromagnet. In deriving Eqs. (6) and (7), we assumed that the correlator $\left\langle S_{\alpha}(t)S_{\beta}\right\rangle$ for a spin on the MI surface can be approximated by the corresponding correlator for a spin deep in the bulk of the MI. The difference between $1/\tau_{\parallel}$ and $1/\tau_{\perp}$ is entirely due to the ordered magnetic state of the local moments at the interface.

To derive the boundary condition for the NM/MI interface, we integrate Eq. (3) over $z$ in the $b$-layer ($-b<z<0$), assuming that $\bm{\mu}_{s}$ is almost constant and independent of time,

[TABLE]

Next we take the limit $b\to 0$ and write the boundary condition in a customary way Brataas et al. (2001); Dejene et al. (2015)

[TABLE]

where we set $\bm{J}_{s,z}=0$ at $z=-b$, because, by construction, the electron does not penetrate into the MI beyond the $b$-layer. The spin dependent conductances read

[TABLE]

It is customary to call the complex quantity $G_{\uparrow\downarrow}=G_{r}+iG_{i}$ spin-mixing conductance Brataas et al. (2001), whereas $G_{s}$ is sometimes called spin-sink conductance Dejene et al. (2015). We note that $G_{s}$ originates entirely from spin-flip processes and can, therefore, be unambiguously associated with magnon emission and absorption. In contrast, $G_{r}$ does not have a physical meaning on its own. However, the combination $G_{r}-G_{s}$ is proportional to the spin decoherence rate ($1/\tau_{\perp}$) of the itinerant electron at the NM/MI interface. It follows from Eq. (7) that a part of $G_{r}-G_{s}$ is due to spin-flip processes ($1/2\tau_{\parallel}$), and hence is identical in nature to $G_{s}$, whereas the other part is due to spin dephasing. The purely dephasing contribution is $G_{r}-\frac{1}{2}G_{s}$ and it originates from almost elastic spin-scattering processes, which do not involve a spin exchange with the MI. Thus, $G_{s}$ and $G_{r}-\frac{1}{2}G_{s}$ correspond to different physical processes and have, therefore, distinct dependences on $B$ and $T$. Finally, $G_{i}$ is a measure of the interfacial exchange field and it is proportional to the MI magnetization.

Results and Discussion- We plot the quantities $G_{i}$, $G_{r}-\frac{1}{2}G_{s}$, and $-\frac{1}{2}G_{s}$ as functions of $B$ for a PM in Figure 2a and as functions of $T$ for a FM in Figure 2b. In the isotropic regime ($\tau_{\parallel}=\tau_{\perp}$), we have $G_{r}=G_{i}=0$, and $G_{s}=-\frac{2}{3}G_{0}S(S+1)$. In the strongly magnetized regime ($\tau_{\parallel}\gg\tau_{\perp}$), we have $G_{r}=G_{0}S^{2}$, $G_{i}=G_{0}S/(\pi\nu_{F}J_{\textit{sd}})$, and $G_{s}\approx 0$. Here, $G_{0}=\frac{\pi}{\hbar}n_{\textit{imp}}^{\textit{2D}}(e\nu_{F}J_{sd})^{2}$ is a characteristic scale of the spin-dependent conductances. We estimate a value of $G_{0}\approx 3.8\times 10^{13}\Omega^{-1}\textrm{m}^{-2}$ for a typical $n_{\textit{imp}}^{\textit{2D}}=5\times 10^{18}\textrm{m}^{-2}$ and $\nu_{F}J_{\textit{sd}}=0.1$. This estimate is compatible with values of spin mixing conductances reported in experiments Vlietstra et al. (2013); Dejene et al. (2015); Das et al. (2019).

Next we consider a ferrimagnet consisting of two species of local moments ($S^{a}$ and $S^{b}$). In the mean-field approximation, no interference terms occur between different species and our results above are modified only by selectively weighting each species by its concentration on the surface ($n_{a}^{\textit{2D}}$ and $n_{b}^{\textit{2D}}$) and taking into account its possibly different coupling strength ($J_{\textit{sd}}^{a}$ and $J_{\textit{sd}}^{b}$). It is possible to obtain a situation in which the interfacial exchange fields of the two local-moment species closely compensate each other, resulting in $G_{i}\ll G_{r}$ —a condition which is believed to hold for Pt thin films deposited on $\textrm{Y}_{3}\textrm{Fe}_{5}\textrm{O}_{12}$ (YIG) Vlietstra et al. (2013) and which would otherwise not be possible in a simple ferromagnet, because $G_{i}$ is the largest spin mixing conductance for $\nu_{F}J_{\textit{sd}}\ll 1$. The $G_{i}$-compensation condition for a ferrimagnet, thus, reads $n_{a}^{\textit{2D}}J_{\textit{sd}}^{a}S^{a}-n_{b}^{\textit{2D}}J_{\textit{sd}}^{b}S^{b}=0$, which differs from the magnetization compensation condition, $n_{a}^{\textit{3D}}S^{a}-n_{b}^{\textit{3D}}S^{b}=0$, and allows for the possibility of having a finite magnetization even when $G_{i}=0$. And vice versa, the Néel order parameter of an antiferromagnet can manifest itself as an interfacial exchange field, provided the $G_{i}$-compensation condition is not fulfilled. We remark that, for YIG, we have $S^{a}=S^{b}=5/2$ and $n_{a}^{\textit{3D}}/n_{b}^{\textit{3D}}=3/2$. And for the $\textrm{Pt}/\textrm{YIG}\textrm{-}[001]$ interface, we have $n_{a}^{\textit{2D}}=n_{b}^{\textit{2D}}$ and $J_{\textit{sd}}^{a}\approx J_{\textit{sd}}^{b}$. A small difference between $J_{\textit{sd}}^{a}$ and $J_{\textit{sd}}^{b}$ may originate from different crystal fields for the $\textrm{Fe}^{3+}$ cation on the tetrahedral ($a$) and octahedral ($b$) sublattice of the garnet.

Despite the fact that YIG has been the material of choice in most experimental studies of SMR, recent experiments started studying also other MIs Isasa et al. (2014, 2016); Vélez et al. (2018); Lammel et al. (2019); Koichi and et al. ; Fontcuberta et al. (2019). Here, we would like to draw attention to an effect due to $G_{i}$ which, to the best of our knowledge, has been overlooked theoretically and which could appear rather puzzling when observed experimentally. This effect consists in a negative, linear-in-$B$ magnetoresistance, which arises from an interplay between SMR and HMR featuring a non-local Hanle effect. And despite the fact that the novel effect is primarily due to $G_{i}$, we keep $G_{r}$ and $G_{s}$ in the expressions below for completeness.

We make use of the boundary condition in Eq. (9) and follow closely the derivation of the SMR and HMR effects Nakayama et al. (2013); Chen et al. (2013); Vélez et al. (2016, 2018), obtaining the corrections to the longitudinal ($\rho_{L}$) and transverse $(\rho_{T})$ resistivity of the Hall-bar setup in Figure 1

[TABLE]

where $\rho_{D}$ is the Drude resistivity and $\omega_{c}\tau$ is the Hall angle, with $\omega_{c}=eB/mc$ being the cyclotron frequency and $\tau$ being the momentum relaxation time. The combined SMR+HMR resistivity corrections read Vélez et al. (2018)

[TABLE]

where $\theta_{\textit{SH}}$ is the spin Hall angle, $\lambda_{s}=\sqrt{\mathcal{D}\tau_{s}}$, with $\mathcal{D}=1/2e^{2}\nu_{F}\rho_{D}$ being the diffusion constant, $G_{\uparrow\downarrow}=G_{r}+iG_{i}$ is the complex spin mixing conductance, $1/\Lambda=\sqrt{1/\lambda_{s}^{2}+i\omega_{B}/\mathcal{D}}$, and the auxiliary function $\mathcal{R}(\mathcal{G},\ell)$ is defined as

[TABLE]

For $\Lambda=\lambda_{s}$, we recover the SMR corrections Nakayama et al. (2013); Chen et al. (2013), whereas for $G_{s}=G_{r}=G_{i}=0$, we recover the HMR corrections Dyakonov (2007); Vélez et al. (2016). We remark that, in general, it is important to take into account $G_{s}$ Dejene et al. (2015); Vélez et al. (2018), which is not negligible in the paramagnetic regime, see Figure 2. The corrections in Eq. (14) were used in Ref. 29 to explain the unusual behavior of SMR in $\textrm{Pt}\textrm{/}\textrm{La}\textrm{Co}\textrm{O}_{3}$ in the high-temperature limit.

For a PM or FM at sufficiently low temperatures, the scale to reach saturation represents only a relatively small portion of the experimentally accessible $B$-field range. The SMR effect develops quickly with increasing $B$ and saturates as shown by the blue solid line in Figure 3a. The SMR effect is dominated by $G_{i}$ for

[TABLE]

which requires that $n_{\textit{imp}}^{\textit{2D}}\gg(\hbar/e^{2}\rho_{D}\lambda_{s})\tanh{\left(d_{N}/\lambda_{s}\right)}$. At the same time, the HMR effect develops gradually and becomes relevant only for large $B$ as shown by the red solid line in Figure 3a. In experiment, the HMR effect is well pronounced at relatively large magnetic fields, $B\lesssim 10\,\textrm{T}$ Vélez et al. (2016). In the region of intermediate $B$, denoted as “interference region” in Figure 3a, the interplay between the SMR and HMR effects can lead to negative differential MR ($\partial\rho_{L}/\partial B<0$). This behavior would not be so surprising if it occurred solely when $G_{i}$ and $\omega_{B}$ had opposite signs. Indeed, $G_{i}$ is a measure of the interfacial exchange field, which is a singular field created at the NM/MI interface by the sd coupling in Eq. (1). The signs of $G_{i}$ and $\omega_{B}$ are equal to each other for $J_{\textit{sd}}>0$ and opposite for $J_{\textit{sd}}<0$. For electrons diffusing over a characteristic length scale $\ell\sim\min(d_{N},\lambda_{s})$, the interfacial exchange field can be smeared near the interface over $\ell$ and superimposed onto $\omega_{B}$, obtaining an average Larmor frequency $\omega_{L}=\omega_{B}+G_{i}/e^{2}\nu_{F}\ell$. One could naïvely expect that the HMR effect, which is proportional to $\omega_{B}^{2}$ for all experimentally relevant $B$-field values, to become proportional to $\omega_{L}^{2}=\left(\omega_{B}+G_{i}/e^{2}\nu_{F}\ell\right)^{2}$, generating, thus, after squaring a cross term proportional to $\omega_{B}G_{i}$. For $J_{\textit{sd}}<0$, this term would then naturally lead to a negative MR. Quite surprisingly, we find a negative MR even for $J_{\textit{sd}}>0$, provided $G_{i}$ exceeds a certain critical value.

To investigate the origin of the anomalous behavior of the MR, we expand $\Delta\rho_{1}$ in Eq. (14) in powers of $\omega_{B}$ at constant $G_{i}$ and set, for simplicity, $G_{r}=G_{s}=0$. The coefficient in front of the linear-in-$\omega_{B}$ term changes sign at the critical value of $G_{i}$ given by

[TABLE]

We find several qualitatively different behaviors of the MR, illustrated by the dashed lines in Figure 3a-b. The lines ➀–➃ correspond to the regions in the parameter space $(d_{N},G_{i})$ shown in Figure 3c, obtained by plotting the magnitude of the linear-in-$\omega_{B}$ term. The critical value in Eq. (17) as a function of $d_{N}$ is shown in Figure 3c by the red dashed line. Thus, for $J_{\textit{sd}}>0$, the anomalous behavior manifests itself in a segment of negative MR on line ➀, marked by the black straight line in Figure 3a. The dependence shown by line ➁ is consistent with the physical picture given above, in which the Zeeman and exchange fields can be superimposed locally with one another, giving rise to a shifted-to-the-left parabolic $B$-field dependence for $\Delta\rho_{1}(B)$, on top of the fully developed SMR gap. The dependence shown by line ➀ cannot be understood in terms of a local interplay between the SMR and HMR effects occurring at the NM/MI interface. We remark that no anomalous behavior occurs in a semi-infinite space, at one interface. Therefore, it is essential to involve in the explanation the NM/vacuum interface, which has a spin accumulation oriented predominantly opposite to, but not strictly anti-aligned with the spin accumulation at the NM/MI interface.

We illustrate the spin accumulations occurring in the SMR effect at both interfaces in Figure 3d. Since the SMR effect suppresses significantly the spin accumulation $\bm{\mu}_{s}(0)$ at the NM/MI interface, the Hanle effect occurring near that interface induces a rather small change of spin accumulation, which represents mainly a rotation about the $B$-field axis, such as $\delta\bm{\mu}_{s}(0)\propto\omega_{B}\left[\bm{n}\times\bm{\mu}_{s}(0)\right]$. In contrast, the Hanle effect occurring near the NM/vacuum interface induces, in the same fashion, a relatively larger change of spin accumulation, $\delta\bm{\mu}_{s}(d_{N})\propto\omega_{B}\left[\bm{n}\times\bm{\mu}_{s}(d_{N})\right]$. By means of diffusion or, in other words, when the film is so thin that the spin accumulations of both interfaces overlap with each other (see orange and green parts of $\mu_{s}^{y}$ in Figure 3d), a non-local interplay between SMR and HMR effects takes place. In particular, for a magnetic field along $z$ as shown in Figure 3d, the Hanle effect at the NM/vacuum interface brings in a $\mu_{s}^{x}$ component generated from a $\mu_{s}^{y}$ component of opposite sign (green part of $\mu_{s}^{y}$). After diffusing across the thin film thickness, the $\mu_{s}^{x}$ component is converted back into a $\mu_{s}^{y}$ component at the NM/MI interface, due to the interfacial exchange field. The longitudinal resistivity correction is governed by the change in the $y$-component of the spin bias across the sample Chen et al. (2013); Vélez et al. (2016), $\Delta\rho_{L}\propto\mu_{s}^{y}(d_{N})-\mu_{s}^{y}(0)$. A negative MR is obtained when the difference $\mu_{s}^{y}(0)-\mu_{s}^{y}(d_{N})$ grows with applying a magnetic field, i.e. when the spin bias across the sample increases with $B$. This usual behavior is obtained also for the quantity $\mu_{s}^{y}(0)$ alone, although we find that the difference $\mu_{s}^{y}(0)-\mu_{s}^{y}(d_{N})$ begins to increase with $B$ at a smaller critical $G_{i}$ than the value at which $\mu_{s}^{y}(0)$ begins to increase. Nevertheless, the physical picture leading to such a striking effect is common to both quantities: The $\mu_{s}^{x}$ component generated from a large negative spin accumulation at the NM/vacuum interface is converted into a $\mu_{s}^{y}$ component at the NM/MI interface due to $G_{i}$, obtaining a non-local contribution $\delta\mu_{s}^{y}(0)\propto-\omega_{B}G_{i}e^{-d_{N}/\lambda_{s}}\mu_{s}^{y}(d_{N})$. This non-local contribution competes with the one generated locally by the Hanle effect at the NM/MI interface, $\delta\mu_{s}^{y}(0)\propto\omega_{B}\mu_{s}^{x}(0)$. Notably, $\mu_{s}^{x}(0)$ is suppressed for large $G_{i}$ as $\propto 1/G_{i}$, which makes the correction generated locally small. From the balance of the local and non-local corrections to $\mu_{s}^{y}(0)$, we recover the exponential dependence of the critical $G_{i}$ in Eq. (17) for large $d_{N}\gg\lambda_{s}$, namely $G_{i,c}\propto e^{-d_{N}/2\lambda_{s}}$. Thus, we conclude that the transition from positive to negative MR for $J_{\textit{sd}}>0$ occurs when the non-local interplay between HMR and SMR dominates over the local one.

In the case of $J_{\textit{sd}}<0$, see Figure 3b, a negative MR is not unusual, since the Zeeman and exchange fields have opposite signs and can, thus, compensate each other to some extent. In this case, one would expect a shifted-to-the-right parabolic $B$-field dependence for $\Delta\rho_{1}(B)$, on top of the fully developed SMR gap. This expectation is, indeed, met when the magnitude of $G_{i}$ is smaller than the critical value in Eq. (17), see line ➃ in Figure 3b. As for line ➂, which corresponds to a large negative $G_{i}$ ($G_{i}<G_{i,c}<0$), its behavior resembles qualitatively that of line ➃, and can not be reliably identified in the absence of the reference curves showing the pure SMR and pure HMR separately. Nevertheless, the anomalous behavior originating from the non-local interplay between SMR and HMR consists here in having a positive slope in the beginning of the interference region, as marked by the straight black line in Figure 3b.

With the help of our theoretical model we explore further several examples that illustrate the non-monotonic behavior of the MR in a realistic system and show how it evolves with temperature. Specifically, we assume that the MI can be described as a Weiss ferromagnetic insulator. It exhibits a spontaneous finite average magnetization, $\langle\hat{S}_{\parallel}\rangle$, at temperatures below the Curie-Weiss temperature $T_{c}$. The $B$- and $T$- dependence of $\langle\hat{S}_{\parallel}\rangle$ is obtained by solving the transcendental equation,

[TABLE]

where $\mathcal{B}_{S}(X)$ is Brillouin function and $J_{m}$ is the coupling constant between nearest neighbours in the Heisenberg model. This expression also describes a PM insulator after setting $J_{m}=0$.

We compute the longitudinal resistivity from Eqs. (13-15). The spin-dependent conductances, Eqs. (10-12), are determined from the relaxation times in Eqs. (6-7), which can be obtained after substitution of magnetization $\langle\hat{S}_{\|}\rangle$ from Eq. (18) and spin-spin correlation function $\langle S^{2}_{\parallel}\rangle$ from relation $\langle S^{2}_{\parallel}\rangle=S(S+1)+\coth[(\hbar\omega_{B}-6\langle\hat{S}_{\|}\rangle J_{m})/2T]\left\langle S_{\parallel}\right\rangle$. Figure 4 summarizes our results for a PM and a FM insulators. The dashed lines in Figure 4a-d correspond to a field applied in $y$-direction, whereas the solid lines to a field in $z$-direction. It is in the latter situation that the predicted anomalous behavior becomes evident.

As one might anticipate, the non-monotonic behavior is best pronounced at low temperatures for which the spin-dependent conductances $G_{s}$, $G_{r}$, and $G_{i}$ saturate after applying a relatively small magnetic field. We have chosen the parameters such that the solid curves in Figure 4a-d correspond to the predicted anomalous behaviors ➀ and ➂ in Figure 3a-b. In the PM case, Figure 4a-b, the anomalous differential MR starts at finite fields when $G_{i}$ is sufficiently large, cf. Figure 2a. In contrast, in the FM case, $G_{i}$ is large enough even at small fields due to the spontaneous magnetization, and the anomalous behaviors are already seen for $B\to 0$ and over a larger range of temperatures below $T_{c}$, see Figure 4c-d. In the FM case one also obtains the SMR gap, defined as $\Delta_{G}:=\rho_{L}(B\equiv B_{z}\rightarrow 0)-\rho_{L}(B\equiv B_{y}\rightarrow 0)$.

In Figure 4e-f, we show $\rho_{L}(B)$ in the FM case for a field in $z$-direction and different values of the NM thickness, $d_{N}$. In accordance to Eq. 17, by changing $d_{N}$ one tunes the critical value of $G_{i}$ and hence the behavior of the MR changes. For the chosen parameters in Figure 4e-f, the thickest film exhibits the normal behavior, see red solid lines in Figure 4e-f, whereas thinner films show the anomaly in the MR, blue and green dashed lines. Thus, our modelling shows that the anomalous behavior is expected to be observed in MIs with sufficiently large values of $G_{i}$. This can be achieved for example in insulating FMs with large local moments, as for example in EuS or EuO Müller et al. (2009); Strambini et al. (2017); Wei et al. (2016).

Conclusions- We have presented a theory of the SMR effect from a microscopic perspective, in which SMR relates to the microscopic processes of spin relaxation at the NM/MI interface. Our theory covers a wide range of MIs and can be used to investigate the effect of a magnetic field and temperature on MR in NM/MI Hall-bar setups and beyond. We found a non-local interplay between SMR and HMR which gives rise to a negative linear-in-magnetic-field MR. Our theory provides a useful tool for understanding present and future experiments and it has the potential to evolve into a full-fledged technique to measure the magnetic properties of the NM/MI interfaces, focusing exclusively on probing the very surface of the MI.

Acknowledgement- This work was supported by Spanish Ministerio de Economia y Competitividad (MINECO) through the Projects No. FIS2014-55987-P and FIS2017-82804-P, and EU’s Horizon 2020 research and innovation program under Grant Agreement No. 800923 (SUPERTED). We thank Felix Casanova, Saul Velez, and Yuan Zhang for useful discussions.