Spin Pumping and Spin Currents in Magnetic Insulators

CHAPTER FIVE

Spin Pumping and Spin Currents in Magnetic Insulators Mathias Weiler*,1, Georg Woltersdorf †, Matthias Althammer*,2, Hans Huebl*, Sebastian T.B. Goennenwein*

*Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany † Department of Physics, Universita¨t Regensburg, Regensburg, Germany 1 Present address: National Institute of Standards and Technology, Boulder, Colorado, USA 2 Present address: Center for Materials for Information Technology MINT and Department of Chemistry, University of Alabama, Tuscaloosa, Alabama, USA

Contents 1. Spin Current Generation 1.1 From Charge to Spin Currents 1.2 Spin Pumping 2. Spin Currents and Magnetization Damping 2.1 Enhanced Gilbert Damping From Spin Pumping 2.2 g"# from Enhanced Damping in Metallic Ferromagnets 2.3 Double Layer Dynamics 2.4 g"# Determined from Line Broadening in YIG 3. Electrical Detection of Spin Currents Generated via Spin Pumping 3.1 Spin Hall Angle 3.2 Electrically Detected Spin Pumping 4. Spin Currents and the Spin-Mixing Conductance Concept 4.1 Spin Hall Magnetoresistance 4.2 Comparison of Spin Pumping, Spin Seebeck Effect, and Spin Hall Magnetoresistance References

125 125 126 128 128 132 132 134 137 137 139 146 147 149 153

Mobile electrical charges are versatile and powerful carriers for energy as well as information. Accordingly, charge transport phenomena have been extensively studied and are well understood. However, charge is not the only conceivable degree of freedom for transport. One attractive alternative option is spin angular momentum, as will be discussed in the context of spin currents in this chapter. Solid State Physics, Volume 64 ISSN 0081-1947 http://dx.doi.org/10.1016/B978-0-12-408130-7.00005-8

#

2013 Elsevier Inc. All rights reserved.

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The term spin current refers to a directed flow of (spin) angular momentum [1,2]. Thus, since only the transport of spin angular momentum is required to realize a finite spin current, the presence of mobile electric charges is not necessary. In other words, spin currents can flow both in electrically conductive materials, as well as in so-called magnetic insulators, that is, in electrically insulating but magnetically ordered materials. In the latter, the localized, exchange-coupled magnetic moments support (propagating) magnonic excitations or spin waves [3,4], which carry angular momentum and thus also a spin current. In conventional, nonmagnetic1 conductors such as semiconductors or normal metals, spin current flow takes place via the transport of mobile charge carriers with a finite spin polarization. Figure 5.1 shows two limiting cases for such mobile electron-based transport processes in the spirit of Mott’s two spin-channel model: Fig. 5.1a depicts spin-unpolarized, pure charge transport arising from the parallel flow of spin-up and spin-down electron currents. In contrast, Fig. 5.1c shows pure spin transport without (net) charge flow. In the two-channel model, the spin current Js is realized via two counter-flowing charge currents of equal magnitude but opposite spin orientation in this case. Finally, Fig. 5.1b schematically depicts one scenario for spin-polarized charge transport, with an unbalanced number of spin-up and spin-down electrons flowing in the same direction.

A

Jc

B

J

c+

Js

C

Js

Figure 5.1 (a) The parallel motion of an identical number of electrons with spin-up and electrons with spin-down results in a spin-unpolarized pure charge current Jc. (b) An unequal number of spin-up and spin-down electrons moving in the same direction gives rise to both, a charge current Jc and a spin current Js. (c) For a counterflow motion of charge carriers with opposite spin orientation, the net electric charge current vanishes. However, a finite pure spin current Js flows.

1

The term nonmagnetic implies that the material under consideration does not exhibit exchange coupling, that is, it is paramagnetic or diamagnetic.

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It also should be emphasized that charge is a conserved quantity, while spin angular momentum can be transferred to, for example, the lattice. Therefore, the spin orientation will randomize on the so-called spin flip time scale [1,2], such that spin and charge currents are qualitatively different in this respect.

1. SPIN CURRENT GENERATION 1.1. From Charge to Spin Currents Charge and spin transport in electrical conductors are connected because of spin–orbit interaction (SOI) [1,2,5–8]. In other words, in materials with finite spin–orbit coupling, a charge current will be accompanied by a spin current, and vice versa. In the literature, the term spin Hall effect (SHE) is now commonly used for the generation of a spin current from a charge current via spin–orbit coupling, while the expression inverse spin Hall effect (ISHE) refers to the reciprocal effect, where a charge current is induced by a spin current [5,6]. The SHE and ISHE are sketched in Fig. 5.2, in a slab of a normal metal where spin-up and spin-down electrons [here assumed to be quantized along z] are the charge carriers. Starting with a pure (spin-unpolarized) charge current Jc || x (c.f. Fig. 5.1a), SOI-induced mechanisms, such as skew scattering [9], side-jump scattering [10], and intrinsic interactions [11,12], result in a force on the electrons, the direction of which is perpendicular to both, the electron propagation direction x and spin orientation s 2 {z, þz}. Hence, spin-up and spin-down electrons are

A

B Js

Jc

z y

J

c

x

J

s

Figure 5.2 (a) Spin Hall effect. When a charge current Jc flows, spin-up and spin-down electrons are deflected in opposite directions due to the spin–orbit interaction. This results in a pure spin current Js orthogonal to Jc. (b) Inverse spin Hall effect. As a direct consequence of the spin Hall effect, a spin current Js gives rise to an orthogonal charge current Jc.

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deflected in opposite directions. In Fig. 5.2a, spin-up electrons are deflected along þy and spin-down electrons along y, which results in an electron current J"s || þ y. Since the spin-down electron current J#s ||  y is antiparallel to the spin-up electron current J"s , the charge current flow cancels and one finds a pure spin current Js ¼ J"s  J#s . The charge current Jc thus gives rise to a spin current Js of electrons perpendicular to both, Jc and s:   ℏ ð5:1Þ J s ¼ aSH  J  s: 2e c In Eq. (5.1), the spin Hall angle aSH is a material parameter that reflects the strength of the spin–orbit coupling, e ¼ |e| is the elementary charge, and ℏ is the reduced Planck constant. We will address the spin Hall angle in more detail in Section 3.1. The ISHE is illustrated in Fig. 5.2b. Consider a pure spin current Js||x with spin-up electrons flowing along þx and spin-down electrons flowing along x. In this case, both the spin direction and the direction of motion are opposite for spin-down and spin-up electrons. They are thus both deflected in the same direction and create a charge current Jc||y given by   2e ð5:2Þ J  s: J c ¼ aSH  ℏ s As can be seen from Eq. (5.2), the ISHE allows for an all-electrical detection of a spin current flowing in a normal metal with a finite spin Hall angle [13].

1.2. Spin Pumping An exchange-coupled, ferromagnetic2 material with magnetization M can also source a spin current if M is driven out of equilibrium. The magnetic excitation can hereby be achieved by a variety of external driving forces, for example, thermal gradients [14–17], (spin-polarized) charge current flow [18], spin torque-related effects [19–22], ferromagnetic resonance (FMR) [13,23–25], phononic or strain-based magnetization motion [26–28], etc. In a macrospin picture, the dynamics of the excited magnetization M is described by the Landau–Lifshitz–Gilbert equation [3,4,29] 2

To enhance readability, we here use the term “ferromagnetic” for all exchange-coupled materials, in particular, also for ferrimagnetic and antiferromagnetic systems.

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  dm dm ¼ gðm  m0 Heff Þ þ a m  : dt dt

ð5:3Þ

The first term on the right hand side of Eq. (5.3) describes the precessional motion of the magnetization orientation m ¼ M/|M| ¼ M/MS perpendicular to the effective magnetic field Heff, with the gyromagnetic ratio g and the saturation magnetization MS [3,4,29]. The relaxation of the magnetization toward its equilibrium orientation is phenomenologically described by the second term of the right hand side in Eq. (5.3), which is a viscous damping term, parameterized by the Gilbert damping parameter a. It is important to realize that the precessional motion of M in a ferromagnetic material is accompanied with a reduction DM of the projection of the magnetization vector along the quantization axis, as sketched in Fig. 5.3. Angular momentum relaxation thus is required for the magnetization to reestablish its equilibrium orientation. This angular momentum change can be achieved by transferring angular momentum to the lattice of the ferromagnet itself. Interestingly, in ferromagnet/normal metal (F/N) heterostructures, another angular momentum transfer process enables magnetization relaxation, as schematically depicted in Fig. 5.6a. In this so-called spin-pumping process, the magnetization relaxes by emitting a flow of angular momentum—that is, a spin current—into the adjacent metal [30,31]. Magnetization damping experiments in F/N heterostructures therefore are a powerful experimental tool for the study of spin current flow across F/N interfaces, as discussed in more detail in Section 2. Moreover, the pumped spin current can be transported within N, and therefore acts as a source of angular momentum (spin battery) [32]. Alternatively, the pumped

DM M

Heff Θ

Figure 5.3 Upon excitation, the magnetization vector M in a ferromagnet will precess around the effective magnetic field Heff with a precession cone angle Y. This motion is accompanied by a reduction DM of the magnetization projection onto the effective magnetic field axis.

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spin current can also be converted back into a charge current via the ISHE [13]. This all-electrical detection of the spin currents arising from spin pumping is addressed in Section 3, and compared to other spin currentrelated effects in Section 4.

2. SPIN CURRENTS AND MAGNETIZATION DAMPING In this section, we address the effect of spin pumping on the magnetization dynamics in the ferromagnetic layer. In particular, we will focus on Gilbert damping in metallic and insulating ferromagnetic materials, since spin current transfer across the F/N interface in F/N heterostructures enhances this type of damping. The efficiency of this interlayer spin current transfer is thereby governed by the interfacial spin-mixing conductance. Accordingly, measurements of the magnetization damping in F/N heterostructures in comparison with the damping in isolated F layers provide direct access to the spin-mixing conductance at the F/N interface. As discussed in more detail in the following, the spin-mixing conductance is furthermore a key parameter for the interpretation of spin pumping/ISHE voltage signals (Section 3), and for spin current flow across F/N interface in more general terms (Section 4).

2.1. Enhanced Gilbert Damping From Spin Pumping As discussed in Section 1, spin pumping is an effect that describes the emission of a pure spin current from F into N when the magnetization vector precesses [31]. Hereby, additional spin relaxation (damping) in the ferromagnet is induced. We follow the approach by Tserkovnyak, Brataas, and Bauer [21,31,33], who use a scattering matrix formalism for quantifying the charge and spin flow over the F/N interface [34]. Imposing that the charge current over the interface vanishes, the spin current density JSP s pumped from the ferromagnet into the normal metal is given by     ℏ dm dm SP SP J s ¼ I s =F ¼ Ar m   Ai , ð5:4Þ 4pF dt dt with the interface area F and the complex-valued spin-pumping conductance A  Ar þ iAi. A can also be expressed as A/F ¼ g"# þ t"#, with the spin-mixing conductance g"# and the spin-dependent transmission t"# [21,31,33]. Hereby, t"# often is considered small and neglected, assuming that the normal metal film is thicker than its transverse spin coherence length [33].

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Note that in literature we find three common definitions for the spinmixing conductance: (i) the integral spin-mixing conductance g"#F, (ii) the interface area normalized spin-mixing conductance g"# given in units m2, which we use throughout this chapter, and (iii) g"# normalized to the conductance quantum e2/h. Inspecting Eqs. (5.3) and (5.4), one finds that spin pumping is closely reminiscent of the Gilbert damping expression. Assuming that the emitted spin current is entirely absorbed in the N layer, one can use the conservation of the total angular momentum of the system to calculate the magnitude of the additional, spin-pumping-induced Gilbert damping in the ferromagnet: I SP s 

1 @Mtot ¼0 g @t

ð5:5Þ

where Mtot ¼ MV is the total magnetic moment and V is the volume of the ferromagnetic layer. Hence the additional damping due to the emission of a spin current across the F/N interface in Eq. (5.4) can be written as: aSP ¼ gmB

g"# 1 , 4pMS tF

ð5:6Þ

where mB is the Bohr magneton, tF is the thickness of F, and  accounts for a backflow of the spin current as discussed below. The inverse dependence of asp ∝ 1/tF on the F film thickness is related to the interfacial origin of the effect. The loss of angular momentum due to JSP s occurs parallelly with the spin relaxation due to intrinsic Gilbert damping in the ferromagnet. For thin metallic ferromagnetic films, the intrinsic damping is mostly caused by magnon–phonon coupling and itinerant electron scattering [35,36]. In the prototype ferromagnetic insulator yttrium iron garnet (Y3Fe5O12, YIG), the intrinsic damping is attributed to magnetoelastic coupling. Magnetic insulators such as YIG are particularly attractive for spin currentrelated experiments, since charge currents cannot propagate in these materials. Thus, charge and spin transport phenomena are separated a priori in magnetic insulators. YIG furthermore exhibits a very low (intrinsic) magnetization damping, such that the impact of spin current transport on the magnetic properties is comparatively easy to detect [37,38], as discussed in the following. When a spin current flows across the F/N interface due to spin pumping, a spin accumulation can build up close to the interface in the normal metal, which itself can cause a back flow of spins Jback into the ferromagnet. Thus, s SP back the total spin current flowing over the interface is given by Jnet . s ¼ Js  Js

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The impact of spin current back flow is of particular importance if the thin N layer does not act as a perfect spin sink, and can be accounted for by the back flow factor [33,39] 

1 e2  ¼ 1 þ 2g"# rN lSD cothðtN =lSD Þ , h

ð5:7Þ

were rN and tN are the resistivity and thickness of N, respectively, and lSD is the spin diffusion length in N. Note that Eq. (5.7) is a simplified correction factor imposing that the FMR precession frequency is smaller than the spin flip time as detailed in Ref. [33,39]. For the net spin current one has SP J net s ¼ J s :

ð5:8Þ

Before interpreting an additional magnetization damping arising in F/N heterostructures in terms of the spin-pumping effect, it is important to confirm the Gilbert nature of the spin relaxation, for example, by frequencydependent measurements of the FMR line width m0DH. For Gilbert damping one expects from Eq. (5.3) o m0 DH ¼ a , g

ð5:9Þ

where o ¼ 2pn is microwave frequency. In real samples, the FMR line width typically is not only caused by Gilbert damping, but also by inhomogeneities of the sample. In addition, line width broadening due to inhomogeneities on short length scales may be strongly magnified by two magnon scattering processes [40,41]. However, even in the presence of significant sample inhomogeneity, the Gilbert damping can be extracted from the evolution of the FMR line width with frequency: o m0 DH ðoÞ ¼ a þ m0 DH0 , g

ð5:10Þ

where DH0 is the zero frequency line width offset, which is caused by magnetic inhomogeneities. If DH0 is large, it is important to measure DH(o) over a wide frequency range [42,43]. Moreover, note that by measuring FMR with the magnetic field applied perpendicular to the film plane, two magnon scattering can be efficiently suppressed due to the modified spin wave dispersion [40,42], such that a more reliable extraction of the Gilbert damping parameter is possible. The measured damping constant is then

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given by the sum of the intrinsic Gilbert damping a0 and the Gilbert damping due to spin pumping aSP: a ¼ aSP þ a0 :

ð5:11Þ

An elegant way to determine aSP is to compare F/N samples with the same F layer, covered by different N materials or N layers of different thicknesses. An example for this procedure is shown in Fig. 5.4a. For a 12-nm thick Ni80Fe20 F layer, m0DH(o) is measured for both an N cap layer with weak spin relaxation (Au) and an N cap layer with large spin relaxation (Pt). From the difference of these two sets of data, the spin-pumping-induced Gilbert damping in Pt can then be determined. Let us finally mention that owing to the so-called magnetic proximity effect, a static magnetic polarization can be induced in the normal metal in the direct vicinity of the F/N interface [44–49]. This complication has been mostly ignored in the spin-pumping literature so far, following the tacit assumption that the proximitized, “magnetic” N surface layer will have magnetic properties similar to the F layer. In this simple picture, a static

Figure 5.4 (a) FMR line width versus frequency measured in 12 nm NiFe films covered by 10 nm Pt (squares) and 6 nm Au (bullets). The spin-pumping induced damping for the thin Au cap is negligible and the difference between the slopes of m0DH versus frequency corresponds to the spin-pumping Gilbert damping caused by the Pt cover layer. (b and c) A cartoon of the dynamic coupling phenomenon. (b) F1 is at resonance and its precessing magnetic moment pumps spin current into the spacer, while F2 is detuned from resonance. (c) Both F1 and F2 resonate at the same effective field, inducing spin currents of equal amplitude in opposite directions. The short arrows in N indicate the instantaneous direction of the spin angular momentum mi  dm dt of the spin currents. The thin layers at the Fi/N interfaces represent the regions where the spin current is absorbed. i

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magnetic proximity effect thus will only slightly increase the F layer thickness, at the expense of tN. If the proximitized layer, however, has magnetic properties differing from those of F, the situation becomes more complex, calling for more detailed corresponding studies in the future.

2.2. g"# from Enhanced Damping in Metallic Ferromagnets Mizukami et al. investigated the enhanced Gilbert damping for very thin Ni80Fe20 films sandwiched between layers of Cu, Pd, and Pt [50,51]. For pure copper layers, no additional damping was detected, while Pd and Pt capping resulted in significant line broadening scaling inversely with tF in agreement with Eq. (5.6). This behavior is attributed to the rapid spin relaxation in Pd and Pt, which is caused by the large SOI in these materials [31]. In contrast, when a capping material with a large spin diffusion length such as Cu is used, even very large capping layer thicknesses can result only in a small additional damping. This effect is caused by a spin accumulation in the normal metal, which leads to a back flow of the spin current injected into N by the spin pumping. The effect of a finite spin diffusion length is quantitatively accounted for with the backflow factor (tN, lSD) (c.f. Eq. 5.7) and therefore the spin diffusion length in the N material can be determined using this model [33]. Careful measurements of the tN thickness dependence of asp by Foros et al. [52] and Ghosh et al. [53] extracted small values for lSD for Pd and Pt, as shown in Table 5.1. Considering that these lSD values are even below the electron mean free path, one has to question the applicability of the spin diffusion theory for data analysis [52]. As one can see from Table 5.1, in metallic systems typical values of g"# are on the order of 1  1019 m2 and in good agreement with corresponding ab initio calculations [56].

2.3. Double Layer Dynamics Instead of using a normal metal with large SOI as an absorber for the spin current, a second ferromagnetic layer can serve as perfect spin sink. In this case, a very thin N spacer can be used and the complications of  in Eq. (5.6) are avoided. The sample then consists of a trilayer structure: F1/N/F2. If the static equilibrium directions of the magnetizations in both ferromagnetic layers are parallel, the spin orientation of the injected spin current (c.f. Eq. 5.4) is transverse to the magnetization orientation of the second ferromagnetic layer, as can be seen in Fig. 5.4b. In this case, one can expect that the spin current generated at the F1/N interface is completely absorbed and creates a spin torque at the N/F2 interface due to the short coherence length

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Table 5.1 Spin-mixing conductances and spin diffusion lengths reported for various normal metal materials deposited on metallic ferromagnets lSD g"# (1019 m2) (nm) Ref. Material System

Pt/Ni80Fe20

Pt/Ni80Fe20/Pt/SiO2 (sputtered)

2.6

–

[50]

Pt/Ni80Fe20

Pt/Ni80Fe20/GaAs (sputtered)

2.5

–

Fig. 5.4a

Pd/Ni80Fe20

Pd/Ni80Fe20/Pd/SiO2 (sputtered)

1.5

–

[50]

Pd/Fe

Pd/Fe/GaAs(001) (MBE)

0.9

9

[52]

Cu/Ni80Fe20

Cu/Ni80Fe20/Cu/SiO2 (sputtered)

1.1

250

[33,51]

Cu/Ni80Fe20

Ni80Fe20/Cu/Co (sputtered)

1.5

–

[53]

Au/Fe

Au/Fe/GaAs(001) (MBE)

1.2

31

[54]

Ag/Fe

Ag/Fe/GaAs(001) (MBE)

1.2

85

[54]

Au/YIG

Fe/Au/YIG/GGG (PLD/MBE)

0.11

–

[37]

Au/YIG

Fe/Au/YIG/GGG (PLD/MBE)

0.49

–

[38]

Pt/YIG

Pt/YIG/GGG (PLD/MBE)

0.97

–

Fig. 5.5b

Ag/YIG

ab inito theory

0.3–0.9

–

[55]

All measurements were performed at 300 K. The values from [50] are extracted using Eq. (5.6) and considering that the ferromagnetic layer is in contact with a metal layer on top and bottom interfaces [50]. In the theoretical work [55], the spin-mixing conductance of YIG/Ag was calculated for different chemical terminations of the YIG interface (O, Fe, and Y) resulting in a range of values for g"#.

of spin-up and spin-down states in F2 [57] (see Fig. 5.4b). Using ab initio methods, Stiles and Zangwill have shown that this effect occurs on a length scale of a few atomic layers for typical 3d ferromagnets [58]. Indeed, in FMR experiments involving magnetic double layer structures, enhanced Gilbert damping was found for epitaxial Fe/Au/Fe structures that were grown on GaAs(0 0 1). In this case, the large uniaxial anisotropy of the Fe/GaAs(0 0 1) interface was used to separate the FMR fields of the first and the second Fe layer, that is, one can prepare a situation, where layer F1 (grown directly onto GaAs) is at FMR while the second layer (F2) is not at resonance [59]. In this situation, the spin current generated at the F1/N interface propagates across the Au spacer and is absorbed in F2 [59]. The advantage of this concept is that if the two interfaces F1/N and N/F2 can be made identical and the spacer is sufficiently thin, one can determine the spin-mixing conductance without the need to model

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the spin accumulation in the spacer material. The obtained value for g"# at the Fe/Au interface is 1  1019 m2. On the other hand, it is also possible to engineer the magnetic anisotropies of F1 and F2 such that both layers resonate at the same field and one has the situation shown in Fig. 5.4c. In this case, layers, F1 and F2, act as mutual spin pumps and spin sinks and the enhanced damping should disappear. For small precession angles, one can write the equation of motion for a F1/N/F2 magnetic double layer as follows [60]:   dmi dmi ¼ gm0 ½mi  Heff   ai mi  dt dt   dm dm j i SP  mj  ð5:12Þ  ai mi  dt dt where m1,2 are unit vectors along the instantaneous magnetization directions of the two ferromagnetic layers F1 and F2. The exchange of spin currents is a symmetric concept and the equation of motion for layer F2 can be obtained by interchanging the indices i > j in Eq. (5.12) [60]. The spin current generated by spin pumping of layer F1 propagates away from the F1/N interface [31] and is absorbed at the N/F2 interface if N is thinner than the spin diffusion length (spin sink effect) [58]. In the experiment it was indeed found that, when both F1 and F2 are at FMR, the spin-pumping-induced line broadening disappears entirely [60]. When the spacer material has a thickness that is comparable to the spin diffusion length, only a part of the injected spin current can reach the N/F2 interface. One can use time-resolved Kerr microscopy to detect the effect of the spin torque caused by the absorbed spin current on F2 [54]. Using this method, the spin diffusion properties of Au and Ag spacers were investigated [54,61]. The results for the spin-mixing conductances and the spin diffusion lengths of all of the discussed experiments are summarized in Table 5.1.

2.4. g"# Determined from Line Broadening in YIG Most spin-pumping experiments involving YIG films are performed using very thick layers that are prepared by liquid phase epitaxy (LPE) [62] on gadolinium gallium garnet (GGG) substrates. These films typically have a thickness of several micrometers. Since the additional damping due to spin pumping is an interface effect (see Eq. 5.6), it scales inversely with the total magnetic moment of the ferromagnetic layer, and it thus is difficult

Spin Pumping and Spin Currents in Magnetic Insulators

135

to observe the additional damping due to spin pumping in very thick ferromagnetic layers. Nevertheless, Reszende et al. [63] observed dramatic changes in the spin-wave spectra and FMR line width when they capped an 8-mm thick LPE-grown YIG film by a Pt layer. Interpreting these data in terms of spin pumping leads to more than one order of magnitude too large values for the spin-mixing conductance. However, some of the recent experiments have been performed using well-defined high-quality ultrathin YIG films grown by pulsed laser deposition (PLD) with thicknesses below 10 nm [37,38]. In this case, the spin-pumping-induced damping and the corresponding g"# can be measured rather easily. In their experiment Heinrich et al. [37] used ultrathin layers of YIG grown on GGG substrates. After ex situ transfer of the ultrathin YIG films, additional polycrystalline layers of Au and Fe were deposited by thermal evaporation in ultra high vacuum without further cleaning of the YIG surface [37]. The sample structure is the following: 5 nm Au/4 nm Fe/2 nm Au/9 nm YIG/GGG(1 1 1). First a thin 2-nm Au spacer was used to explore the presence of interlayer exchange coupling (IEC) between the ferromagnetic layers. The determined value of 1.5  104 J/m2 for the IEC between the metallic Fe layer and the insulating YIG layer indicates the presence of a spin-dependent electronic potential at the YIG/Au interface. In fact, the IEC coupling between Fe and YIG caused a splitting of the resonance line and revealed that the YIG surface was not homogenous leading to three different coupling constants. Using a thicker Au spacer (6 nm) suppresses the IEC but hardly affects the spin transport between YIG and Fe [61]. With such a sample, the additional damping due to spin pumping was measured to determine the spin-mixing conductance. In this case (as discussed in Section 2.3), the metallic Fe layer acts as a perfect spin sink. The increased damping was determined by measuring the FMR line width in samples with and without the Fe-layer acting as a spin sink. In both cases, the damping constant was determined by measuring DH between 10 and 40 GHz. From the difference in line width versus frequency observed in these two different samples, the spin-pumping-induced damping is determined using Eq. (5.10). The result of this procedure is a spin-mixing conductance for the YIG/Au interface of g"# ¼ 1.4  1018 m2. This value is about one order of magnitude lower than typical values for g"# in metallic systems and only a fraction of the theoretically predicted value of g"# for YIG/Ag interfaces (3  9  1018 m2) [55]. In a follow-up experiment, Burrowes et al.

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investigated the influence of surface cleaning by Argon bombardment of the YIG film prior to the deposition of the metallic cap layers [38]. Upon optimizing the sample treatment, the value of the spin-mixing conductance increased to g"# ¼ 3–4.9  1018 m2. We also prepared Pt/YIG bilayer structures in order to measure the enhanced Gilbert damping. In contrast to [37,38], the metallic spin sink layer is deposited in situ by thermal evaporation subsequent to the PLD growth of YIG/GGG(1 1 1). As shown in Fig. 5.5a, one observes a significant broadening that is induced by the Pt-capping layer. Note that the uncapped single layer of YIG shows two FMR lines. The resonance at lower magnetic field has much larger amplitude and only this line will be considered in the following. In order to determine the additional Gilbert damping, the frequency dependence of the FMR line width needs to be analyzed (c.f. Eq. 5.10). The measured line width is plotted as a function of frequency for 27 nm YIG and 27 nm YIG/7 nm Pt samples in Fig. 5.5a. From the difference of the slope of the line width for both samples, one finds by using Eq. (5.6) a value g"# ¼ 9.7  1018 m2. This result for the mixing conductance of the Pt/YIG interface can be compared to spin pumping/ISHE experiments (see Section 3) which were performed using samples grown with the same recipe. In this way, the spin-mixing conductance for in situ prepared YIG/Pt interfaces is inferred from two independent schemes, using Eqs. (5.6) and (5.13). From the ISHE measurement, discussed in more detail in the next

Figure 5.5 (a) FMR spectra of an uncapped 27-nm thick YIG film (solid line) and a 27-nm thick YIG film covered by a 7-nm Pt (dashed line) (b) FMR line width versus frequency measurement of a 27-nm YIG covered by a 7-nm Pt (squares) and no cover layer (bullets).

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137

section, one finds g"# ¼ 1  1019 m2, which is in excellent agreement with the result determined by the enhanced Gilbert damping.

3. ELECTRICAL DETECTION OF SPIN CURRENTS GENERATED VIA SPIN PUMPING In Section 2, we have seen that spin currents arising from resonant magnetization dynamics provide an additional damping channel for the magnetization in F/N heterostructures. This allows to infer the magnitude of the spin current, and to quantify the spin-mixing conductance g"# from broadband FMR experiments (c.f. Table 5.1). A complimentary, second approach to quantify the spin currents in F/N hybrids exploits the ISHE in the normal metal layer. In this scheme, the net spin current is converted into a charge current via the ISHE, such that conventional (charge-based) electronics can be used to measure Jnet s . The conversion of the spin current Jnet to a detectable charge current J s c is governed by the spin Hall angle aSH. We therefore first review the efficiency of the SHE in various electrical conductors, before addressing the electrical detection of pumped spin currents in more detail.

3.1. Spin Hall Angle As evident from Eqs. (5.1) and (5.2), the spin Hall angle aSH of a material describes its efficiency in converting charge to spin currents or vice versa. In general, both intrinsic [64] and extrinsic [65,66] contributions affect aSH. The spin Hall angle can be determined in several ways. Typical experimental approaches include electrically detected spin pumping (Section 3.2 below), spin transfer torque FMR (STT-FMR) [20,67,68], and nonlocal electrical measurements [69–72]. These three measurement schemes are sketched in Fig. 5.6. In magnetic insulator/normal metal hybrid structures, the spin Hall magnetoresistance (SMR) effect [73–77], discussed in more detail in Section 4, is another option. Note, however, that the magnetoresistance observed in such magnetic insulator-based hybrid structures has also been attributed to a magnetic proximity effect in the normal metal [48,49]. In electrically detected spin-pumping experiments, as schematically depicted in Fig. 5.6a, one takes advantage of the spin-pumping process described in Section 2 to inject a net spin current Jnet into a normal metal s with finite spin–orbit coupling. Because of the ISHE, this spin current is converted into a charge current Jc in the normal metal layer. Either Jc, or the corresponding electric field EISHE ¼ VDC/L building up along the

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B

C

z

E ISHE

M

Jc

Js

M

FM

H

Js H

y x

C D

V

I

rf

A

J

E ISHE

H Jc

s

NM

Figure 5.6 Schematic depiction of typical ferromagnet/normal metal (F/N) samples used for (a) spin pumping, (b) spin transfer torque, and (c) nonlocal spin Hall measurements, respectively.

sample under open circuit conditions, is then recorded in the experiment as a measure for Jnet s . In STT-FMR experiments [20,67], one uses the direct SHE in F/N bilayers (c.f. Fig. 5.6b) to convert a radio frequency charge current in the normal metal into a radio frequency spin current that penetrates into the ferromagnet. The magnetization in F is then driven by both, the AC Oersted magnetic field stemming from the AC charge current, and the AC spin transfer torque stemming from the AC spin current. The resulting resonant magnetization precession can be probed via a rectification voltage, appearing due to the anisotropic magnetoresistance (AMR) of the ferromagnet in FMR [20,67]. Analyzing the field-swept line shape of this DC voltage allows extraction of the ratio of charge and spin current, viz., the spin Hall angle. Note that it is also possible to apply a DC charge current to the normal metal to induce magnetization dynamics above a certain current density [18]. If aSH is large enough, it even becomes possible to switch the magnetization orientation in the ferromagnet via a charge current pulse [19,67,78,79]. Finally, in nonlocal measurements, the (inverse) SHE effect is probed in lateral nanostructures as schematically depicted in Fig. 5.6c. To this end, one generates a diffusive spin current in the normal metal by driving a charge current Jc through the F/N interface. This spin current is then converted into a charge current by the ISHE, giving rise to a measurable electric field EISHE. To measure the direct SHE, one drives the charge current through the normal metal along y and probes the voltage that develops between the ferromagnet and the normal metal [69,71]. The maximal lateral dimensions of these structures are limited by the spin diffusion length lSD of the normal metal. Note that one does not necessarily need to use the same normal metal for the spin current transport and detection/generation, but may as well use different materials, for example, as in Ref. [80].

Spin Pumping and Spin Currents in Magnetic Insulators

139

An efficient spin-to-charge current interconversion requires materials with large spin Hall angle aSH. A selection of literature values for aSH and related parameters in elemental metals, alloys, and semiconductors is compiled in Tables 5.2 and 5.3. Large spin Hall angles |aSH|  0.1 have been reported in several materials, for example, in b–W (aSH ¼ 0.33, Ref. [78]), Cu–Bi (aSH ¼  0.24, Ref. [80]), b–Ta (aSH ¼  0.15, Ref. [67]), Au (aSH ¼ 0.113, Ref. [71]), and Pt (aSH ¼ 0.16, Ref. [79]). Note, however, that different groups report very different spin Hall angles for one and the same material, as evident from Tables 5.2 and 5.3. This large spread in the reported aSH values observed in nominally identical materials could be interpreted as evidence that extrinsic effects play a major role in the SHE. More detailed and systematic experiments will be required to rule out experimental shortcomings and uncertainties, and to identify the microscopic mechanisms resulting in the exact magnitude of aSH in a given normal metal sample.

3.2. Electrically Detected Spin Pumping We now quantitatively address the electrical detection of spin pumping in F/N hybrids sketched in Fig. 5.6a [13,23]. The net spin current pumped across the F/N interface in FMR is thereby converted into a charge current Jc ¼ aSH(2e/ℏ)Jnet s  s via the ISHE (c.f. Eq. 5.2). For open circuit conditions, Jc results in charge accumulation and a corresponding electric field EISHE ¼ VDC/L (Fig. 5.6a), where L is the separation between the voltage contacts. One thus expects a DC electrical voltage VDC to appear across an F/N hybrid when the F layer is driven into FMR. Note that, while the additional Gilbert damping caused by spin pumping is isotropic (independent of the magnetization M orientation), VDC follows the symmetry of the ISHE and thus is proportional to the cross product of the interface normal (the direction of Jnet s ) and the spin polarization s ‖ M. Thus, to observe maximal VDC one employs the measurement geometry depicted in Fig. 5.6a. Figure 5.7 shows FMR and VDC data recorded using the geometry sketched in Fig. 5.6a in a cavity-based FMR setup operating at n ¼ 9.3 GHz [92]. The data were acquired using two qualitatively different F/N samples, namely, an Fe(10 nm)/Pt(7 nm) bilayer and a YIG(10 nm)/ Pt(7 nm) bilayer. In both samples, the magnetic layer and the Pt layer were deposited in situ, without breaking the vacuum [75,92]. The room temperature FMR traces obtained using magnetic field modulation and

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Table 5.2 Spin Hall angle aSH and electrical conductivity s for various materials (room temperature values unless otherwise stated) s 1/(mVm) Technique Ref. Material aSH (%)

Al

0.032

10.5

Nonlocal

[69]

Al

0.016

17

Nonlocal

[69]

Au

11.3

37

Nonlocal

[71]

Au

0.35

25.2

Spin pumping

[81]

Au

0.16

25.2

Spin pumping

[23]

Au-C

9.5

Theory (0 K)

[65]

Au-Pt

12

Nonlocal

[72]

Au-Zn

2.2

Theory (0 K)

[65]

Cu-Bi

24

10

Nonlocal

[80]

Cu-Ir

2.1

7.14

Nonlocal

[82]

Cu-Pt

2.8

Theory (0 K)

[65]

Cu-Zn

0.2

Theory

[65]

In0.07Ga0.93As

0.01

0.00286

Optical (30 K)

[83]

Nb

0.87

1.1

Nonlocal (10 K)

[84]

Mo

0.8

2.8

Nonlocal (10 K)

[84]

Mo

0.2

0.67

Nonlocal (10 K)

[85]

Mo

0.05

4.66

Spin pumping

[81]

Mo

0.028

4.66

Spin pumping

[23]

Pd

1.2

2.2

Nonlocal (10 K)

[84]

Pd

1

1.97

Spin pumping

[86]

Pd

0.64

4

Spin pumping

[81]

Pt

0.24–16

2–10

(see Table 5.3)

Si

0.01

0.02

Spin pumping

[91]

b-Ta

15

0.53

STT-FMR

[67]

b-Ta

12

0.53

STT-FMR

[67]

Ta

0.37

0.3

Nonlocal (10 K)

[84]

b-W

33

0.3

STT-FMR

[78]

16.7

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Spin Pumping and Spin Currents in Magnetic Insulators

Table 5.3 Spin Hall angle aSH, spin diffusion length lSD and electrical conductivity s reported for various Pt thin films (room temperature values unless otherwise stated, sorted by the magnitude of aSH) Material aSH (%) lSD (nm) s 1/(mVm) Technique Ref.

Pt

16

1.4

2.8

Spin torque

[79]

Pt

11

1.5

2.4

SMR

[75]

Pt

8

7

6.4

Spin torque

[87]

Pt

8

3.7

2.42

Spin pumping

[88]

Pt

7

<6

STT-FMR

[67]

Pt

5.6

<6

5

STT-FMR

[20]

Pt

4

10

2

Spin pumping

[89]

Pt

2.1

11

8.1

Nonlocal (10 K)

[84]

Pt

1.3

10

2.4

Spin pumping

[81]

Pt

1.2

8.3

4.3

Spin pumping

[90]

Pt

0.67

10

2.42

Spin pumping

[23]

Pt

0.37

3

6.41

Nonlocal

[70]

Pt

0.24

10

Theory-intrinsic

[64]

phase-sensitive detection in the Fe(10 nm)/Pt(7 nm) bilayer for H ‖ x and H ‖ x are depicted in panel (a), while the DC voltage recorded simultaneously is shown in Fig. 5.7c. In FMR, one observes VDC  80 mV in this sample. The sign change of VDC with reversal of H direction hereby is a mandatory condition for the detection of a spin current via the ISHE, since the spin current spin orientation s changes sign when H is inverted, such that VDC must also change sign according to Eq. (5.2). As evident from Fig. 5.7b and d, a very similar behavior is observed in the YIG(10 nm)/ Pt(7 nm) sample. The external magnetic field required to excite FMR is different in the two samples, due to their different magnetic anisotropies and saturation magnetizations [24]. It is important to note that a DC voltage VDC appearing upon FMR in an F/N hybrid structure can have different origins. On the one hand, VDC can arise due to spin pumping in combination with the ISHE in the normal metal, as just described. On the other hand, VDC may also stem from microwave rectification [20,23,90,93–95]. In the latter process, radio-frequency currents (induced in the sample by the radio-frequency radiation used to

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B

A H

x x

E

hRF

D

V C

C

H

H

EISHE

D

Js

DV SP

V

L

M Heff

w

m H

Figure 5.7 Electrically detected spin-pumping data, recorded in the experiment geometry sketched in Fig. 5.6(a). Panels (a) and (b) show the FMR signal at n ¼ 9.3 GHz, while panels (c) and (d) depict the simultaneously recorded DC voltage VDC of a Fe(10 nm)/ Pt(7 nm) bilayer and a YIG(10 nm)/Pt(7 nm) bilayer, respectively. A schematic of the experimental configuration is shown in panel (e).

drive the FMR) are rectified by the M-dependent bilayer resistance, which also periodically oscillates in FMR because of the AMR or the anomalous Hall effect of the ferromagnet. In cavity-based FMR setups, rectification voltages can be efficiently suppressed by positioning the sample in a region of the cavity in which the microwave electric field is minimal. Along the same line of thought, broadband FMR measurements require a careful sample design to rule out or at least minimize rectification contributions to the observed VDC [23,96]. Based on the purely symmetric line shape of VDC in Fig. 5.7, the VDC sign inversion with inversion of H, and the positioning of the sample in a node of the electric field in the microwave cavity, we attribute the VDC observed in both samples entirely to the combined action of the spin pumping and the ISHE. Taken together, Fig. 5.7 thus shows that electrically conductive as well as electrically insulating ferromagnets can pump a spin current into an adjacent normal metal, as also inferred from the changes in magnetization damping in F/N heterostructures made using both metallic and insulating ferromagnetic materials in Section 2. To quantify the interfacial spin current or the spin-mixing conductance from these ISHE VDC measurements, one must take into account that VDC is short-circuited both by the normal metal and the conductive ferromagnet

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Spin Pumping and Spin Currents in Magnetic Insulators

[23,24]. In other words, in an F/N bilayer sample with width w and length L, the total conductance G ¼ wLðsF tF þ sN tN Þ (calculated in a simple parallel resistor picture) impacts the magnitude of VDC. Here, sF and tF are the conductivity and the thickness of the ferromagnetic (F) layer, while sN and tN are the corresponding properties of the normal metal (N) layer. Furthermore, spin backflow effects parameterized by  as discussed in the context of Eq. (5.7) can be important for the magnitude of VDC, if the normal metal layer does not act as a perfect spin current sink. The magnitude of the DC voltage arising due to spin pumping in combination with the ISHE is given by [23,24] t N 2 RwnP sin Y , VSP ¼ eg"# aSH lSD tanh ð5:13Þ 2lSD where R ¼ 1/G is the bilayer resistance, Y is the precession cone angle of M around Heff (Fig. 5.3), and P is a factor to account for noncircular magnetization precession [97]. One can use Eq. (5.13) to extract the product g"# aSH lSD tanh2lt from electrically detected spin-pumping measurements. Using literature values for lSD and aSH, and given that R, P, and sin2 Y are known, this allows calculating the spin-mixing conductance [23,24]. It is straightforward to measure the sample resistance R, and the ellipticity parameter P can be calculated from the ferromagnet’s magnetic anisotropy [97] (0.5 P 1.5 in typical thin film samples [92]). The precession cone angle of the magnetization in FMR Yres ¼ 2hMW/(DHFWHM) can be determined from the full width at half maximum magnetic field line width DHFWHM of the FMR line and the microwave magnetic field hMW used to drive the resonant magnetization dynamics [98]. In cavity-based FMR, hMW typically is inferred from calibration experiments, while in broadband FMR, the magnitude of hMW can also be calculated from the microwave current distribution derived from electromagnetic modeling [99,100]. Alternatively, magnetotransport experiments in combination with FMR also enable a quantification of the precession cone angle, as described, for example, in [101]. Alternatively, it also is possible to infer the spin current density N

SD

JsSP ¼

DVSP 1 C L

ð5:14Þ

pumped across the F/N interface in FMR from electrically detected spinpumping experiments [23,24]. Due to the inclusion of the backflow

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correction factor , JSP s is not the net spin current density but the bare pumped spin current density (the spin current density flowing from the ferromagnet into the normal metal). C is the open-circuit spin Hall conversion efficiency given by [24]   2e t N rN C ¼ aSH lSD tanh : 2lSD tN ℏ

ð5:15Þ

Figure 5.8a shows the spin current densities according to Eq. (5.14), extracted from electrically detected spin-pumping measurements in a variety of conductive ferromagnet/Pt as well as YIG/Pt bilayers. JSP s hereby is plot2 1 ted against the parameter ESP ¼ 2hnP sin Y, which corresponds to the energy introduced into the magnetic subsystem (the ensemble of YIG magnetic moments) by the nonequilibrium magnetization dynamics. To evaluate Eq. (5.14), we have used the material parameters lSD ¼ 1.5 nm and aSH ¼ 0.11 for Pt. These parameters are in accordance to those extracted

B

10-6

10

(b)

YIG Ni Co Fe

10-7 10-8

10

g↑↓

YIG/Pt

-10

10

20

/ 2p

1019

Heusler Fe3O4

10-11

Py 18

10-29

10-28

10-27

ESP (J)

10-26

10-25

-2

g↑↓ ≈1 × 10 m YIG/Pt

-12

10-30

lPt = 1.5 nm aSH,Pt = 0.11

10-9

10

21

g↑↓ (m-2)

Spin current density Jssp (J/m²)

A

19

10

10

100

1000

R (Ω)

Figure 5.8 (a) The spin current induced by spin pumping across a F/N interface, JSP s , scales linearly with the nonequilibrium energy ESP in the magnetic subsystem (see text). The full symbols correspond to the conductive ferromagnet/Pt bilayers also discussed in Ref. [24], with the Heusler-based bilayers made using Co2FeSi or Co2FeAl. The Py data were taken from Refs. [23,102]. The open symbols depict data from YIG/Pt bilayers (c.f. Ref. [75] and Section 4). All experimental data were taken at room temperature. (b) The g"# extracted from the data in panel (a) is comparable for conductive ferromagnet/Pt and insulating ferromagnet/Pt bilayers. R is the resistance of the respective samples.

Spin Pumping and Spin Currents in Magnetic Insulators

145

via SMR measurements in the same set of YIG/Pt samples [75], and will be consistently used throughout this chapter. Following the arguments presented in Sections 1 and 2, the pumped DC spin current is given by [23,31] JsSP ¼

g"# 1 g"# hnP sin 2 Y ¼ ESP : 2p 2 2p

ð5:16Þ

The dashed line in Fig. 5.8a indicates the constant of proportionality g2p ¼ 110 m determined in our thin film YIG/Pt samples as discussed in more 2p detail in Section 4.3 As evident from Fig. 5.8b, g"# for the conductive ferromagnet/Pt bilayers generally is somewhat larger than g"# for the YIG-based magnetic insulator/Pt bilayers, as also suggested from theoretical calculations [39]. We would like to emphasize that g"# ¼ 9.4  1018 m2 inferred in one of our YIG/Pt samples from the magnetization damping analysis as described in Section 2 quantitatively matches with g"#  1  1019 m2 derived from spin pumping in combination with the ISHE. Furthermore, the spin-mixing conductance g"# 4.9 1018 m2 experimentally determined from magnetization damping experiments in YIG/Au samples in Refs. [37,38] and the spin-mixing conductance values determined in YIG/Pt samples in Ref. [103] also compare very well to the above values. Finally, we would like to note that the conductive ferromagnet/Pt samples in Fig. 5.8 are exactly the ones analyzed in Ref. [24]. More precisely, the electrically detected spin pumping raw data discussed in Ref. [24] and in Fig. 5.8 are identical. However, in Ref. [24], the data were analyzed disregarding the effect of spin backflow and using lSD ¼ 10 nm [104] and aSH ¼ 0.013 [23] from literature, while in Fig. 5.8, spin backflow is taken into account in the analysis and lSD ¼ 1.5 nm and aSH ¼ 0.11 are used for Pt. Nevertheless, very similar values for g"# were found in Ref. [24], owing to the small ( 15%) change in the product aSHlSD for the parameters used here and for those in Ref. [24] (with  ¼ 1). This demonstrates that the simultaneous extraction of aSH and lSD from electrically detected spinpumping experiments is not straightforward. The quantitative comparison of electrically detected spin pumping, spin Seebeck, and SMR data presented in the section 4.2, however, enables a more reliable approach for the extraction of these values, corroborating the values lSD ¼ 1.5 nm and aSH ¼ 0.11 for Pt quoted above. "#

19

3

2

2 Since g"# is in units of m2, and since JSP s is in units of J/m (angular momentum ℏ per time per area, viz., J s/s/m2), ESP must be an energy in units of J.

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4. SPIN CURRENTS AND THE SPIN-MIXING CONDUCTANCE CONCEPT Spin currents are not only important for spin pumping; rather, the generation and detection of spin currents are of key importance also for the spin Seebeck effect (SSE) [14,15,105,106] and the SMR [73–77] in ferromagnetic insulator/normal metal hybrid systems. Each of these effects has been studied extensively in its own respect. However, spin pumping, SSE, and SMR are all governed by one and the same interface parameter: the spin-mixing conductance g"#. The latter is a measure for the efficiency of the spin current Js generation in the presence of a nonequilibrium perturbation of the magnetic subsystem (the ensemble of magnetic moments in the ferromagnet). In the spirit of Ohm’s law for electrical currents, one can write a relation Js ¼ g2p E, with E being the nonequilibrium energy introduced into the magnetic subsystem. Thus, according to the spin-mixing conductance concept [107], the spin current generation efficiency should be identical for spin pumping, spin Seebeck, and SMR if probed in one and the same sample. In the following, we experimentally test this conjecture, using the set of thin film YIG/Pt samples described in more detail in Ref. [75]. In the case of spin pumping, the pumped DC spin current density JSP s was already introduced in Section 3 in the context of Eq. 5.14. As evident from Eq. (5.16), JsSP ¼ g2p ESP can be written in the above form. The SSE (see Chapter 3) describes the generation of a spin current [105] "#

"#

JsSSE ¼

g"# gℏ kB DT , 2p Ms Va

ð5:17Þ

in a F/N bilayer due to the presence of a temperature difference DT between the magnons in F and the electrons in N. Here [105],   2 4pD 3=2 Va ¼ 3zð5=2Þ kB T

ð5:18Þ

is the temperature-dependent magnetic coherence volume with the Riemann zeta function z and the spin wave stiffness D at temperature T. In analogy to spin pumping, the spin Seebeck spin current JSSE provides a s relaxation channel for the—in this case thermally—excited magnetization. Note that, as in the case of spin pumping, JSSE is the bare pumped spin s current with the net spin current flow being JSSE  with  from Eq. (5.7). s

147

Spin Pumping and Spin Currents in Magnetic Insulators

Interfacial spin currents in conjunction with the SHE furthermore lead to a novel magnetoresistance effect [74]. This so-called SMR is observed in ferromagnetic insulator/N bilayers that are exposed to an in-plane charge current Jc. The SMR is a direct consequence of the interconversion of charge and spin currents via the SHE. To show that JsSMR ¼ g2p ESMR , we briefly digress to introduce the SMR effect in the following, before coming back to the comparison of spin current magnitudes arising from the three effects mentioned earlier. "#

4.1. Spin Hall Magnetoresistance Due to the SMR, the resistance of a normal metal (N) layer in contact to a magnetic insulator depends on the magnetization orientation in the latter. The SMR is caused by the combined action of the SHE and the ISHE, together with a magnetization-orientation dependent absorption or reflection of a spin current at the magnetic insulator/N interface. The SMR effect was first observed in YIG/Pt hybrid structures [73,108], but further experiments have shown that the SMR also is present in YIG/Ta [77], magnetite/Pt, and nickel ferrite/Pt hybrids [75]. For the sake of simplicity, we here, however, limit the discussion to YIG as the magnetic insulator. To understand the SMR effect, it is instructive to consider the two different steady states that are achieved for the two qualitatively different orientations of the magnetization in YIG (the magnetic insulator) with respect to the charge current density Jc [74]. As illustrated in Fig. 5.9a, a constant Jc is applied to the N layer. Due to the SHE, this charge current generates a spin current density Js flowing toward the magnetic insulator/N interface (c.f. Eq. 5.1). This spin current has a spin polarization s oriented perpendicular to both Jc and Js. At the magnetic insulator/normal metal interface, we distinguish between two cases: Js is reflected at the interface if the magnetization M is oriented collinear to s (Fig. 5.9a). This leads to a zero net spin current flow, since the reflected spin current Js,refl compensates the spin current Js generated by the SHE. On the other hand, if M is oriented perpendicular to s (Fig. 5.9b), only a part of Js is reflected and a part JSMR is s absorbed by the magnetic insulator, since it now is possible to transfer angular momentum across the interface via spin transfer torque. Thus, the orientation of M in the magnetic insulator determines the magnitude of the spin current flow JSMR across the F/N interface into F, and the dissipation assos SMR ciated with Js results in a change of the electric resistance of the normal metal layer.

148

Mathias Weiler et al.

A

B

J

M||s

M⊥s

c

Pt

J

c

Pt

fl

fl

Js,re

Js,re YIG

YIG

Js M

z y

x

M

Js

Figure 5.9 Schematic illustration of the spin Hall magnetoresistance effect in N/magnetic insulator (Pt/YIG) hybrids. The orientation of the magnetization M in the magnetic insulator controls the amount of back-reflected spin current Js,back and the amount of spin current JSMR absorbed by the magnetic insulator. Due to the combined action of s SHE and ISHE, the resistance of the N layer depends on the orientation of M in the magnetic insulator. Panel (a) depicts the case for the spin current spin polarization s parallel to M and panel (b) corresponds to orthogonal s and M.

A quantitative model of the SMR based on diffusive spin transport has been put forward in Ref. [74]. Within this model, the spin-dependent electrochemical potential and the associated spin currents in N are analytically calculated with respect to the M-dependent boundary conditions at the magnetic insulator/N interface. Assuming that Jc is oriented along the x-direction and the surface normal is parallel to the z-direction (same coordinate system as in Ref. [74]), we can describe the longitudinal resistivity rlong as a function of the projections of the magnetization orientation m ¼ M/M onto the Cartesian coordinate system defined by x, y, and z as rlong ¼ r0 þ Drm2y ,

ð5:19Þ

with the m-independent resistivity contribution r0 and the m-dependent contribution Dr. The magnitude of the SMR is then given by Dr/r0. The SMR is qualitatively different from the AMR effect in conductive polycrystalline ferromagnets, since the latter obeys r0 þ Drm2x. Thus, AMR and SMR can be distinguished in magnetotransport experiments via an analysis of the dependence of the magnetoresistance on the (out-of-plane) magnetization orientations [73,75]. According to Ref. [74], the SMR can be expressed as
1 2 Dr a2SH 2lSD rN ðtN Þ e2 h1 g"# tanh2
¼ r0 1 þ 2lSD rN e2 h1 g"# coth lt N

SD

tN 2lSD


:

ð5:20Þ

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Spin Pumping and Spin Currents in Magnetic Insulators

The SMR effect thus scales with a2SH. In addition, the SMR effect will be sizeable only if the thickness tN of the N layer is comparable to its spin diffusion length lSD. Moreover, the spin-mixing conductance g"# and the resistivity rN of N also directly influence the SMR effect magnitude. To compare the SMR with spin pumping and spin Seebeck type of experiments, we need to determine the amount of spin current JSMR that s is flowing across the magnetic insulator/N interface. The maximum transfer of spin current occurs for M parallel to the x-(mx ¼ 1, my ¼ mz ¼ 0) or z-(mz ¼ 1, mx ¼ my ¼ 0) directions, while there is no spin current flowing across the interface for M parallel to the y-direction. Using Eq. (5.14) from Ref. [74] together with the spin accumulation at the interface in the absence of spin transfer m0s ¼ 2elSDrNaSHJc tanh[tN/(2lSD)] also introduced there, we obtain JsSMR ¼

g"# tN : 2elSD rN aSH Jc tanh 2p 2lSD

ð5:21Þ

Here,  again is the backflow parameter defined in Eq. (5.7). Due to to the net spin current continuity at the interface, we can also relate JSMR s spin current flow in N at the interface (z ¼ 0) by using Eq. (5.16) from Ref. [74] and substituting Eq. (5.20) into it (and neglecting the imaginary part of the spin-mixing conductance). This yields a second expression for JSMR : s JsSMR ¼ Jc

Dr ℏtN : r0 aSH elSD tanh2lt N

ð5:22Þ

SD

We will use Eq. (5.22) below to derive JSMR from conventional DC s magnetoresistance measurements.

4.2. Comparison of Spin Pumping, Spin Seebeck Effect, and Spin Hall Magnetoresistance Returning to the comparison of spin pumping, spin Seebeck, and SMR, it is straightforward to see that Eqs. (5.16), (5.17), and (5.21) all have the same form Js ¼ g2p E. Thus, according to theory, one should be able to experimentally observe one and the same scaling of the spin current magnitude Js with the appropriate energy E for spin pumping, SSE, and SMR. In all three cases, the nonequilibirum energy E can be deduced from experiment or simulation. For spin pumping, ESP ¼ 12hnP sin 2 Y is given mainly by the precession cone angle Y, and can be determined from "#

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experiment as detailed in Section 3 in the context of Eq. (5.13). For the SSE, the determination of ESSE ¼ gℏkBDT/(MsVa) requires the calculation of Va according to Eq. (5.18). Furthermore, one needs to quantify DT. In thin films, this is a substantial experimental challenge, but one may use finite element simulation to do so [109]. In the case of the SMR, ESMR ¼ 2elSD rPt aSH Jc tanh2lt  is given by a set of material parameters and the charge current Jc. We finally need to determine the respective Js in order to test the spinmixing conductance scaling conjecture. Exploiting the ISHE (see Section 1.1) to this end, one thus detects a spin-pumping voltage DVSP in FMR (see Section 3). A typical DVSP trace obtained in a YIG/Pt bilayer is shown in Fig. 5.10a. The associated spin current density is given by Eq. (5.14). In longitudinal spin Seebeck experiments, one establishes a thermal perturbation along the normal interface of a magnetic insulator/N bilayer by heating N, while the magnetic insulator is in contact with a thermal reservoir (or vice versa). Both, resistive [111] and laser heating [108] have been used to do so. The electrical Js detection is identical to that used in spin-pumping experiments, so that one can find the spin current from Pt

SD

B

h

RF

He

E

Pt YIG

M

R

VSM

ati

VSS

VSP Pt YIG

C ng

J

c

M

M

Pt YIG

t Ho d l Co

w

w

L

L

L

M w

A

0° 500 nV

ΔVSSE

ΔVSP

Δr

0.1 nΩm

10 μV

90° 220

240

260

-50

0 m0H (mT)

50

-30 -15

0

15

30

r0 = 408.98 nΩm

Figure 5.10 (a) Measurement geometry and data obtained in a typical spin-pumping experiment. (b) Heating the Pt side of the YIG/Pt bilayer sample results in a DC voltage due to the spin Seebeck effect. (c) The spin Hall magnetoresistance causes a change Dr in the longitudinal resistivity when rotating M from M k Jc (0 ) to M ? Jc (90 ). All experimental data were acquired using the same YIG(20 nm)/Pt(7 nm) bilayer sample. Data taken from Ref. [110].

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JsSSE ¼

DVSSE 1 : C L

ð5:23Þ

Here, DVSSE is the DC voltage recorded in the presence of a thermal perturbation DT. Typical experimental data is displayed in Fig. 5.10b. To detect the SMR experimentally, one applies an in-plane charge current Jc to a magnetic insulator/N bilayer and records the change in bilayer resistivity Dr/r0 when rotating M‖Jc (r0) to M ? Jc (r0 þ Dr). From the measurement data (c.f. Fig. 5.10c), one can then calculate the spin current density via Eq. (5.22). The spin currents extracted from experimental spin pumping, spin Seebeck, and SMR data of a series of YIG/Pt bilayer samples4 with the help of Eqs. (5.14), (5.23), and (5.22) are plotted as a function of the respective nonequilibrium energies ESP, ESSE, and ESMR in Fig. 5.11. As expected from the above g"# conjecture, the magnitude Js of the spin current density scales linearly with E. This scaling holds for all three effects and over more than five orders of magnitude, providing strong evidence that spin current flow across the F/N interface indeed is the physical phenomenon responsible for the spin pumping, the spin Seebeck, and the SMR effects. In other words, spurious effects, such as a magnetic proximity polarization of the Pt close to the YIG interface [48], can be ruled out. This, in particular, also implies that the magnetoresistance observed in YIG/Pt hybrids indeed is due to spin current physics. Figure 5.11 shows data obtained from over 20 different YIG/Pt samples, with the thickness tN of the Pt layer ranging between 2 and 18 nm. The g"# values extracted from these data do not vary systematically with tN, as evident from Fig. 5.12. This implies that the tN dependence of Js predicted by Eqs. (5.14), (5.23), and (5.22) is consistent with experiment. Furthermore, the material parameters (lSD ¼ 1.5 nm, aSH ¼ 0.11, g"# ¼ 1  1019 m2) extracted from Fig. 5.11 are within the range of those found for Pt using other methods and other samples (c.f. Sections 2 and 3). Most importantly, one finds that this single set of parameters is sufficient to quantitatively model spin pumping, spin Seebeck, and SMR. Thus, the proposed scaling relation Js ¼ g2p E indeed does hold for all three effects. Moreover, since the functional dependence of the SMR on aSH is different from the dependence of the SSE and spin pumping on aSH, the "#

4

The samples for this study have been fabricated by pulsed laser deposition of the YIG layer, followed by an in situ electron beam evaporation of the Pt. Both gadolinium gallium garnet and yttrium aluminum garnet single crystal substrates were used. See Ref. [75] for more details.

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Spin current density Js (J/m2)

10-4 10-5

g

lPt = 1.5 nm

10-6

aSH,Pt = 0.11 g

10-7

↑↓

↑↓

/2p

= 1 × 1019m-2

10-8 10-9 Spin pumping

10-10

Spin Seebeck

10-11

SMR

10-12 10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

Energy E (J)

Figure 5.11 The experimentally observed spin current density Js scales linearly with the nonequilibrium energy E over more than five orders of magnitude. This scaling is observed while invariably using the indicated set of parameters (lSD ¼ 1.5 nm, aSH ¼ 0.11, g"# ¼ 1  1019 m2) for all three effects.

1019

g

↑↓

(m-2)

1020

1018 0

5

10

15

20

tN (nm)

Figure 5.12 The spin-mixing conductance does not exhibit a systematic dependence on the normal metal thickness tN.

scaling analysis in Fig. 5.11 allows for a reliable, separate extraction of both aSH and lSD. Furthermore, the spin backflow factor  is a function of g"# and enters both the expressions for ESMR and JSMR for the SMR, while only JSP s s SSE and Js depend on  in the case of spin pumping and the SSE. Thus, a single set of values for aSH, lSD, and g"# can be extracted from the quantitative comparison of the three effects, with much higher confidence than from an analysis of any of these effects alone. Finally, the data presented above show no indication for systematic spurious effects, such as a static magnetic proximity polarization in Pt close to the YIG/Pt interface [47–49]. In conclusion, the spin-mixing conductance concept enables a quantitative and consistent modeling of the experimental data acquired in electrically

Spin Pumping and Spin Currents in Magnetic Insulators

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detected spin pumping, SMR and SSE experiments in thin film YIG/Pt bilayer samples [110]. This corroborates the notion that the spin-mixing conductance concept yields a robust and coherent picture of the physics of spin current transport across F/N interfaces—in F/N bilayers based on both metallic as well as insulating ferromagnets. On this basis, more sophisticated spin transport schemes and devices can be envisaged in the future.

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