Controlled magnon spin transport in insulating magnets

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Controlled magnon spin transport in insulating magnets Liu, Jing

DOI:

10.33612/diss.97448775

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2019

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Liu, J. (2019). Controlled magnon spin transport in insulating magnets: from linear to nonlinear regimes.

University of Groningen. https://doi.org/10.33612/diss.97448775

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Controlled magnon spin transport in insulating magnets

From linear to nonlinear regimes

Jing Liu

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ISBN: 978-94-034-1991-6

ISBN: 978-94-034-1990-9 (electronic version)

The work described in this thesis was performed in the research group Physics of Nanode- vices of the Zernike Institute for Advanced Materials at the University of Groningen, the Netherlands. This work was supported by NanoLab NL and the Zernike Institute for Ad- vanced Materials. This thesis is part of research program Magnon Spintronics financed by the Netherlands Organization for Scientific Research (NWO).

Printed by: Gildeprint, Enschede

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Controlled magnon spin transport in insulating magnets

From linear to nonlinear regimes

PhD thesis

to obtain the degree of PhD at the University of Groningen

on the authority of the Rector Magnificus Prof. E. Sterken

and in accordance with the decision by the College of Deans.

This thesis will be defended in public on Monday 30 September 2019 at 9:00 hours

by

Jing Liu

born on 16 May 1991 in Hunan, China

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Co-supervisor Prof. G. E. W. Bauer

Assessment committee Prof. P. J. Kelly

Prof. T. Banerjee Prof. W. Han

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to Lihua & Baoju, my parents

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Contents

1 Introduction 1

1.1 Magnetism . . . . 1

1.2 Magnetic excitation: Magnons or spin waves . . . . 2

1.3 Magnon spintronics . . . . 3

1.4 A gem: Yttrium iron garnet (YIG) . . . . 3

1.5 Motivation and thesis outline . . . . 4

Bibliography . . . . 5

2 Theoretical background 7 2.1 Magnons and spin waves . . . . 7

2.2 Ferrimagnetic insulator: YIG . . . . 9

2.3 Magnon spectra . . . 11

2.4 Magnon injection and detection . . . 21

2.4.1 Spin injection and detection . . . 21

2.4.2 (Inverse) spin Hall effect . . . 24

2.4.3 Electrical method . . . 27

2.4.4 Microwave method . . . 30

2.5 Magnon transport theory . . . 33

2.5.1 Nonlocal magnon transport set-up . . . 33

2.5.2 Magnon chemical potential and spin diffusion equation . . . 33

2.5.3 Energy-dependent magnon transport . . . 35

2.5.4 From the linear to nonlinear response regime . . . 35

2.5.5 Summary . . . 36

Bibliography . . . 37 vii

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3 Experimental set-up and methods 41

3.1 Device Fabrication . . . 41

3.1.1 Pt-YIG nonlocal device . . . 42

3.1.2 Electron beam lithography (EBL) . . . 43

3.2 Electrical generation and detection of magnons . . . 46

3.2.1 Set-up . . . 46

3.2.2 Lockin technique for nonlocal measurement . . . 48

3.2.3 Typical results . . . 49

3.3 Micowave excitation of magnons . . . 51

3.3.1 Setup . . . 52

3.3.2 Microwave reflection measurement . . . 54

3.3.3 Spin pumping measurement . . . 56

Bibliography . . . 57

4 Magnon planar Hall effect and anisotropic magnetoresistance in a magnetic insulator 59 4.1 Introduction . . . 59

4.2 Experimental details . . . 61

4.2.1 Devices . . . 61

4.2.2 Measurement techniques . . . 62

4.3 Results and discussion . . . 64

4.4 Conclusions . . . 67

4.5 Supplementary Material . . . 67

4.5.1 Origin of the angle shift in the MPHE and MAMR measurement 67 4.5.2 Derivation for the magnitude of MPHE and MAMR signals . . . 69

4.5.3 Double detector MPHE measurements . . . 73

4.5.4 In-plane magnetocrystalline anisotropy of YIG (111) . . . 73

4.5.5 Out-of-plane misalignment of the sample plane with respect to the applied magnetic field . . . 77

4.5.6 Sign and magnitude of the MPHE and MAMR . . . 82

4.5.7 Reciprocity and linearity of the MPHE and MAMR . . . 85

5 Magnon transport in YIG with Ta and Pt spin injection and detection elec- trodes 89 5.1 Introduction . . . 89

5.2.1 Devices . . . 91

5.2.2 Measurement techniques . . . 92

5.3 Results and discussion . . . 92 viii

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Contents

5.4 Conclusions . . . 100

6 Microwave control of thermal magnon spin transport 103 6.1 Introduction . . . 103

6.2 Experimental setup . . . 105

6.3 Results . . . 106

6.3.1 Nonlocal signals under an rf field . . . 106

6.3.2 Rf-power dependency . . . 107

6.3.3 Rf-power reflection and spin pumping . . . 109

6.4 Discussion . . . 110

6.5 Conclusion . . . 113

6.6 Supplementary Material . . . 114

6.6.1 Sample preparation . . . 114

6.6.2 Experimental setup . . . 114

6.6.3 Local measurement . . . 114

6.6.4 Comparison between the rf-power dependent first and second harmonic signals . . . 117

6.6.5 Rf-power reflection measurement . . . 117

6.6.6 Spin-pumping measurement . . . 118

6.6.7 Excitation current dependency . . . 119

6.6.8 Onsager reciprocity . . . 119

6.6.9 Rf power calibration . . . 120

6.6.10 Temperature effect due to the rf power . . . 121

6.6.11 Rf field strength and precession cone angle . . . 122

6.6.12 Comparison of 210-nm- and 100-nm- thick YIG results . . . 123

6.6.13 Extracting the nonlocal resistances . . . 125

7 A ”magnon transistor” on 10 nm thick YIG film 131 7.1 Introduction . . . 131

7.3 Results . . . 134

7.4 Discussion . . . 134

7.4.1 Angle dependent analysis . . . 134

7.4.2 Polynomial analysis of nonlocal signals at specific angles . . . . 138

7.4.3 Magnitude of the nonlocal signals . . . 141

7.5 Summary and outlook . . . 142

Bibliography . . . 144 ix

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Summary 145

Samenvatting 150

Acknowledgements 156

Publications 169

Curriculum Vitae 171

x

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1

Chapter 1

Introduction

1.1 Magnetism

M

agnetism is a discovery of the ancient times since people found that lodestones, naturally magnetized stones, attract iron. This magic force without touching has puzzled many philosophers and scientists, transcending location and culture:

Since the time of classical Greece, people have been using the term of a magnet. In ancient Indian medical texts, it is suggested to use magnets to remove arrows from people’s bodies. In ancient China, a lodestone was made into a spoon shape, where the handle of the spoon was found to always point to the direction of the south, thereby discovered the compass. This curiosity ended up having profound conse- quences for human discoveries. Magnetism was employed to navigate, to explore new lands, and to figure out the stars and the relationship between humans and the universe.

In modern times, after the discovery of matter’s microscopic structure people started to wonder: What makes a magnet magnetic? This ancient topic became magnif- icent again. After hundreds of years of exploration, we now know that the building blocks of magnetic materials are atoms or molecules with nonzero magnetic moments due almost entirely to the orbital motion and spin of electrons. When the magnetic atoms get together and form a solid, the force which calls the magnetic moments to order is the exchange interaction^†. The alignment of the magnetic moments is disrupted by thermal fluctuations so that above a certain temperature, called the Curie temperature Θc, the net magnetic moment is zero. However, below Θc, the magnetic moments are aligned in the absence of an applied field so that the material is spontaneously magnetized. This emergent phenomenon is caused by spontaneous symmetry breaking: The chaotic sea of magnetic moments containing all the rotational symmetries is suddenly left with only one rotational symmetry axis by freez- ing all the moments in one single direction just like other phase transitions such as from water to ice.

Based on their magnetic characteristics, materials can be roughly classified into

†The importance of exchange interaction was first pointed out by Ya. I. Frenkel, Ya. G Dorfman and W. Heisenberg in 1928 [1]

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five categories depending on the properties of the building blocks and binding forces [2]: When the constituent atoms or molecules do not possess permanent magnetic moments, i.e. unpaired electrons, the material is diamagnetic. By contrast, the materials composed of magnetic atoms but without long-range ordering as a result of the exchange interaction show paramagnetism. If however, the exchange energy is able to align the neighboring magnetic moments parallel to each other, this gives rise to ferromagnetic long-range ordering. Alternatively, the neighbouring moments can be aligned antiparallel to each other due to a different type of exchange interaction.

Depending on the numbers of the parallel and antiparallel moments, the materials are antiferromagnetic or ferrimagnetic when the numbers are equal or unequal, respectively. Among different types of exchange interaction, direct exchange interaction makes most ferromagnets conducting and superexchange interaction makes most of the ferri- or anti-ferromagnets insulating.

In reality, there are more complicated magnetic order arrangements. Also, for a given magnetic specimen, the energy is minimized by dividing it into many domains, within which magnetic moments are aligned. However, the neighbouring domains have magnetic moments that are not aligned. Nevertheless, people have figured out quite a bit about the arrangement of the magnetic moments in different materials, which is not the end of the story. For a full understanding, one needs to know how the magnetic moments move, i.e. the dynamics of magnetism.

1.2 Magnetic excitation: Magnons or spin waves

”Magnon” is a concept to describe the thermodynamics of magnetism, proposed by Bloch [3] in 1930 when he was in Utrecht, the Netherlands. They are bosons, whose statistical properties suggest that their number is proportional to T^3/2[4, 5].

This temperature dependence matches the T^3/2-dependent reduction of spontaneous magnetization at low temperatures and that of the specific heat, which shows that magnons carry both spin and heat. Deviations from that are attributed to the phonon contribution. For more than half a century, the study of magnons has always been fo- cused on the long wavelength GHz magnons, which have dominating wave (rather than particle) properties [6]. Thanks to the rise of nanotechnology and crystal fabrication techniques, it has become possible to study magnons at a different scale:

namely those found in ultra-thin films and nanodevices [7–12]. Fundamentally, this provides a better chance to explore the quasiparticle properties of magnons. Practi- cally, this makes it possible to scale down the magnon-based device.

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1.3. Magnon spintronics 3

1.3 Magnon spintronics

Unlike the spin transport in metals which is mainly contributed by the mobile electrons [13], in insulators the spins are mainly carried by magnons. Therefore, the magnon emerges as another player on the playground of spintronics, leading to magnon spintronics [6], which is appealing for information technology for several reasons:

First, the motion of the information carrier magnons is not accompanied by any Joule heating like in the field of electronics or conventional electron spintronics. Therefore, magnon-based data processing can be an antidote to the thermodynamic bottleneck of Moore’s law [14–16] due to overheating from Ohmic dissipation produced by electron motion in conducting circuits. Secondly, magnons possess different statistics than fermions, i.e. Bose-Einstein statistics. One of the most profound results is that the whole band of magnons contribute to the transport. This provides a possibility to operate the device at different frequencies. Thirdly, when the number of magnons is large enough, the Bose-Einstein condensation can be realized even at room temperature [17]. The resulting state may lead to the holy grail of dissipationless information transport for next-generation logic devices. More attention has been drawn to the spin transport in insulating materials since the experimental demonstration of the spin Seebeck effect [18] in 2008. This opened up a field called spin caloritronics [19, 20], where the spin degree of freedom is connected with heat. In the case of magnon spins, abundant physics are explored and even efficient use of waste heat can be applied to manipulate magnon transport. To facilitate this transport in magnon spintronics, the desired materials are those with low damping, which generally means long magnon life time.

1.4 A gem: Yttrium iron garnet (YIG)

Yttrium iron garnet (YIG) is a ferrimagnetic oxide, which was first synthesized in 1956^† by Bertaut and Forrat [21]. Synthesizing garnets was a trendy thing at that time, because their high refractive index makes them good candidates for economi- cal jewelry. Surprisingly, this synthetic gemstone shows magnetic properties, especially its lowest magnetic damping properties, a record which still holds after more than half a century. Because of this, YIG has become a real gem in the field of magnetism. Kittel made the analogy between YIG in magnetism and fruit flies in biol- ogy in the 1960s. In 1993, one of the most comprehensive reviews dedicated to YIG [22, 23] compared the role of YIG to that of germanium in semiconductor physics, water in hydrodynamics and quartz in crystal acoustics. With the development of material synthesis [24] and nanotechnology, people still keep being surprised by the

†Coincidentally, YIG shares the year of its birth with my parents.

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knowledge and applications we can obtain from this gem.

1.5 Motivation and thesis outline

The aim of this thesis is to study the transport of magnons. For this purpose, YIG is used as the platform or the transport channel. All the experiments have been con- ducted in atmosphere at room temperature. The effects observed in this thesis pro- vide ways to steer the flow of information carried by the magnon spins in magnonic devices. This thesis is made up of the following chapters, of which a brief overview is given below:

• Chapter 2 gives an overview of the theoretical background, which is necessary for understanding the effects studied in this thesis.

• Chapter 3 systematically explains the fabrication process and measurement methods used for the experiments presented in this thesis.

• Chapter 4 provides experimental evidence for the magnon planar Hall effect and anisotropic magnetoresistance. The relative difference between magnon current conductivities parallel (σ^m_∥) and perpendicular (σ⊥^m) to the magnetization is found to be approximately 5%.

• Chapter 5 uses different heavy-metal paramagnetic electrodes to study the magnon spin transport in a nonlocal experiment: Pt and Ta on YIG. For both electrodes, a similar magnon relaxation length of ∼ 10 µm is extracted. However, since Pt and Ta have opposite sign of the spin Hall angle and different properties at the YIG interface, changes in nonlocal transport are observed.

• Chapter 6 demonstrates that an rf microwave field strongly influences the transport of incoherent thermal magnons. Transport can be suppressed by 95% or experience an enhancement as large as 800%. This study shows the interplay between coherent and incoherent spin dynamics.

• Chapter 7 documents the properties of 3-terminal magnon transistors on ultra- thin YIG.

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1

Bibliography 5

Bibliography

[1] L. D. Landau, J. S. Bell, M. J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of continuous media, vol. 8, Elsevier, 2013.

[2] S. Blundell, Magnetism in condensed matter, Oxford University Press, 2001.

[3] F. Bloch, “Zur Theorie des Ferromagnetismus,” Zeitschrift f ¨ur Physik 61(3-4), pp. 206–219, 1930.

[4] C. Kittel and P. McEuen, Introduction to solid state physics, John Wiley & Sons, Inc., 8th ed., 2005.

[5] D. D. Stancil and A. Prabhakar, Spin waves, Springer, 2009.

[6] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, “Magnon spintronics,” Nature Physics 11(6), p. 453, 2015.

[7] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, “Long-distance transport of magnon spin information in a magnetic insulator at room temperature,” Nature Physics 11(12), pp. 1022–1026, 2015.

[8] S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn, M. Althammer, R. Gross, and H. Huebl, “Non-local magnetoresistance in YIG/Pt nanostructures,” Applied Physics Letters 107(17), p. 172405, 2015.

[9] J. Li, Y. Xu, M. Aldosary, C. Tang, Z. Lin, S. Zhang, R. Lake, and J. Shi, “Observation of magnon- mediated current drag in Pt/yttrium iron garnet/Pt (Ta) trilayers,” Nature Communications 7, p. 10858, 2016.

[10] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and M. Kl¨aui, “Tunable long-distance spin transport in a crystalline antiferromagnetic iron oxide,”

Nature 561(7722), p. 222, 2018.

[11] K. S. Das, J. Liu, B. J. van Wees, and I. J. Vera-Marun, “Efficient injection and detection of out-of-plane spins via the anomalous spin Hall effect in permalloy nanowires,” Nano Letters 18(9), pp. 5633–5639, 2018.

[12] H. Liu, C. Zhang, H. Malissa, M. Groesbeck, M. Kavand, R. McLaughlin, S. Jamali, J. Hao, D. Sun, R. A. Davidson, et al., “Organic-based magnon spintronics,” Nature Materials 17(4), p. 308, 2018.

[13] I. ˇZuti´c, J. Fabian, and S. D. Sarma, “Spintronics: Fundamentals and applications,” Reviews of Modern Physics 76(2), p. 323, 2004.

[14] G. E. Moore et al., “Cramming more components onto integrated circuits,” 1965.

[15] G. E. Moore et al., “Progress in digital integrated electronics,” in Electron Devices Meeting, 21, pp. 11–

13, 1975.

[16] M. M. Waldrop, “The chips are down for Moores law,” Nature News 530(7589), p. 144, 2016.

[17] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, “Bose–einstein condensation of quasi-equilibrium magnons at room temperature under pumping,” Nature 443(7110), p. 430, 2006.

[18] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, “Obser- vation of the spin Seebeck effect,” Nature 455(7214), p. 778, 2008.

[19] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, “Spin caloritronics,” Nature Materials 11(5), p. 391, 2012.

[20] S. R. Boona, R. C. Myers, and J. P. Heremans, “Spin caloritronics,” Energy & Environmental Sci- ence 7(3), pp. 885–910, 2014.

[21] F. Bertaut and F. Forrat, “Structure des ferrites ferrimagnetiques des terres rares,” Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences 242(3), pp. 382–384, 1956.

[22] V. Cherepanov, I. Kolokolov, and V. L’vov, “The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet,” Physics Reports 229(3), pp. 81–144, 1993.

[23] A. A. Serga, A. V. Chumak, and B. Hillebrands, “YIG magnonics,” Journal of Physics D: Applied Physics 43(26), p. 264002, 2010.

[24] C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Br ¨uckner, and J. Dellith, “Sub-micrometer yt-

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trium iron garnet LPE films with low ferromagnetic resonance losses,” Journal of Physics D: Applied Physics 50(20), p. 204005, 2017.

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2

Chapter 2

Theoretical background

Abstract

The following Chapter gives an overview of the basic physical concepts needed to under- stand the work presented in this thesis.

2.1 Magnons and spin waves

M

agnons or spin waves are the elementary excitations of magnetic order [1, 2].

They are quasiparticle representations of quantized spin waves: Particle and wave characters of these magnetic excitations manifest themselves on small and large scales, respectively. Bloch proposed the idea of magnons in the 1930s to find a consistent theory for the thermodynamic behavior of magnets [3].

The concept of magnons can be conveniently explained by the toy model shown

Ground state

T = 0 K a

M0

First excited state

T ≠ 0 K

(energetically unfavorable)

First excited state c

T ≠ 0 K

(energetically favorable)

b 2μ _B

M1

Figure 2.1: aMagnetic ground state at zero kelvin with a total magnetization of M0. b First excited magnetic state with collectively excited magnetic moments. This is an energetically favorable configuration, which gives rise to a total magnetization of M1 (M0− M1 = 2µB).

c”Naive” first excited magnetic state with one spin flipping its sign from the ground state.

This configuration also gives rise to a total magnetization of M1, but is energetically highly unfavorable.

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in Fig. 2.1: In the ground state of an isotropic single-domain ferromagnet, all magnetic moments statically align parallel to each other at a temperature of zero kelvin, which gives rise to a net static magnetization of M0 as shown in Fig. 2.1a. With increasing temperature, all magnetic moments are collectively excited by thermal fluctuations: They precess around their ground states, which results in a reduced static magnetization of M1as shown in Figs. 2.1b. This excitation state can be under- stood to arise as follows: Owing to the quantum mechanical nature of magnetism, the difference between the magnetization of the ground and first excited state, i.e.

∣M0−M1∣, equals to two Bohr magnetons, which corresponds to a change of angular momentum of ̵h. This means that with increasing temperature the reduction of the magnetization is quantized. A naive way to realize this change in magnetization is to flip one magnetic moment from the ground state as depicted in Fig. 2.1c. How- ever, this is energetically highly unfavorable because the strong exchange energy^† between neighboring spins prefers to align them parallel to each other in a ferro- magnet. By contrast, the configuration in Fig. 2.1b is energetically favorable, because neighboring spins are precessing with only a small phase shift with respect to their neighbors. This results in a propagating spin wave in a magnet. In the quasiparticle language, one can say that a magnon is excited in the magnetic system.

Each magnon carries the same amount of angular momentum ∼ ̵h, which corresponds to a reduction of the net magnetization of ∼ 2µB. However, every magnon can have different energy

E = ̵hω(k), (2.1)

where ̵h is the reduced Planck constant, ω(k) is the angular precession frequency of each individual spin in a spin wave. The precession frequency ω depends on the wavevector k based on the magnon dispersion relation ω(k) (see section 2.3), from which the magnon group velocity, effective magnon mass and magnon density of states g(E) can be solved.

As spin-1 quasiparticles, multiple magnons can occupy the same energy level, namely without following the Pauli exclusion principle. Thus, magnons are bosons, which obey Bose-Einstein statistics. If one views the absolute zero temperature case in Figs. 2.1a as a ”vacuum state” of the magnet, the excited state can be seen as ”a gas of magnons”. Unlike a defined system with bosonic atoms such as⁴He, the number of magnons (quasiparticles) in a magnet is not conserved. With rising the temperature, the overall population of magnons increases and the distribution obeys the Bose-Einstein statistics

f (E) = 1

e^β(E−µ)−1, (2.2)

†If one compares the ordering effect of exchange interaction with the effect of a magnetic field at room temperature, the power of the exchange interaction is comparable with a field strength as enormous as tens of millions of Oersteds or thousands of Teslas.

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2.2. Ferrimagnetic insulator: YIG 9

where f (E) is the distribution function, or the probability that a magnon has an energy of E, β = 1/kBT is the inverse of the product of Boltzmann constant and temperature and µ is the chemical potential, or the energy cost of adding a particle to the system. The magnon density N is given by

N =∫

∞

0 n(E)dE (2.3)

= ∫

∞ 0

g(E)f (E)dE, (2.4)

where n(E) is the number of magnons per unit volume with energy E. g(E) is the density of states of magnons, or the number of energy states per unit volume with energy E, which relates the distribution function to the particle density. In a magnon system at thermal equilibrium, µ equals zero and Eq. 2.2 turns into the Planck distribution function:

f (E) = 1

e^βE−1. (2.5)

2.2 Ferrimagnetic insulator: YIG

Yttrium iron garnet (YIG) is magnetically ordered and electrically insulating; therefore, it is called a magnetic insulator^†. Due to these properties, one can study magnetic excitations, i.e. magnons or spin waves, in YIG without having to consider the influence of mobile electrons such as in the case of conventional spin current in ferromagnetic metals [4].

Crystal structure and magnetic ordering

YIG is a ferrimagnetic oxide, whose stoichiometric formula is Y3Fe5O12. Its crys- tal structure is body-centered cubic (bcc), and rather complex as shown in Fig. 2.2a.

One primitive cell consists of 80 atoms, of which there are 20 distinct magnetic ones, i.e. Fe³⁺. There are three sublattices: 8 iron ions, 12 iron ions and 12 yttrium ions occupy octahedral (a), tetrahedral (d) and dodecahedral (c) sites, respectively, i.e.

open spaces in the O²⁻scaffolding, as shown in Fig. 2.2b. The long-range magnetic ordering originates from the antiferromagnetic coupling, i.e. super-exchange interaction, between two neighboring (a)- and (d)-type Fe³⁺ions via an oxygen ion. Since the number of these two types of Fe³⁺ions is different, the resulting magnetization is non-zero, which makes YIG a ferrimagnet with a magnetic moment of around 20µB††

†To clarify, historically Faraday cages were also called magnetic insulators, since they block magnetic fields.

††5µB× (12 − 8): Magnetic moments (per primitive cell) associated with Fe³⁺times the number difference between the majority (d)-type and minority (a)-type Fe³⁺.

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^{(d) Fe} ^{(a) Fe}

(c) Y O

3+ 3+

3+ 2-

a b

Figure 2.2: Crystal structure of YIG. a A bcc unit cell of YIG adapted from Ref [4]. Only 1/8 of the unit cell is filled with ions for convenience of visualization; the body diagonal is sketched as a dashed line indicating the orientation. The remaining 7/8 parts of the unit cell have the same composition but different orientations, as indicated by the dashed lines. Note that a bcc unit cell contains two lattice points (8 times of the sketched part in a), whereas the primitive cell has only one lattice point (4 times of the sketched part in a). b A zoomed-in view of different ions belonging to the three sublattices, depending on their O²⁻environment.

At a tetrahedral (d) site, there is an Fe³⁺ion, which has the majority spin. By contrast, the antiferromagnetically coupled minority spin is carried by an Fe³⁺located at an octahedral (a) site. Y³⁺is located at a dodecahedral (c) site.

in one primitive cell. The bcc lattice constant of YIG is around 12.38 ˚A, which is larger than the separation between the nearest interacting magnetic ions, 3.46 ˚A. The for- mer roughly sets the Brillouin zone boundary (more precisely, it is determined by the Wigner-Seitz cell), whereas the latter is the lower bound for spin-wave wavelength.

Weak spin-orbital coupling

The ground state of Fe³⁺ions is given by S = 5/2, L = 0 (⁶S_5/2^†). The half-filled d-shell (3d⁵) makes Fe³⁺ions have no net orbital angular momentum in their ground state.

Thus, the magnetic properties of YIG are due entirely to spin and the gyromagnetic ratio ∣γ/2π∣ = ∣γS/2π∣ = 28GHz/T (2.8 MHz/G) [2], which is why the orbital angular momentum is said to be quenched. In addition, YIG has a very symmetric cubic structure, which makes the scenario of orbital quenching apply just as well as in the

†The ground states of atoms and ions are often indicated with the notation^2S+1XJ, where 2S + 1 is the number of states with a given S (called multiplicity) and X is a letter corresponding to the value of L according to the convention: L = 0, 1, 2, 3... corresponds to S, P, D, F...

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2.3. Magnon spectra 11

case of atomic ions. This is the reason why YIG ordinarily has small or negligible spin-orbit coupling.

A gem for magnon spintronics

YIG is such an important material in the study and applications of magnetism for the following reasons: First, YIG has the lowest magnetic damping among all magnetic materials, corresponding to the narrowest ferromagnetic resonance linewidth among those observed. This means that it has the longest magnon or spin wave life time, which is ideal for magnon transport studies. Section 2.4.1 discusses the concept of damping in detail. Second, it has a high magnetic ordering temperature (Curie temperature Tc) of around 560 K. Even though it is a ferrimagnet, its thermodynamic behavior resembles that of a ferromagnet, especially in the low temperature regime (T < 260 K) [4]. High Tcmakes it convenient to study and utilize its magnetic properties at room temperature. Third, even though YIG is a man-made material with a complex crystal structure, the matching of the Fe³⁺ions and the space in between the O²⁻scaffold is extraordinarily good. This means that there are very few distortions so that the acoustic damping of YIG is even lower than that of quartz [4].

Therefore, the magnetic oxide YIG has been widely used in microwave devices [4, 5].

2.3 Magnon spectra

In order to study magnons, especially their transport properties in solids, the magnon spectrum, i.e. the dispersion relation or the magnon frequencies as a function of the wavevector, ω(k), needs to be considered [2]. From the dispersion relation, abundant information about magnons can be derived, such as their density of states (DOS), group velocities and effective masses. By combining the DOS with a distribution function of magnons (cf. Eq. 2.2), one can construct a general transport model based on a spin diffusion equation, from which the main transport properties can be obtained. Therefore, the following section discusses magnon spectra. Special attention is paid to the lowest lying dipolar-exchange magnon dispersion relation for YIG film with the thicknesses used in this thesis.

Exchange interaction

The exchange interaction, which is quantum mechanical in nature, governs the long- range magnetic ordering. It describes the interaction between two electrons which carry spin angular momentum S. The exchange energy between neighboring spins

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λ=2π/k -π/a 0 π/a

Wavevector k Ferromagnetic coupling: J > 0

ħω

~k2 4JS(1-

co s k

a)

π/a

-π/a 0

Wavevector k

ħω

λ=2π/k

Antiferromagnetic coupling: J < 0

~k -4JS |sin k a|

8JS

-4JS

a b

c d

Figure 2.3: One-dimensional magnetic chains and corresponding magnon dispersion relation with ferromagnetic (upper panel) and antiferromagnetic (lower panel) coupling .

in a lattice can be expressed following the Heisenberg model:

H_exchange= −J ∑

n

(Sn⋅Sn+1), (2.6)

where J is an exchange coefficient which quantifies the strength of the exchange interaction. This interaction is mainly due to the contribution between nearest neigh- bor spins at sites n and n+1. The spins are associated with magnetic moments m = γS.

The sign of J tells the type of coupling: J > 0 corresponds to ferromagnetic coupling where mn and mn+1tend to align parallel to each other, while J < 0 is the case for antiferromagnetic coupling where mn and mn+1prefer to align antiparallel to each other. Even though exchange coupling is a short-range magnetic effect, it gives rise to long-range magnetic ordering.

Building upon this model, a dispersion relation can be constructed for a one- dimensional ferromagnetic chain (J > 0) as shown in Fig. 2.3a with magnetic mo- ments pointing along the z-axis in the ground state [2]. The excitation spins at each site n are described as an approximate ground state (S_n^z) plus small excitations (Sn^y, S^z_n), i.e. magnons which do not interact with each other. This is the result of a

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semiclassical ansatz, i.e. an equation of motion for spins S = (S^x_n, S_n^y, S_n^z), in which

S^x_n=ue^i(kna−ωt) (2.7)

S_n^y=ve^i(kna−ωt) (2.8)

S_n^z≈S, (2.9)

where a is the separation between two spins, k is the magnon wavevector, u and v are small constants compared to ∣S∣. The resulting dispersion relation reads

̵hω = 4J S(1 − cos ak), (2.10)

so that at the edges of Brillouin zone (k = ±π/a) the corresponding magnon fre- quency is 8JS (cf. Fig. 2.3b). This dispersion relation can also be obtained by apply- ing the Holstein-Primakoff transformation to the Heisenberg Hamiltonian Eq. 2.6, i.e. writing spin operators as a function of magnon creation and annihilation operators. In the limit of small excitation numbers, this results in a non-interacting magnon Hamiltonian with the same dispersion as Eq. 2.10. If the wavelength λ is long compared to the spacing between two spins a so that ka ≪ 1, the dispersion is approximated by the quadratic relation

hω ≈ 2J Sa̵ ²k²=Dk² (2.11)

where D is called spin wave stiffness and behaves like an inverse of the effective magnon mass,

m^∗=

̵h² d²E(k)/dk² =

h̵²

2J a². (2.12)

A large mass corresponds to a small group velocity and small contributions to transport.

Next, moving from a one-dimensional chain to a three-dimensional crystal, the dispersion relation of a simple cubic spin arrangement is a function of the three components kx, kyand kzof the wavevector:

̵hω(k) = 24JS(1 −1

3(cos ak_x+cos ak_y+cos ak_z)), (2.13) which in the small-wavevector regime can be approximated as ̵hω ≈ 4JSa²k². There- fore, the exchange-dominated magnons or spin waves obtained when considering only the Heisenberg exchange interaction exhibit a cosine-shaped dispersion in the first Brillouin zone, which is nearly isotropic for small wave numbers. This dispersion is isotropic due to the isotropic character of the exchange energy.

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In an antiferromagnetic Heisenberg chain (J < 0) as sketched in Fig. 2.3c, the re- sulting dispersion relation reads

hω = −4J S∣ sin ak∣,̵ (2.14)

which is approximated as ̵hω ≈ −4JSa∣k∣ in the small wavevector regime, i.e. ka ≪ 1 as shown in Fig. 2.3d. Without the presence of an external field and anisotropy energy, this dispersion corresponds to two degenerate branches related to the two antiferromagnetically coupled sublattices. By contrast, in a ferrimagnet, these two branches are non-degenerate, because the number of the magnetic ions in the two sublattices are unequal. The lower and higher branches are called acoustic and optical modes, which correspond to the excitation of majority and minority spin lattices, respectively. At low energies, the excitation spectrum of a ferrimagnet is determined by the lower-lying acoustic mode which has the shape of ferromagnetic dispersion as seen in Fig. 2.8 later in this section.

Dipole-dipole interaction

Classically, a magnetic moment is a magnetic dipole. Thus, it has a magnetic field around it, which can act on other magnetic moments. When two magnetic moments are situated rather far away, with large rij, they can still interact with each other via the dipole-dipole interaction instead of the exchange interaction. Under the magnetostatic approximation, the resulting energy of the dipolar coupling between the two magnetic moments at i-th and j-th sites is given by

H_dipolar=g ∑

i,j

(mi⋅mj) −3(mi⋅rij)(mj⋅rij)

r_ij³ . (2.15)

For small wave vectors, one looks at the properties of electromagnetic waves in sat- urated magnetic insulators under the magnetostatic approximation [2]. Since these waves also describe the precessing motion of magnetic moments, they can be viewed as spin waves. Unlike the spin waves or magnons coupled by exchange energy, the coupling between the magnetic moments is dominated by the dipolar energy. Thus, they are called dipolar spin waves or magnetostatic waves. In general, both dipole and anisotropy energy should be taken into account, which together gives rise to the magnetostatic modes and anisotropic dipolar magnon dispersion relation. The dispersion relation splits into three modes as sketched in Fig. 2.4, depending on the relative directions of k, M and the plane of the thin film [2]:

• In-plane M, k ∥ M: The Backward volume mode propagates in the whole volume of a magnet with negative group velocity (∂ω/∂k < 0), namely the wave packet

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M k

Forward volume mode

M k Backward volume mode

M k

Damon-Eshbach surface mode

Wavevector k (108 m-1)

Magnon frequency ω/2π (GHz)

M k=0 Out-of-plane uniform precession mode

M In-plane uniform precession mode

k=0

Figure 2.4: Overview of dispersion characteristics for magnetostatic waves or dipolar spin waves in an infinite YIG film of 210 nm thickness. The exchange energy is not included. De- pending on the relative direction of the out-of-plane or in-plane magnetization with respect to the direction of the wavevector, one encounters the forward volume mode (FVM, out-of- plane M⊥ k), backward volume mode (BWV, in-plane M ⊥ k) and Damon-Eshbach surface mode (DESM, in-plane M∥ k). BWV and FVM extend over the whole volume of the film.

Their spectral range begins at a wavevector of zero, where they possess negative and positive group velocities, respectively. On the other hand, DESMs are localized at the surfaces of the film. Uniform precess modes with zero wavevector can be both in-plane and out-of-plane.

moves opposite to the propagation direction of the waves. Thus, with increasing wavevector, the magnon frequency ω decreases, which implies a magnon band minimum different from that of the uniform in-plane precession mode.

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Hex

H0

φ k θ

θex

x y z

Figure 2.5: xyz-coordinate system used for calculating the dipolar-exchange magnon dispersion relation. The thin film magnet (gray) lies in the xy-plane. An external magnetic field Hexhas a magnitude of Hexand an angle of θexwith respect to the z-axis. This gives rise to an internal magnetic field H0with a magnitude of H0 and an angle of θ with respect to the z-axis, which defines the ground state of the magnetization, i.e. the static magnetization of the magnet. The wavevector k of the propagating spin waves or magnons is in the plane of the film at an angle of ϕ with respect to the y-axis.

• In-plane M, k ⊥ M: The Damon-Eshbach surface mode amplitudes are significant mainly at the surfaces of the film and propagate in only one direction. When k → 0, the surface mode merges together with the backward volume mode at the uniform precession mode (k = 0).

• Out-of-plane M, in-plane k (k ⊥ M): The Forward volume mode propagates in the whole volume of a magnet with positive group velocity (∂ω/∂k > 0). Unlike the case of in-plane magnetization, the magnon band minimum coincides with the uniform precession mode (k = 0) for the out-of-plane magnetization.

Dipolar-exchange magnon dispersion relation for YIG

Herring and Kittel [6], as well as Kalinikos and Slavin [7] proposed a master formalism for the magnon dispersion relation where both dipolar and exchange energies are taken into account.

In a thin-film magnet, the magnon dispersion relation ω(k) is anisotropic espe- cially in the low wavevector regime where dipolar energy dominates, meaning that magnon frequencies depend not only on the magnitude but also on the direction of the wavevector k. In the high wavevector regime where exchange dominates, the dispersion relation is isotropic with a parabolic shape or more precisely, a cosine shape. The following part constructs the lowest-lying dipolar-exchange dispersion relation for a YIG film.

In an xyz-coordinate system (cf. Fig. 2.5 ), a thin film magnet is taken to lie in

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the xy-plane. An external field Hexis applied in the yz-plane at an angle of θexwith respect to the z-axis. Due to the demagnetization field (shape anisotropy) generated by the out-of-plane component of the magnetization, the internal magnetic field H0

inside the magnet differs from Hex, and obeys

H0cos θ = H_excos θ_ex−M_scos θ (2.16)

H0sin θ = H_excos θ_ex, (2.17)

where Ms is the saturation magnetization of the magnet and θ corresponds to the orientation of the static magnetization. After obtaining the internal field, the magnon dispersion can be calculated following Kalinikos’ and Slavin’s formalism [7],

ω(k) =

√ (ω_H+

D_s

h̵ k²)(ω_H+ D_s

h̵ k²+ω_MF (k)), (2.18) where ωH = γµ₀H₀, ωM = γµ₀M_s, γ is the gyromagnetic ratio, µ0 is the vacuum permeability, Dsis the spin stiffness of the magnet, ̵h is the reduced Planck constant and

F (k) = P (k) + sin²(θ)(1 − P (k)(1 + cos²ϕ) + ωM

P (k)(1 − P (k)) sin²ϕ

ωH+^D̵h^sk² ), (2.19) where P (k) = 1 −^1−e_kt^−kt and t is the thickness of the film.

Dipole-dipole interactions are dominant in the small-wavevector regime (∣k∣ ≲ 10⁷m⁻¹for a 210 nm thick YIG film with µ0M_s =170mT), while exchange interac- tions are significant in the large wavevector regime (∣k∣ ≳ 10⁸m⁻¹). Between 10⁷m⁻¹ and 10⁸m⁻¹), lies the dipolar-exchange regime where both interactions matter. To compare, the first Brillouin zone boundary of YIG has a wavevector with a magnitude of about 5 × 10⁹m⁻¹(∣k∣ = 2π/a where a ≈ 12 ˚A is the lattice parameter of a unit cell YIG).

A 210 nm thick YIG film with in-plane magnetization exhibits a dipolar-exchange magnon dispersion relation as sketched in Fig. 2.6. A field in the plane of the film is applied to align the magnetization, i.e. θ = 0 in Fig. 2.5. This is easily achievable since the in-plane coercive field of such a YIG film is only about 0.1 mT. Here one assumes that it is an infinite film. For ϕ = 0 (k∥M ≠0and k⊥M =0) one obtains the pure backward volume mode, while for ϕ = π/2 (k∥M =0and k⊥M ≠0) it gives rise to pure surface mode. However, in between these two limiting cases, magnons have both characters. Moreover, an important feature of the dispersion in Fig. 2.6 is that there are two magnon band minima, which both belong to the backward volume mode with opposite wavevector.

With decreasing thicknesses of the YIG films, the following changes occur in the spectra, which is visualized in the linecut plots of Fig. 2.7:

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Figure 2.6: Dispersion relation of a 210 nm thick YIG film with in-plane magnetization at an external field of 10 mT. Magnon frequencies (ω/2π) are plotted in the space of k⊥M and k∥M, i.e. wavevectors perpendicular and parallel to the in-plane magnetization, corresponding to the magnetostatic surface mode and backward volume mode. The colour gradient indicates the magnon frequencies increasing from purple to apricot. The peanut-shaped isofrequency line in blue corresponds to the magnon frequency of 2 GHz in Fig. 2.7c. The blue and red lines indicate the crosssections of k∥M = 0 and k⊥M = 0, respectively. Correspondingly, blue and red linecuts of the dispersion relation of k⊥ M and k ∥ M are drawn in Fig. 2.7a. There are two magnon-band minimum points with magnon frequency of ωmin, which only has a zero k⊥M and a non-zero k∥M component. The lowest-lying magnon dispersion relation is drawn with parameters obtained from the fit of rf power reflection measurement [8]: Gyromagnetic ratio (γ= 27.3 GHz/T) and saturation magnetization (µ0Ms= 170 mT). An exchange stiffness of 1× 10⁻³⁹J m²is used [7].

• Surface and bulk modes merge and backward volume modes lose its back- ward character (cf. Fig. 2.7b). The band minimum moves towards the zero-

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Magnon frequency ω/2π (GHz)

210 nm thick YIG film at an in-plane field of 10mT

k

M k M

kM (107 m-1)

k M (107 m-1)

kM (107 m-1) 210nm

100nm

k M k M

10 nm

100 nm

210 nm

210nm

100nm10nm

ω_min

a b

c d

Figure 2.7:Cross section of the magnon dispersion relation for different film thicknesses. For a 210 nm thick in-plane magnetized YIG film at an external field of 10 mT, cross section at con- dition of a, k⊥M = 0 (red), k∥M = 0 (blue), and c, ω/2π = 2 GHz. In a, the resulting magnon frequencies as a function of k∥M(red) and k⊥M(blue), i.e. wavevectors perpendicular and parallel to the in-plane magnetization, correspond to the pure magnetostatic backward volume mode and surface mode, respectively. In c, it shows the peanut-shaped isofrequency line at 2 GHz. The two global magnon band minima belong to the backward volume mode and are indicated by ωminin a and two red dots in c. In b and d, the similar cross section are shown for YIG film with thicknesses of 210 nm, 100 nm and 10 nm. We draw the lowest magnon dispersion relation with parameters obtained from the Kittel fit of rf power reflection measurement:

Gyromagnetic ratio (γ = 27.3 GHz/T) and saturation magnetization (µ0M_s = 175 mT). We used exchange stiffness of 1× 10⁻³⁹J m²[7].

momentum mode (cf. Fig. 2.7d).

• The dispersion becomes less anisotropic (cf. Fig. 2.7d).

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Optical mode: M0 increases

Acoustic mode: M0 decreases

0 20 40 60 80 100

E (meV)

0 300 600 900 1200

T (K)

N Γ H

a

b

c

Figure 2.8: Acoustic and optical magnon modes in a ferrimagnet. Schematic illustration of aoptical and b acoustic modes. Excitation of an acoustic mode causes a reduction of the net magnetization, whereas generation of an optical mode leads to an enhanced net magnetiza- tion. The precessing red and blue arrows are the majority and minority spins. c Full magnon dispersion relation of YIG at 300K, where the blue and red curves correspond to the optical and acoustic modes. Figure c is adapted from Ref. [9].

Ferrimagnon dispersion relation for YIG

YIG is often treated as a ferromagnet with ”ferromagnons” to make its complex magnetic structure and properties more accessible. This simplified approach could ex- plain the thermodynamic properties of YIG at low temperature but is not so success- ful at high temperatures. So far in this section, only the lowest lying magnon branch has been discussed. Since YIG has two sub-lattices with 20 magnetic ions in a unit cell, this gives rise to 20 branches in a magnon spectrum, including both acoustic and optical modes as shown in Fig. 2.8. The magnon spin currents associated with these two modes have opposite signs.

Neutron scattering experiments in the 1970s analyzed the lowest 3 branches [10]

and a theoretical calculation from 1993 [4] estimated the spectra including all 20 branches. However, the magnon spectrum of YIG has been recently revisited both theoretically and experimentally [9, 11, 12] with the so far most accurate and com- plete information about this complex oxide. Particular attention has been paid to the impact of the optical modes, especially on the room-temperature magnetic properties of YIG.