Theory of Spin Hall Magnetoresistance from a Microscopic Perspective

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Theory of Spin Hall Magnetoresistance from a Microscopic Perspective XIAN-PENG ZHANG, F. Sebastian Bergeret, and Vitaly N. Golovach Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b02459 • Publication Date (Web): 05 Aug 2019 Downloaded from pubs.acs.org on August 6, 2019

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Theory of Spin Hall Magnetoresistance from a Microscopic Perspective Xian-Peng Zhang,†,‡ F. Sebastian Bergeret,‡,† and Vitaly N. Golovach∗,¶,‡,†,§ †Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastián, Basque Country, Spain ‡Centro de Física de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, 20018 Donostia-San Sebastián, Basque Country, Spain ¶Departamento de Física de Materiales UPV/EHU, 20018 Donostia-San Sebastián, Basque Country, Spain §IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain E-mail: [email protected] Phone: +34 943018453

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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 behavior of the resistance due to an interplay between the Hanle effect and the spin mixing at the interface. We describe an unusual negative magnetoresistance originating from a non-local Hanle effect. Our theory provides a useful tool for understanding the experiments on heavy metals in contact with magnetic insulators of different kinds, and it enables the spin Hall magnetoresistance effect to be used as a technique to study magnetism at interfaces.

Keywords: Spin Hall effect, magnetoresistance, magnetic insulator, spin mixing conductance, Hanle effect

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. 1–13 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. 14–16 This effect, called the spin Hall magnetoresistance (SMR), has been observed in several experiments. 17–21 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. 18,22 The latter can be controlled by magnetic fields.

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Although the theory of SMR 18,22 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. 23–27 Recent experiments 28–30 show, however, that the SMR effect depends both on B 29 and on T , 28–30 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), 31,32 which has an identical angular dependence to SMR, 32 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.

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Figure 1: Sketch of a Hall bar fabricated from a thin metallic film (blue) deposited on the surface of a magnetic insulator (brown). The longitudinal (VL ) and Hall (VH ) voltages are sensitive to variations of the charge current occurring under the influence of the spindependent scattering at the interface. The inset shows the basic process responsible for SMR: an electron with charge −e and Fermi velocity vF moving randomly in the metal scatters off the surface of the magnetic insulator and interacts with a local moment S i . The SMR corrections are expressed in terms of the interfacial exchange field, the spin-flip rate, and the spin dephasing rate, which all depend on the magnetic state of the local moments, and thus, can be controlled by magnetic fields and temperature.

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 = HNM + Vsd + HMI , where HNM is the Hamiltonian of a disordered metal with SOC and Zeeman field, HMI is the Heisenberg Hamiltonian in a magnetic field, and Vsd describes the coupling at the NM/MI interface. We model the interface by an exchange interaction between local moments and itinerant electrons

Vsd = −Jsd

X

S i · s(r i ),

i

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(1)

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where S i is the spin operator of the local moment, s(r i ) is the operator of the itinerant spin density at position r i , and Jsd 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 dN and the spin-relaxation length λs . 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 λs . 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 Vsd in Eq. (1) as perturbation. Although the thickness b should be kept small (b ∼ l), we obtain physically meaningful results by sending first l → 0 in the diffusive limit, and only in a second step b → 0, going thus through an intermediate stage of the calculation in which l b λs . 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 f1 . To simplify the magnetic problem, we employ the Weiss mean-field theory for HMI . In this approximation, the state of the magnetic system is a product state of individual local f1

It is important to remark that the coupling in Eq. (1) acts more efficiently when the spin 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 s(r i ). We should, therefore, reduce Jsd in Eq. (1) by a factor n(z > λF )/n(z = 0), where n(r) is the average charge density in the metal and λF 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 Jsd hereafter.

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moments, yielding hSαi (t)Sγj i = δij hSαi (t)Sγi i + (1 − δij )hSαi ihSγj i.

(2)

The equilibrium properties of the local moments are determined by the spin expectations hSα i and hSα2 i (α = 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 µs in the metal (including the b-layer) following a standard derivation 33 µ˙ αs −

1 α ∂i Js,i − ωL αγκ nγ µκs = −Γακ µκs , eνF

(3)

where superscript Greek indices denote spin projections (α = {x, y, z}) and subscript Latin indices denote current directions (i = {x, y, z}), n = B/B is the unit vector of the B-field, e is the elementary charge (e > 0), νF is the density of states per spin species at the Fermi level, αβγ is the antisymmetric tensor, and repeated indices are implicitly summed over. α α The spin current Js,i has units of electrical current, with −Js,i /2e giving the amount of spin

with polarization α transported in direction i through a unit cross section and per unit of time. Both the Larmor precession frequency, ωL , and the spin relaxation tensor, Γακ are inhomogeneous in space due to the b-layer insertion. Specifically, for the geometry in Figure 1, we have n2D imp Jsd ˆ hSk iδb (z), ωL (z) = ωB − ~

(4)

where ωB = gµB B/~, with g ≈ 2 being the electron g-factor and µB the Bohr magneton, ˆ ˆ n2D imp is the number of local moments per unit area at the MI/NM interface, Sk = S · n is the longitudinal spin operator of a representative local moment, and δb (z) equals to 1/b in the b-region and zero elsewhere. In the limit b → 0, δb (z) tends to the Dirac δ-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 6


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splitting of the density of states in the superconducting regime. 34,35 The spin relaxation tensor in Eq. (3) reads δακ 1 1 δακ + + − nα nκ δb (z), Γακ (z) = τs τ⊥ τk τ⊥

(5)

where τs 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 . 10 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, νF↑ ≈ νF↓ ≡ νF , owing to the large Fermi energy of the NM. In Eq. (5), τk and τ⊥ denote, respectively, the longitudinal and transverse spin relaxation times per unit length for the itinerant electron in the b-region. In our notations, T1 = bτk is the relaxation time of the longitudinal spin component µk = µs · n, and T2 = bτ⊥ is the decoherence time of the transverse spin components µ⊥ = n × (µs × n). Within the Born-Markov approximation, 36 we obtain 1 2π 2D 2 m n νF Jsd ωL nB (ωLm ) [1 + nB (ωLm )] |hSˆk i|, = τk T imp 1 1 π 2 ˆ2 νF Jsd hSk i, = + n2D τ⊥ 2τk ~ imp

(6) (7)

where nB (ω) = 1/(e~ω/T − 1) is the Bose-Einstein distribution function and ωLm = ωB − P hSˆk i j Jij /~, with Jij being the coupling constant of the Heisenberg ferromagnet. In deriving Eqs. (6) and (7), we used the Weiss theory approximation in Eq. (2) and have additionally assumed that the correlator hSˆα Sˆβ i(ω) for a spin on the MI surface can be approximated by the corresponding correlator for a spin deep in the bulk of the MI. We remark that 1/τk is due to spin-flip processes in which the itinerant electron is scattered, say, from a state k ↑ to a state k0 ↓ and a spin excitation is created in the MI. This spin excitation is actually a magnon in general, although it is equivalent to a local moment excitation in the Weiss theory, because the magnon band is flat (Einstein model). The spin-flip correlators hSˆ± Sˆ∓ i(ω) have

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been expressed through the spin expectation value hSˆk i, see Supporting Information. Furthermore, we remark that 1/τ⊥ contains two distinct terms. One term (1/2τk ) is again due to spin-flip processes, whereas the other is due to spin dephasing. The dephasing contribution comes from scattering processes during which the electron spin, being in a superposition of spin-up and spin-down, acquires a precession phase about the n-direction. Notably, it is the term Ski sk (r i ) in the scalar product S i · s(r i ) in Eq. 1 that is responsible for such random kicks of the spin precession phase, i.e. for dephasing. Therefore, the magnetic state of the MI is not changed in the Weiss-theory approximation, since it is an eigenstate of Ski . It is important to note also that the difference between 1/τk and 1/τ⊥ is entirely due to the ordered magnetic state of the local moments at the interface. In the unordered state, we have 2 1/τk = 1/τ⊥ = 2πn2D imp νF Jsd S(S + 1)/3~ and no distinction between dephasing and spin-flip

processes can be made. 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 µs is almost constant and independent of time, b 1 1 υ z=0 κ J = bωL υγκ nγ µs − + µυs − eνF s,z z=−b τs τ⊥ 1 1 − − nυ (n · µs ). τk τ⊥

(8)

Next we take the limit b → 0 and write the boundary condition in a customary way 21,37

−eJ s,z (0) = Gs µs + Gr n × (n × µs ) + Gi n × µs ,

(9)

where we set J s,z = 0 at z = −b, because, by construction, the electron does not penetrate

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into the MI beyond the b-layer. The spin dependent conductances read 1 Gs = −e2 νF , τk 1 1 2 − , Gr = e νF τ⊥ τk e2 Gi = − n2D νF Jsd hSˆk i. ~ imp

(10) (11) (12)

It is customary to call the complex quantity G↑↓ = Gr + iGi spin-mixing conductance, 37 whereas Gs is sometimes called spin-sink conductance. 21 We note that Gs originates entirely from spin-flip processes and can, therefore, be unambiguously associated with magnon emission and absorption. In contrast, Gr does not have a physical meaning on its own. However, the combination Gr − Gs is proportional to the spin decoherence rate (1/τ⊥ ) of the itinerant electron at the NM/MI interface. It follows from Eq. (7) that a part of Gr − Gs is due to spin-flip processes (1/2τk ), and hence is identical in nature to Gs , whereas the other part is due to spin dephasing. The purely dephasing contribution is Gr − 21 Gs ∝ hSˆk2 i and it originates from almost elastic spin-scattering processes, which do not involve any exchange of angular momentum with the MI. Thus, Gs and Gr − 12 Gs correspond to different physical processes and have, therefore, distinct dependences on B and T . Finally, Gi is a measure of the interfacial exchange field and it is proportional to the MI magnetization.

Results and Discussion We plot the quantities Gi , Gr − 21 Gs , and − 12 Gs 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 (τk = τ⊥ ), we have Gr = Gi = 0, and Gs = − 23 G0 S(S + 1). In the strongly magnetized regime (τk τ⊥ ), we have Gr = G0 S 2 , 2 Gi = G0 S/(πνF Jsd ), and Gs ≈ 0. Here, G0 = π~ n2D imp (eνF Jsd ) is a characteristic scale of the

spin-dependent conductances. We estimate a value of G0 ≈ 3.8 × 1013 Ω−1 m−2 for a typical 18 −2 n2D and νF Jsd = 0.1. This estimate is compatible with values of spin mixing imp = 5 × 10 m

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Figure 2: The spin dependent conductances Gx (x = s, r, i) arranged in combinations of − 21 Gs , Gr − Gs , and Gi to describe, respectively, the spin flips (magnon emission), the spin dephasing (no spin transfer to MI), and the interfacial exchange field. (a) The dependence on B for a PM insulator at T = 1 K. (b) The dependence on T for an FM insulator with a Curie 2 temperature of TC = 100 K. The conductances are measured in units of G0 = π~ n2D imp (eνF Jsd ) . 38 −3 The sd -coupling constant is Jsd ac = 0.2 eV, which is parameterized by the lattice constant 2 of the NM, ac = 0.4 nm. Other parameters: S = 2, νF Jsd ' 0.08, and n2D imp ac = 0.5. conductances reported in experiments. 21,30,39 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 2D on the surface (n2D a and nb ) and taking into account its possibly different coupling strength a b (Jsd and Jsd ). 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 Gi Gr —a condition which is believed to hold for Pt thin films deposited on Y3 Fe5 O12 (YIG) 39 and which would otherwise not be possible in a simple ferromagnet, because Gi is the largest spin mixing conductance for νF Jsd 1. The Gi -compensation condition for a ferrimagnet, a a 2D b b thus, reads n2D a Jsd S − nb Jsd S = 0, which differs from the magnetization compensation a 3D b condition, n3D a S − nb S = 0, and allows for the possibility of having a finite magnetization

even when Gi = 0. And vice versa, the Néel order parameter of an antiferromagnet can manifest itself as an interfacial exchange field, provided the Gi -compensation condition is

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3D not fulfilled. We remark that, for YIG, we have S a = S b = 5/2 and n3D a /nb = 3/2. And for a b 2D and Jsd ≈ Jsd . A small difference between the Pt/YIG-[001] interface, we have n2D a = nb a b Jsd and Jsd may originate from different crystal fields for the Fe3+ 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. 14,29,40–42 Here, we would like to draw attention to an effect due to Gi 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 Gi , we keep Gr and Gs 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, 18,22,29,32 obtaining the corrections to the longitudinal (ρL ) and transverse (ρT ) resistivity of the Hall-bar setup in Figure 1 ρL ' ρD + ∆ρ0 + ∆ρ1 1 − n2y , ρT ' −ρD ωc τ nz + ∆ρ1 nx ny + ∆ρ2 nz ,

(13)

where ρD is the Drude resistivity and ωc τ is the Hall angle, with ωc = eB/mc being the cyclotron frequency and τ being the momentum relaxation time. The combined SMR+HMR resistivity corrections read 29

2 ρD [2 − R(Gs , λs )], ∆ρ0 = θSH 2 ∆ρ1 = θSH ρD {R(Gs , λs ) − Re [R(Gs − G↑↓ , Λ)]} , 2 ∆ρ2 = θSH ρD Im [R(Gs − G↑↓ , Λ)] ,

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(14)

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√ Dτs , with D = 1/2e2 νF ρD being the diffusion p = Gr + iGi is the complex spin mixing conductance, 1/Λ = 1/λ2s + iωB /D,

where θSH is the spin Hall angle, λs = constant, G↑↓

and the auxiliary function R(G, `) is defined as 2` R(G, `) = tanh dN

dN 2`

1 − ρD G` coth 1 − 2ρD G` coth

dN 2` . dN `

(15)

For Λ = λs , we recover the SMR corrections, 18,22 whereas for Gs = Gr = Gi = 0, we recover the HMR corrections. 31,32 We remark that, in general, it is important to take into account Gs , 21,29 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 Pt/LaCoO3 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 Gi for Gr − Gs G2i λs ρD coth (dN /λs ),

(16)

2 which requires that n2D imp (~/e ρD λs ) tanh (dN /λs ). 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 . 10 T. 32 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 (∂ρL /∂B < 0). This behavior would not be so surprising if it occurred solely when Gi and ωB had opposite signs. Indeed, Gi 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 Gi and ωB are equal to each other for Jsd > 0 and opposite for Jsd < 0. For electrons diffusing over a characteristic length scale ` ∼ min(dN , λs ), the interfacial exchange field can 12


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(a)

(c )

Jsd > 0

Δρ1

HMR regime

SMR regime

interference region 2 1

1

2

3

4

∂ρ/∂B0

3

4

HMR regime

interference region

SMR regime

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µs(0) Hanle

MI/N

y

p

θSMR

µs

M

µsy

overla

NM/v

acuum

x

B

x

z=0

z

µsy µs(dN)

B

z=d N

Hanle

Figure 3: (a) Interplay between SMR and HMR effects for Jsd > 0 and MI in the paramagnetic regime. The blue and red solid lines show, respectively, the SMR and HMR effects in the absence of one another. Between the SMR regime (small B) and the HMR regime (large B) an intermediate “interference” region occurs, where anomalous behavior, marked by the straight black line, is possible due to a non-local Hanle effect and its interplay with SMR. The dashed line À shows the qualitative behavior of ∆ρ1 (B) for the non-local interference, whereas line Á shows it for the local interference. (b) Same as in (a), but for Jsd < 0 and with lines Â and Ã corresponding, respectively, to the non-local and local regimes of interference. (c) Separation of the parameter space (dN , Gi ) into different regimes of interference. The red dashed line shows the critical value of Gi in Eq. (17) as a function of dN . The regions À-Ã correspond to the four kinds of behavior shown in (a) and (b). The color code shows the sign of ∂ρL /∂ωB at constant Gi and ωB → 0. (d) Sketch of the spin accumulation at the thin-film interfaces as created by the SHE and altered by the SMR effect. The SMR effect suppresses the spin density at NM/MI interface and rotates it by a finite angle, θSMR , about the magnetization direction. In the absence of overlap between the two spin accumulations (orange and green), the Hanle effect acts locally at each interface and alters the spin accumulation in an expected manner, quite similarly to the SMR effect, see text. The overlap between the two spin accumulations makes it possible for the Hanle effect from the NM/vacuum interface to affect significantly the spin accumulation at the NM/MI interface, especially when the latter is strongly suppressed due to SMR. With applying a B field, the component µys (z = 0) can exhibit an increase instead of the decrease which one could naïvely expect from the local Hanle effect.

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be smeared near the interface over ` and superimposed onto ωB , obtaining an average Larmor frequency ωL = ωB + Gi /e2 νF `. One could naïvely expect that the HMR effect, which is proportional to ωB2 for all experimentally relevant B-field values, to become proportional to 2

ωL2 = (ωB + Gi /e2 νF `) , generating, thus, after squaring a cross term proportional to ωB Gi . For Jsd < 0, this term would then naturally lead to a negative MR. Quite surprisingly, we find a negative MR even for Jsd > 0, provided Gi exceeds a certain critical value. To investigate the origin of the anomalous behavior of the MR, we expand ∆ρ1 in Eq. (14) in powers of ωB at constant Gi and set, for simplicity, Gr = Gs = 0. The coefficient in front of the linear-in-ωB term changes sign at the critical value of Gi given by G2i,c

λs dN sinh2 (dN /2λs ) sinh −1 . = 2 2 2λs ρD cosh(dN /λs ) dN λs

(17)

We find several qualitatively different behaviors of the MR, illustrated by the dashed lines in Figure 3a-b. The lines À–Ã correspond to the regions in the parameter space (dN , Gi ) shown in Figure 3c, obtained by plotting the magnitude of the linear-in-ωB term. The critical value in Eq. (17) as a function of dN is shown in Figure 3c by the red dashed line. Thus, for Jsd > 0, the anomalous behavior manifests itself in a segment of negative MR on line À, marked by the black straight line in Figure 3a. The dependence shown by line Á 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 Bfield dependence for ∆ρ1 (B), on top of the fully developed SMR gap. The dependence shown by line À 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

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in Figure 3d. Since the SMR effect suppresses significantly the spin accumulation µs (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 δµs (0) ∝ ωB [n × µs (0)]. In contrast, the Hanle effect occurring near the NM/vacuum interface induces, in the same fashion, a relatively larger change of spin accumulation, δµs (dN ) ∝ ωB [n × µs (dN )]. 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 µys 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 µxs component generated from a µys component of opposite sign (green part of µys ). After diffusing across the thin film thickness, the µxs component is converted back into a µys 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, 22,32 ∆ρL ∝ µys (dN ) − µys (0). A negative MR is obtained when the difference µys (0) − µys (dN ) 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 µys (0) alone, although we find that the difference µys (0) − µys (dN ) begins to increase with B at a smaller critical Gi than the value at which µys (0) begins to increase. Nevertheless, the physical picture leading to such a striking effect is common to both quantities: The µxs component generated from a large negative spin accumulation at the NM/vacuum interface is converted into a µys component at the NM/MI interface due to Gi , obtaining a non-local contribution δµys (0) ∝ −ωB Gi e−dN /λs µys (dN ). This non-local contribution competes with the one generated locally by the Hanle effect at the NM/MI interface, δµys (0) ∝ ωB µxs (0). Notably, µxs (0) is suppressed for large Gi as ∝ 1/Gi , which makes the correction generated locally small. From the balance of the local and non-local corrections to µys (0), we recover the exponential dependence of the critical Gi in Eq. (17) for large dN λs , namely Gi,c ∝ e−dN /2λs . Thus, we conclude that the transition from positive

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to negative MR for Jsd > 0 occurs when the non-local interplay between HMR and SMR dominates over the local one. In the case of Jsd < 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 ∆ρ1 (B), on top of the fully developed SMR gap. This expectation is, indeed, met when the magnitude of Gi is smaller than the critical value in Eq. (17), see line Ã in Figure 3b. As for line Â, which corresponds to a large negative Gi (Gi < Gi,c < 0), its behavior resembles qualitatively that of line Ã, 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, hSˆk i, at temperatures below the Curie-Weiss temperature Tc . The B- and T -dependence of hSˆk i is obtained by solving the transcendental equation, " hSˆk i = −SBS

S T

!# ~ωB − hSˆk i

X

Jij

,

(18)

j

where BS (X) is Brillouin function. Equation (18) also describes a PM insulator after setting Jij = 0. We remark that, for sufficiently small values of Jij (including the case of a PM insulator), the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction 43 between local moments on the MI surface may dominate the surface magnetism. The RKKY coupling constant, 2 JijRKKY ∼ νF Jsd , becomes on average exponentially suppressed over distances larger than the

mean-free path, 44 although the fluctuations of this coupling constant remain long-ranged. 45,46

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It is important to note, however, that this interaction leads to an effectively two-dimensional Heisenberg model on the surface of the MI and that the Weiss mean-field theory can then P be applied only for sufficiently large temperatures, T j JijRKKY , to avoid discrepancies with the Mermin-Wagner theorem. 47

Figure 4: (a-d) Relative longitudinal resistivity, [ρL (Bi ) − ρL (By → 0)]/ρD , as a function of magnetic field B applied along main directions (B ≡ Bi with i = x, y, z). Solid lines show the case B ≡ Bz (or B ≡ Bx , which is equivalent), whereas dashed lines shows the case B ≡ By . Panels (a,b) correspond to a PM insulator, whereas panels (c,d) to a FM insulator with a Curie temperature of TC = 100 K. Panels (a,c) and (b,d) correspond, respectively, to positive 38,48 and negative sd -coupling constant, Jsd a−3 respectively. The NM thickness c = ±0.1 eV, is chosen as dN = 2 nm and the different curves correspond to different temperatures. (e-f) Different to above, relative longitudinal resistivity, [ρL (Bz ) − ρL (Bz → 0)]/ρD , as a function of magnetic field B applied in z-direction, for a FM insulator which couples to NM with a coupling constant: (e) Jsd a−3 = +0.1 eV and (f) Jsd a−3 = −0.1 eV. The temperature c c is chosen as T = 10 K and the different curves correspond to different thickness. In all panels we have chosen the following values of other parameters: θSH = 0.1, λs = 3.0 nm, 2 ρD = 1.0 × 10−6 Ω m, n2D imp ac = 0.5, S = 2, |νF Jsd | ' 0.04, and ac = 0.4 nm. 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 hSˆk i from Eq. (18) and spin-spin correlation

P function hSk2 i from the relation hSk2 i = S(S + 1) + coth[(~ωB − hSˆk i j Jij )/2T ] Sk , see 17


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Supporting Information. 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 Gs , Gr , and Gi 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 À and Â in Figure 3a-b. In the PM case, Figure 4a-b, the anomalous differential MR starts at finite fields when Gi is sufficiently large, cf. Figure 2a. In contrast, in the FM case, Gi is large enough even at small fields due to the spontaneous magnetization, and the anomalous behaviors are already seen for B → 0 and over a larger range of temperatures below Tc , see Figure 4c-d. In the FM case one also obtains the SMR gap, defined as ∆G := ρL (B ≡ Bz → 0) − ρL (B ≡ By → 0). In Figure 4e-f, we show ρL (B) in the FM case for a field in z-direction and different values of the NM thickness, dN . In accordance to Eq. 17, by changing dN one tunes the critical value of Gi 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 Gi . This can be achieved for example in insulating FMs with large local moments, as for example in EuS or EuO. 35,49,50

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

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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.

Supporting Information Available The model Hamiltonian in second quantization, the longitudinal and transverse spin-relaxation rates expressed via spin-spin correlators, the Weiss mean-field theory.

Author Information Corresponding Author ∗

Email: [email protected]

Notes The authors declare no competing financial interest.

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.

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Theory of Spin Hall Magnetoresistance from a Microscopic Perspective

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