university of groningen controlled magnon spin transport

University of Groningen

Controlled magnon spin transport in insulating magnetsLiu, Jing

DOI:10.33612/diss.97448775

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Citation for published version (APA):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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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

Magnons 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 andlarge scales, respectively. Bloch proposed the idea of magnons in the 1930s to find aconsistent 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)

cFirst excited state

T ≠ 0 K(energetically favorable)

b 2μ B

M1

Figure 2.1: a Magnetic ground state at zero kelvin with a total magnetization of M0. b Firstexcited magnetic state with collectively excited magnetic moments. This is an energeticallyfavorable 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 highlyunfavorable.

2

8 2. Theoretical background

in Fig. 2.1: In the ground state of an isotropic single-domain ferromagnet, all mag-netic 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. Withincreasing temperature, all magnetic moments are collectively excited by thermalfluctuations: They precess around their ground states, which results in a reducedstatic magnetization of M1 as 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 angularmomentum of h. This means that with increasing temperature the reduction of themagnetization is quantized. A naive way to realize this change in magnetization isto 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, becauseneighboring spins are precessing with only a small phase shift with respect to theirneighbors. This results in a propagating spin wave in a magnet. In the quasiparticlelanguage, one can say that a magnon is excited in the magnetic system.

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

E = hω(k), (2.1)

where h is the reduced Planck constant, ω(k) is the angular precession frequency ofeach individual spin in a spin wave. The precession frequency ω depends on thewavevector k based on the magnon dispersion relation ω(k) (see section 2.3), fromwhich the magnon group velocity, effective magnon mass and magnon density ofstates 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 casein Figs. 2.1a as a ”vacuum state” of the magnet, the excited state can be seen as ”agas of magnons”. Unlike a defined system with bosonic atoms such as 4He, thenumber of magnons (quasiparticles) in a magnet is not conserved. With rising thetemperature, the overall population of magnons increases and the distribution obeysthe Bose-Einstein statistics

f(E) =1

eβ(E−µ) − 1, (2.2)

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

2

2.2. Ferrimagnetic insulator: YIG 9

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

N = ∫

∞

0n(E)dE (2.3)

= ∫

∞

0g(E)f(E)dE, (2.4)

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

f(E) =1

eβE − 1. (2.5)

2.2 Ferrimagnetic insulator: YIG

Yttrium iron garnet (YIG) is magnetically ordered and electrically insulating; there-fore, it is called a magnetic insulator†. Due to these properties, one can study mag-netic excitations, i.e. magnons or spin waves, in YIG without having to considerthe influence of mobile electrons such as in the case of conventional spin current inferromagnetic 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 ionsoccupy 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 magneticordering originates from the antiferromagnetic coupling, i.e. super-exchange inter-action, between two neighboring (a)- and (d)-type Fe3+ ions via an oxygen ion. Sincethe number of these two types of Fe3+ ions is different, the resulting magnetization isnon-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 magneticfields.

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

2


(a) Fe(d) 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]. Only1/8 of the unit cell is filled with ions for convenience of visualization; the body diagonal issketched as a dashed line indicating the orientation. The remaining 7/8 parts of the unit cellhave the same composition but different orientations, as indicated by the dashed lines. Notethat a bcc unit cell contains two lattice points (8 times of the sketched part in a), whereas theprimitive cell has only one lattice point (4 times of the sketched part in a). b A zoomed-inview 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, theantiferromagnetically 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 largerthan 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 theWigner-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 (6S5/2†). 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 gyromagneticratio ∣γ/2π∣ = ∣γS/2π∣ = 28 GHz/T (2.8 MHz/G) [2], which is why the orbital angularmomentum is said to be quenched. In addition, YIG has a very symmetric cubicstructure, 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 isthe number of states with a given S (called multiplicity) and X is a letter corresponding to the value of Laccording to the convention: L = 0,1,2,3... corresponds to S,P,D,F...

2

2.3. Magnon spectra 11

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

A gem for magnon spintronics

YIG is such an important material in the study and applications of magnetism forthe following reasons: First, YIG has the lowest magnetic damping among all mag-netic materials, corresponding to the narrowest ferromagnetic resonance linewidthamong those observed. This means that it has the longest magnon or spin wavelife time, which is ideal for magnon transport studies. Section 2.4.1 discusses theconcept 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 thermo-dynamic behavior resembles that of a ferromagnet, especially in the low temperatureregime (T < 260 K) [4]. High Tc makes it convenient to study and utilize its magneticproperties at room temperature. Third, even though YIG is a man-made materialwith a complex crystal structure, the matching of the Fe3+ ions and the space in be-tween the O2− scaffold is extraordinarily good. This means that there are very fewdistortions 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 mag-non spectrum, i.e. the dispersion relation or the magnon frequencies as a functionof 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 distri-bution function of magnons (cf. Eq. 2.2), one can construct a general transport modelbased on a spin diffusion equation, from which the main transport properties can beobtained. Therefore, the following section discusses magnon spectra. Special atten-tion is paid to the lowest lying dipolar-exchange magnon dispersion relation for YIGfilm 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 whichcarry spin angular momentum S. The exchange energy between neighboring spins

2


λ=2π/k π/a-π/a 0Wavevector k

Ferromagnetic coupling: J > 0

ħω

~k2

4JS(1

- co

s k

a)

π/a-π/a 0Wavevector k

ħω

λ=2π/k

Antiferromagnetic coupling: J < 0

~k-4

JS |s

in k

a|

8JS

-4JS

a b

c d

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

in a lattice can be expressed following the Heisenberg model:

Hexchange = −J∑n(Sn ⋅ Sn+1), (2.6)

where J is an exchange coefficient which quantifies the strength of the exchange in-teraction. 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 couplingwhere mn and mn+1 tend to align parallel to each other, while J < 0 is the case forantiferromagnetic coupling where mn and mn+1 prefer to align antiparallel to eachother. Even though exchange coupling is a short-range magnetic effect, it gives riseto 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 ateach site n are described as an approximate ground state (Szn) plus small excitations(Syn, S

zn), i.e. magnons which do not interact with each other. This is the result of a

2


semiclassical ansatz, i.e. an equation of motion for spins S = (Sxn, Syn, S

zn), in which

Sxn = uei(kna−ωt) (2.7)

Syn = vei(kna−ωt) (2.8)

Szn ≈ 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 = 4JS(1 − cosak), (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 op-erators. In the limit of small excitation numbers, this results in a non-interactingmagnon Hamiltonian with the same dispersion as Eq. 2.10. If the wavelength λ islong compared to the spacing between two spins a so that ka ≪ 1, the dispersion isapproximated by the quadratic relation

hω ≈ 2JSa2k2 =Dk2 (2.11)

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

m∗=

h2

d2E(k)/dk2=

h2

2Ja2. (2.12)

A large mass corresponds to a small group velocity and small contributions to trans-port.

Next, moving from a one-dimensional chain to a three-dimensional crystal, thedispersion relation of a simple cubic spin arrangement is a function of the three com-ponents kx, ky and kz of the wavevector:

hω(k) = 24JS(1 −1

3(cosakx + cosaky + cosakz)), (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 consideringonly the Heisenberg exchange interaction exhibit a cosine-shaped dispersion in thefirst Brillouin zone, which is nearly isotropic for small wave numbers. This disper-sion is isotropic due to the isotropic character of the exchange energy.

2


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

hω = −4JS∣ sinak∣, (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 anisotropyenergy, this dispersion corresponds to two degenerate branches related to the twoantiferromagnetically coupled sublattices. By contrast, in a ferrimagnet, these twobranches are non-degenerate, because the number of the magnetic ions in the twosublattices are unequal. The lower and higher branches are called acoustic and opti-cal modes, which correspond to the excitation of majority and minority spin lattices,respectively. At low energies, the excitation spectrum of a ferrimagnet is determinedby the lower-lying acoustic mode which has the shape of ferromagnetic dispersionas 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 fieldaround it, which can act on other magnetic moments. When two magnetic momentsare situated rather far away, with large rij, they can still interact with each other viathe dipole-dipole interaction instead of the exchange interaction. Under the magne-tostatic approximation, the resulting energy of the dipolar coupling between the twomagnetic moments at i-th and j-th sites is given by

Hdipolar = g∑i,j

(mi ⋅mj) − 3(mi ⋅ rij)(mj ⋅ rij)

r3ij. (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 thesewaves also describe the precessing motion of magnetic moments, they can be viewedas spin waves. Unlike the spin waves or magnons coupled by exchange energy, thecoupling between the magnetic moments is dominated by the dipolar energy. Thus,they are called dipolar spin waves or magnetostatic waves. In general, both dipole andanisotropy energy should be taken into account, which together gives rise to themagnetostatic modes and anisotropic dipolar magnon dispersion relation. The dis-persion relation splits into three modes as sketched in Fig. 2.4, depending on therelative 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 volumeof a magnet with negative group velocity (∂ω/∂k < 0), namely the wave packet

2


Mk

Forward volume mode

Mk

Backward volume mode

Mk

Damon-Eshbach surface mode

Wavevector k (108 m-1)

Ma

gn

on

fre

qu

en

cy ω

/2π

(G

Hz)

Mk=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 spinwaves 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 respectto 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 surfacemode (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 positivegroup velocities, respectively. On the other hand, DESMs are localized at the surfaces of thefilm. 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 increas-ing wavevector, the magnon frequency ω decreases, which implies a magnonband minimum different from that of the uniform in-plane precession mode.

2


Hex

H0

φ

k

θ

θex

x

y

z

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

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

• Out-of-plane M, in-plane k (k ⊥ M): The Forward volume mode propagates inthe whole volume of a magnet with positive group velocity (∂ω/∂k > 0). Unlikethe case of in-plane magnetization, the magnon band minimum coincides withthe 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 formal-ism for the magnon dispersion relation where both dipolar and exchange energiesare 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 thatmagnon frequencies depend not only on the magnitude but also on the direction ofthe wavevector k. In the high wavevector regime where exchange dominates, thedispersion relation is isotropic with a parabolic shape or more precisely, a cosineshape. The following part constructs the lowest-lying dipolar-exchange dispersionrelation for a YIG film.

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

2


the xy-plane. An external field Hex is applied in the yz-plane at an angle of θex withrespect to the z-axis. Due to the demagnetization field (shape anisotropy) generatedby the out-of-plane component of the magnetization, the internal magnetic field H0

inside the magnet differs from Hex, and obeys

H0 cos θ =Hex cos θex −Ms cos θ (2.16)

H0 sin θ =Hex cos θex, (2.17)

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

ω(k) =

√

(ωH +Ds

hk2)(ωH +

Ds

hk2 + ωMF (k)), (2.18)

where ωH = γµ0H0, ωM = γµ0Ms, γ is the gyromagnetic ratio, µ0 is the vacuumpermeability, Ds is the spin stiffness of the magnet, h is the reduced Planck constantand

F (k) = P (k) + sin2(θ)(1 − P (k)(1 + cos2 ϕ) + ωM

P (k)(1 − P (k)) sin2 ϕ

ωH +Dshk2

), (2.19)

where P (k) = 1 − 1−e−kt

ktand t is the thickness of the film.

Dipole-dipole interactions are dominant in the small-wavevector regime (∣k∣ ≲107 m−1 for a 210 nm thick YIG film with µ0Ms = 170 mT), while exchange interac-tions are significant in the large wavevector regime (∣k∣ ≳ 108 m−1). Between 107 m−1

and 108 m−1), lies the dipolar-exchange regime where both interactions matter. Tocompare, the first Brillouin zone boundary of YIG has a wavevector with a magni-tude of about 5 × 109 m−1 (∣k∣ = 2π/a where a ≈ 12 A is the lattice parameter of a unitcell YIG).

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

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

2


Figure 2.6: Dispersion relation of a 210 nm thick YIG film with in-plane magnetization at anexternal 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 tothe magnetostatic surface mode and backward volume mode. The colour gradient indicatesthe magnon frequencies increasing from purple to apricot. The peanut-shaped isofrequencyline in blue corresponds to the magnon frequency of 2 GHz in Fig. 2.7c. The blue and red linesindicate the crosssections of k∥M = 0 and k⊥M = 0, respectively. Correspondingly, blue andred linecuts of the dispersion relation of k ⊥ M and k ∥ M are drawn in Fig. 2.7a. There aretwo magnon-band minimum points with magnon frequency of ωmin, which only has a zerok⊥M and a non-zero k∥M component. The lowest-lying magnon dispersion relation is drawnwith parameters obtained from the fit of rf power reflection measurement [8]: Gyromagneticratio (γ = 27.3 GHz/T) and saturation magnetization (µ0Ms = 170 mT). An exchange stiffnessof 1 × 10−39 J m2 is 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-

2


Ma

gn

on

fre

qu

en

cy ω

/2π

(G

Hz)

210 nm thick YIG "lm at an in-plane "eld of 10 mT

k M

k M

k M (107 m-1)

k

M (1

07

m-1)

k M (107 m-1)

210 n

m10

0 nm

k Mk M

10

nm

10

nm

10

0 n

m2

10

nm


21

0 n

m

10

0 n

m

10

nm


ωmin

a b

c d

Figure 2.7: Cross section of the magnon dispersion relation for different film thicknesses. Fora 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 magnonfrequencies as a function of k∥M (red) and k⊥M (blue), i.e. wavevectors perpendicular and par-allel to the in-plane magnetization, correspond to the pure magnetostatic backward volumemode and surface mode, respectively. In c, it shows the peanut-shaped isofrequency line at2 GHz. The two global magnon band minima belong to the backward volume mode and areindicated by ωmin in a and two red dots in c. In b and d, the similar cross section are shown forYIG film with thicknesses of 210 nm, 100 nm and 10 nm. We draw the lowest magnon disper-sion relation with parameters obtained from the Kittel fit of rf power reflection measurement:Gyromagnetic ratio (γ = 27.3 GHz/T) and saturation magnetization (µ0Ms = 175 mT). Weused exchange stiffness of 1 × 10−39 J m2 [7].

momentum mode (cf. Fig. 2.7d).

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

2


Optical mode: M0 increases

Acoustic mode: M0 decreases

0

20

40

60

80

100

E (

me

V)

0

300

600

900

1200

T (K

)

ΓN H

a

b

c

Figure 2.8: Acoustic and optical magnon modes in a ferrimagnet. Schematic illustration ofa optical and b acoustic modes. Excitation of an acoustic mode causes a reduction of the netmagnetization, 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 magnondispersion relation of YIG at 300K, where the blue and red curves correspond to the opticaland 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 mag-netic 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 branchhas been discussed. Since YIG has two sub-lattices with 20 magnetic ions in a unitcell, this gives rise to 20 branches in a magnon spectrum, including both acousticand optical modes as shown in Fig. 2.8. The magnon spin currents associated withthese 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 20branches. However, the magnon spectrum of YIG has been recently revisited boththeoretically 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 theimpact of the optical modes, especially on the room-temperature magnetic proper-ties of YIG.

2

2.4. Magnon injection and detection 21

2.4 Magnon injection and detection

In order to study their transport properties, magnons are excited at site A by a stim-ulus, such as an electrical current, a heat source or a microwave field. The generatedmagnons propagate in a magnet and are then detected at site B. By varying the dis-tance betweenA and B, one can obtain the distance-dependent behavior, from whichthe transport properties of magnons is studied. This section explains the couplingmechanism between magnons and the stimuli so that magnons can be injected anddetected, especially using the electrical method.

In section 2.4.2, a basic introduction into spin injection and detection is given: Theequation of motion of a magnetic moment and the Landau-Lifshitz-Gilbert equationare discussed to describe the magnetization dynamics, after which the concept of spintransfer torque (STT) is used to explain the transfer of angular momentum betweentwo media.

In addition, a pure spin current source is crucial to have a clear identification ofmagnon excitation and detection. Section 2.4.2 introduces the (inverse) spin Hall effect,which is used in this thesis to convert charge current and pure spin current into eachother. In this way, magnon injection and detection can be realized fully electrically.

Building upon the above knowledge, the following sections explain the theoryof the specific magnon excitation and detection methods used in this thesis. First,as introduced in section 2.4.3, the electrical approach uses an electrical current as astimulus, where spin accumulation of mobile electrons is used to generate magnonspin current in an attached magnetic insulator. This is the most important methodused throughout all the work presented in this thesis. Besides, as an effect of electri-cal current, Joule heating provides a thermal stimulus for the magnon system, whichis discussed by introducing the spin Seebeck effect (SSE). Second, section 2.4.4 intro-duces a microwave field as a stimulus to excite magnons in the GHz range. Thismethod is employed in Chapter 6.

2.4.1 Spin injection and detection

Slonczewski [13] and Berger [14] spearheaded the development of spin transfer torquein ferromagnetic multilayers, where the spin-polarized electrical current from a fer-romagnet can apply a torque on another ferromagnetic metal via direct transfer ofspin angular momentum. As a result, the magnetic moments of the second magnetprecess with a larger or smaller precession angle depending on the polarization ofthe spin current. When the current is large enough, the direction of the magnetiza-tion can be reoriented or even switch sign. In the following the theoretical buildingblocks for this phenomenon are laid out.

2


M

He�

-M×Heff

-M×dM/dt

Figure 2.9: Illustration of the LLGequation.

Equation of motion for a magnetic momentum

The equation of motion for a magnetic moment m in the presence of an effectivemagnetic field is

dmdt

= γ(m × µ0Heff), (2.20)

where γ = gµB/h is the gyromagnetic ratio of the electron with Lande factor g ≈ −2,µ0 is the vacuum permeability and Heff is the effective internal magnetic field whichmay include the exchange bias field, dipolar field, magnetocrystalline anisotropy, anexternally applied static and dynamic small field.

Landau-Lifshitz-Gilbert equation

Based on the macrospin model, i.e. neglecting the spatial variations of individualmagnetic moments, and multiplying both sides of equation 2.20 by the density ofmagnetic ions in a solid N (M = Nm), the Landau-Lifshitz equation [15] is obtained,

dMdt

= γ(M × µ0Heff), (2.21)

which describes the motion of magnetization under an effective field. However, thisequation implies that the precession around the effective field continues endlessly.This is an idealized model. In order to capture the damping feature that occurs inreality, a phenomenological term has been included in eq. 2.21 for small damping,which is known as the Gilbert damping term. This gives rise to the well-known

2


Landau-Lifshitz-Gilbert (LLG) equation,

dMdt

= γ(M × µ0Heff) +α

Ms(M ×

dMdt

), (2.22)

whereMs is the saturation magnetization, which is the magntiude of the vector M. αis the Gilbert damping parameter: It is a material-dependent dimensionless param-eter. So far YIG has been found to have the smallest α, for the best samples with amagnitude of 10−5. This means that after an excitation, it takes about 105 precessionperiods before it aligns with the effective field. The damping parameters for ironand permalloy are around 0.003 and 0.01, respectively. The theory and method tomeasure α will be introduced in sections 2.4.4 and 3.3. The relaxation time of a modeis the time required for the amplitude of the small-signal magnetization to decay bythe factor 1/e after removal of an excitation. The relaxation time T0 of the uniformprecession mode in an infinite medium is related to the damping parameter α by

1

T0= ωFMRα, (2.23)

where ωFMR is the ferromagnetic resonance frequency of the uniform precessionmode. Alternatively, T0 is related to the full FMR line width ∆H of the uniformprecession mode by

1

T0= −

γµ0∆H

2, (2.24)

which is used in the microwave absorption experiment to determine α in section 3.3.

Spin transfer torque

When a spin current flows towards a magnet, it can transfer angular momentum tothe magnet via spin transfer torque (STT) [13, 14]:

τ STT ∝ M × (M ×µs), (2.25)

where M is the magnetization of the magnet of interest, µs is the spin polarization ofthe incoming electron spin current. When µs ⊥M, τ STT is maximal. By contrast, τ STT

is zero when µs is parallel or antiparallel to M. STT tends to orient or tilt M alongthe direction of µs. Therefore, M and µs need to have components perpendicular toeach other in order to have non-zero τ STT and effective angular momentum transfer.

Spin mixing conductance

The spin current js across an interface between a normal metal and a ferromagnetdepends on the orientation of the spin polarization µs in the normal metal and the

2


magnetization of the ferromagnet M [16]. The precise dependence is governed bythe spin mixing conductance as follows [17]:

js =Gr

eM2s

M × (M ×µs) +Gi

eMs(M ×µs) +

Gs

eµs (2.26)

where Gr and Gi are the real and imaginary parts of the spin mixing conductance(G↑↓ = Gr + iGi), while Gs is known as the effective spin mixing conductance. Thefirst term corresponds to the STT. The second term is related to a field-like torque,which effectively makes M precess around µs. Finally, the last term is remarkable,because it permits spin transfer through the interface when µs and M are parallel toeach other, which has been studied recently [18–20].

2.4.2 (Inverse) spin Hall effect

The spin Hall effect (SHE) and the inverse spin Hall effect (ISHE) occur in non-magnetic heavy metals such as platinum, tantalum and tungsten where spin-orbitcoupling is significant. The SHE describes the generation of a pure transverse purespin current by a source of charge current. By contrast, the ISHE is the Onsagerreciprocal process of the SHE, where a source of pure spin current gives rise to atransverse charge current. After their discovery, these phenomena quickly becamepopular as tools to inject and detect spin current [21–24], in addition to the methodof using the intrinsically spin-polarized current in ferromagnets [25–27].

Fig. 2.10 illustrates the SHE: A source of charge current propagating along the x-axis with a current density of jc

x causes transverse spin currents in the yz-plane js. Forexample, when the spin polarization of the electron is along the y-axis, the electrondrifts in the direction of the z-axis. In the reciprocal process, namely the ISHE, givena source of spin current with polarization in the y-axis propagating along the z-axisa transverse charge current along the x-axis is produced. The relation between thesource current (on the right) and the induced transverse current (on the left) can beexpressed for the SHE and the ISHE respectively as following

js= θSHσ × jc

x (2.27)

jcx = θSHσ × js, (2.28)

where θSH is the spin Hall angle which captures the sign and conversion efficienciesbetween the charge and spin current and σ is the spin polarization of the electrons.

The symmetry of the (I)SHE, i.e. the orthogonality and sign [28], can be conve-niently summarized by the hand rule depending on the sign of θSH: Right (θSH > 0,e.g. Pt) or left (θSH < 0, e.g. Ta) hand rules are applied to identify the direction of thespin polarization (thumb), electron movement direction of the source current (indexfinger) and direction of the induced transverse electron current (middle finger) [29].

2


jx

c

jyz

s

x

y

z

Figure 2.10: Schematic illustration of the spin Hall effect (SHE) or the inverse spin Hall ef-fect (ISHE). In the SHE, a charge current with current density of jcx produces transverse spincurrents due to a strong spin-orbit coupling in a heavy metal. For example, one resultingspin current jsyz with a polarization along the y-axis flows in the direction of the z-axis. Dueto the unpolarized charge current source, spin currents propagate in all the directions in theyz-plane. At the surface of a non-magnetic metal the spin current is blocked. The spin Hallcurrent is then compensated by a diffuse spin current in the opposite direction, or a spin ac-cumulation with same polarization as shown in the red-colored cross-section. The red arrowsindicate the direction of the electron spin polarization. The example here describes the situa-tion for the heavy normal metal with positive spin Hall angle (θSH), such as platinum, wherethe right hand rule can be applied to identify the direction of the electron spin polarization(thumb), the electron movement of the source current (index finger), i.e. opposite to jcx, andthe direction of induced electron current (middle finger). For heavy metals with negative θSH,the resulting spin polarization possesses the opposite sign. This hand rule also applies forthe ISHE, i.e. the reciprocal process of the SHE: A spin current propagating in the yz-planegenerates a charge current along the x-axis.

The following part explains two relevant examples involving the (I)SHE in a het-erostructure of heavy metal and magnetic insulator such as Pt∣YIG. One is the spinmagnetoresistance, and the other is electrical magnon injection and detection. Theformer provides information about the coupling between the electron and magnonspins. The latter is the central set-up used in the transport measurement.

Spin magnetoresistance: SHE and ISHE

When a metal exhibiting the SHE is in contact with a magnetic insulator, the resis-tance of this metal depends on the magnetization of the magnet. This effect is calledspin magnetoresistance (SMR). Therefore, by varying the magnetization of YIG in a

2


jx

c

xy

z

YIG

Pt

jx

c

YIG

Pt

a bLow Pt resistance High Pt resistance

Min

ima

l ST

T

Ma

xim

al S

TT

μs

Pt

M0

YIG

μs

Pt

M0

YIG

Figure 2.11: Schematic illustration of the spin magnetoresistance (SMR). A Pt∣YIG het-erostructure is used to illustrate two limiting cases for SMR when the polarization of the spinaccumulation in Pt µPt

s and the static magnetization of YIG MYIG0 are a parallel and b perpen-

dicular to each other.

Pt∣YIG heterostructure, the resistance of the Pt can be tuned [30–33]. Besides, a spinmixing conductanceG↑↓ can be obtained, which is related to the efficiency of the spininjection at the interface, the effective spin-mixing conductance Gs (Gs/G

↑↓ < 1) [20].Also, the magnetization of the materials can be probed locally via SMR [34–36].

When a charge current Jc passes through Pt, a resulting spin accumulation dueto the SHE at the interface of Pt exerts a STT on the magnetization of YIG as dis-cussed in section 2.4.1, depending on the relative orientation of the polarization ofthe spin accumulation µPt

s and magnetization MYIG [37]. An external field Hex is ap-plied to orient the magnetization of YIG. In the limit of a static magnetization MYIG

0 ,corresponding to zero temperature, the STT as a function of relative orientation isbounded by the following extreme cases as shown in Fig. 2.11 [38, 39]:

• µPts ∥ MYIG

0 : No spin will be transferred via STT, which gives rise to reflectionof the electrons with their spins unchanged. The scattered electrons experi-ence the ISHE, which contributes to the electrical current; therefore, this cor-responds to a low resistance state of the heavy metal and (for constant currentbias) a low voltage signal is measured across the Pt strip.

• µPts ⊥MYIG

0 : Maximal spin transfer torque is achieved, resulting in a spin flip ofthe back-scattered electrons, i.e. a spin current is absorbed. Under the influenceof the ISHE, they move against the direction of the applied electron current;therefore, this corresponds to a high resistance state and a high voltage signal ismeasured.

However, at elevated temperatures thermal equilibrium magnons in YIG cause in-

2


dividual magnetic moments to precess around the net magnetization with randomphases. In other words, each magnetic moment has both static and dynamic com-ponents (MYIG

= mYIGdc + mYIG

ac ). This makes the difference between the low and highresistance states in the SMR effect smaller than that in the static situation. For ex-ample, when µPt

s ∥ mYIGdc , the spin angular momentum transfer between the Pt and

YIG is not zero, because mYIGac is perpendicular to µPt

s . This is parametrized by theeffective spin mixing conductance, cf Eq. 2.26. As a result, magnons can be injectedand detected electrically.

2.4.3 Electrical method

Magnon injection (SHE) and detection (ISHE)

Using a Pt∣YIG∣Pt heterostructure, magnons may be electrically injected and detectedat elevated temperatures (room temperature is the condition used in the work pre-sented in this thesis). When an electrical current is sent through the first Pt strip, theresulting spin-polarized electrons scatter with the localized orbitals in YIG. This iscalled s-d scattering, where the s-orbital of the mobile electrons and the d-orbital ofthe iron moments are coupled via the exchange interaction. This idea has been for-mulated using second quantization as magnon creation and annihilation operatorsto first predict electrical magnon generation [19]. The resulting magnons possess anenergy of kBT , which is ∼ 6 THz at room temperature. They can propagate in YIGuntil they reach the second Pt electrode, where the reciprocal process occurs so thatmagnons are detected. With the ISHE, an electrical voltage response is measured.In comparison with the SMR measurement where a voltage is measured at the Ptwith current source, this geometry is called nonlocal measurement, since voltage ismeasured not at the Pt strip with the source current but instead at the second one.Analogously, SMR is referred to as local measurement. As depicted in Fig. 2.12, thesymmetry of the nonlocal results are as following [38–40]:

• µPts ∥ mYIG

dc : Magnons are generated and detected most efficiently (µPts ⊥ mYIG

ac ),which gives rise to a high voltage signal in the nonlocal measurement (V high

nl ).By contrast, in the local geometry the SMR shows low voltage response (V low

local)at this condition.

• µPts ⊥mYIG

dc : The final result of magnon injection and detection shows the poor-est efficiency, which results in a low voltage signal in the nonlocal measurement(V low

nl ), while SMR shows high voltage response (V highlocal ).

From the torque point of view, in the first case where µPts ∥ mYIG

dc , µPts at the interface

cannot exert a torque on mYIGdc but on mYIG

ac . In contrast, the pioneering nonlocal

2


YIG

Pt

a High nonlocal voltage

Ma

xim

al m

ag

no

n

ele

ctri

cal i

nje

ctio

n

μs

Pt

M0

YIG

V

YIG

Pt

b Low nonlocal voltage

Min

ima

l ma

gn

on

ele

ctri

cal i

nje

ctio

n

Pt

V

μs

Pt

M0

YIG

Ma

xim

al m

ag

no

n

ele

ctri

cal d

ete

ctio

n

Min

ima

l ma

gn

on

ele

ctri

cal d

ete

ctio

n

Pt

Figure 2.12: Schematic illustration of the nonlocal measurement for electrical magnon in-jection and detection. A typical nonlocal device with two Pt strips on top of YIG film is usedto illustrate two boundary conditions for magnon transport experiment when the polarizationof the spin accumulation in Pt µPt

s and the static magnetization of YIG MYIG0 are a parallel and

b perpendicular to each other. In a, the generated magnons (⊕) diffusively propagate insidethe YIG and are measured at the detector.

experiment on YIG film [21] showed the possibility of electrical magnon generationand detection but on a much larger length scale of several millimeters.

Thermal magnon injection: Spin Seebeck effect

Beyond the electrical spin injection, the Joule heating caused by a current excitesmagnons thermally, which gives rise to a magnon spin current. This effect is relatedto the spin Seebeck effect [41], which describes how a temperature gradient leads toa magnon spin current. In the case of magnetic insulators, magnons and phononsare the carriers for heat transport.

2


YIG

a High nonlocal voltage

Ma

xim

al m

ag

no

n

ele

ctri

cal d

ete

ctio

n

μs

Pt

M0

YIG

V

Pt

YIG

b Low nonlocal voltage

Th

erm

al m

ag

no

n

inje

ctio

n

Pt

V

Pt

μs

Pt

M0

YIG

Min

ima

l ma

gn

on

ele

ctri

cal d

ete

ctio

n

Th

erm

al m

ag

no

n

inje

ctio

n

Pt

Figure 2.13: Schematic illustration of the nonlocal measurement for thermal magnon in-jection and electrical detection. The two boundaries correspond to the maximal and minimalmagnon electrical detection efficiencies when the static magnetization of YIG MYIG

0 and thevoltage detection direction of the Pt strip are a perpendicular and b parallel to each other. Inboth a and b, due to the Joule heating of the Pt heater and the resulting spin Seebeck effect,a magnon current is driven by the temperature gradient. This results in a magnon depletionregion (⊖) near the Pt heater and a magnon accumulation (⊕) region at the bottom of theYIG film. Magnon depletion and accumulation lead to magnon currents with opposite signs,which both propagate in the YIG and can be detected by the detector.

In the nonlocal set-up illustrated in Fig. 2.13, the Joule heating of the injector Ptstrip sets a temperature gradient, which gives rise to a magnon current along thetemperature gradient both in the film plane and perpendicular to the film plane.The resulting temperature gradient stretches beyond the boundary of YIG films, butmagnons are limited to the inside of the magnetic sample, which can lead to a mag-non accumulation at the bottom of thin YIG films [39] (thickness⩽ 210 nm in this

2


thesis). Correspondingly, there is a magnon depletion region near the Pt heater. Thisgives rise to a diffusive magnon current from the accumulation region to the deple-tion region.

The profile of the magnon accumulation/depletion distribution depends mainlyon the thickness of the YIG films and the interface spin opacity of YIG and Pt [42].Thus, the ISHE voltages measured by the Pt detector can have opposite signs de-pending on the distance between the Pt heater and the detector, i.e. whether thedetector is in the depletion or accumulation regime. Besides, the diffusive magnoncurrents resulting from the magnon accumulation at the boundary of the YIG filmcan be detected as well [43]. Thermal magnon generation is less localized in compar-ison with the electrical magnon excitation.

2.4.4 Microwave method

In addition to the electrical approach, one of the most popular stimuli to excite mag-nons is a microwave field, i.e. a radio-frequency (rf) GHz alternating magnetic field.By exposing the magnetic insulator to an rf field, magnons in the GHz regime, i.e.magnetostatic waves, are pumped inductively by the radiation of the rf field. Un-like the electrically generated magnons, microwave excited magnons have definedfrequencies. Thus, the microwave method is magnon-frequency-selective or energy-resolved.

The pumping of magnons via the microwave radiation can be classified into twocategories depending on the relation between the rf field hrf and the static magne-tization MYIG

0 [44, 45]: Perpendicular pumping (hrf ⊥ MYIG0 ) and parallel pumping

(hrf ∥ MYIG0 ). The focus is on the perpendicular pumping configuration, which is

used in Chapter 6. Next, a kinetic instability, which is also known as a Suhl instabil-ity, is discussed for both configurations.

Perpendicular pumping

The microwave field couples with the uniform precession mode (k = 0), where oneobserves a ferromagnetic resonance (FMR) absorption. At this condition, the rf fre-quency ωrf coincides with the uniform precession frequency ωFMR. From the magnondispersion relation (c.f. Eqs. 2.18 and 2.19) one obtains the resonance condition foran in-plane magnetized film as [2]

ωFMR =√ω0(ω0 + ωM), (2.29)

where ω0 = −γµ0H0, ωM = −γµ0Ms, H0 is the internal field calculated from the ap-plied external field (see Eqs. 2.16 and 2.17). Once Eq. 2.29 is fulfilled, all the magneticmoments precess around H0 with the same frequency of ωFMR and the same phase.

2


Three-magnon scattering

c

Two-magnon

scattering

ω2, k2defect

a Four-magnon

scatteringb

ω1, k1

ω1, k1

ω2, k2

ω3, k3

ω1, k1

ω2, k2

ω3, k3

ω4, k4

ω1, k1ω2, k2

ω3, k3

d

Figure 2.14: Illustration of various kinetic magnon processes. Depending on the numberof magnons involved in the scattering events, they are classified into a two-, c, d three- and bfour-magnon scattering processes.

However, when the rf field amplitude reaches a critical value hcrf, kinetic processes, i.e.

magnon-magnon scattering processes, start to play a role and magnons with k ≠ 0 areexcited without direct coupling with the microwave radiation. Low damping mate-rials such as YIG have small hc

rf at a given pumping frequency: For example, a YIGfilm with FMR linewidth of 0.01 mT at 10 GHz has a critical field of 0.03 mT. On theother hand, in this inductive method the coupling strength between the magneticinsulator and the microwave field generator, i.e. a stripline, decays exponentiallywith distance. The following part introduces three types of magnon kinetic pro-cesses, which are named based on the number of magnons involved in the scatteringevents:

• Two-magnon scattering describes the scattering of a magnon with a defect assketched in Fig. 2.14a. This process is energy conserving (ω1 = ω2) but not mo-mentum conserving (k1 ≠ k2). This has been seen as one of the most importantmechanisms for the damping, linewidth for the FMR.

• Four-magnon scattering is shown in Fig. 2.14b. For example, two FMR modespumped by the microwave field scatter with each other (k1 = k2 = 0, ω1 =

ω2 = ωp), which results in two magnons with opposite momentum (k3 = −k4),but one with higher frequency ω3 > ωp and the other with lower frequencyω4 < ωp. Similarly, this process obeys momentum and energy conservation

2


(k1 + k2 = k3 + k4, ω1 + ω2 = ω3 + ω4), which is referred as a second-order Suhlprocess. For a given YIG film and external field, the possible kinetic processesdepend on the dispersion relation as shown in Fig. 2.6. A specific example offour-magnon scattering is shown in Chapter 6.

• Three-magnon scattering is sketched in Figs. 2.14c and 2.14d. A microwave-pumped uniform precession mode (k1 = 0) with frequency of ω1 = ωp candrive two magnons with opposite momenta (k2 = −k3) and frequencies ofω2 = ω3 = ωp/2 (cf. Fig. 2.14d). This process is also referred to as first-orderSuhl process, in which both momentum and energy are conserved (k1 = k2+k3,ω1 = ω2 + ω3). Three-magnon interaction is common in micrometer-thick YIGfilm, whose magnon band minimum frequency ωmin is significantly lower thanthe uniform precession frequency ωFMR, so that ωFMR ∼ ω1 and ω2 = ω3 = ωmin

is energetically possible. The YIG films used in this thesis have submicrometerthicknesses, specifically 210 nm or thinner, where three-magnon interaction isnot so prominent.

Parallel pumping

The precession of a magnetic moment is not necessarily a circle as shown in Fig.2.9,but more close to an ellipse due to magnetic anisotropies: In a thin film the ac-component of the magnetic moment perpendicular to the plane is smaller than in theplane. As a result, the dc component of the magnetic moment acquires a double fre-quency component. Thus, when a parallel rf field couples to this double-precession-frequency mode and the rf field is strong enough, this mode splits into a pair ofmagnons with the opposite momenta with half of the oscillating frequency of thedc-component. This process is referred to as parametric pumping, which is very ef-ficient when the resulting pairs of magnons have the frequency of the magnon bandminimum. Under this condition, a magnon condensed state has been generated atroom temperature.

Spin pumping

In addition to the microwave spectroscopy, spin pumping has been introduced as anelectrical way to characterize the microwave pumped magnon system. Microwave-generated uniform precession acts as a source of pure spin current which is pumpedinto an attached medium in which a pure spin current can be detected using a heavymetal and the SHE [30, 46–53]. This process is called spin pumping. Therefore, witha heterostructure of Pt∣YIG, by bringing YIG into resonance with a microwave field,an ISHE voltage is generated in Pt.

2

2.5. Magnon transport theory 33

Comparing electrical and microwave methods

The major difference between the electrical and microwave methods lies in the typeof magnons they excite: At room temperature, the electrical method generates mag-nons with various frequencies up to THz, which are incoherent. By contrast, themicrowave field can produce magnons with defined GHz frequency, i.e. coherentmagnons, which have been studied for decades [54]. Local thermomagnonic torquetheory [55] describes the interplay between the coherently generated GHz magnonsand the incoherent thermal magnons, which is fundamentally important to under-stand magnon spin transport. In Chapter 6, a transport experiment of magnons gen-erated by the electrical method is conducted in the presence of a microwave fieldwhich simultaneously produces GHz coherent magnons, in order to shed light onthe nature of the electrically generated magnons.

2.5 Magnon transport theory

2.5.1 Nonlocal magnon transport set-up

The theory discussed here is based on nonlocal magnon injection and detection set-ups [39, 40, 56]: Two heavy-metal strips are fabricated on top of a YIG magneticinsulator film. A current is sent through one contact, turning it into a magnon injec-tor, where SHE-induced electron spin accumulation excites magnons in the magnet,as introduced in section 2.4.2. The resulting magnons propagate in the magnet untilthey reach the other contact, the magnon detector, where the magnon signals can bemeasured as an inverse spin Hall voltage.

Since the contacts are generally much longer than the distance between the in-jector and detector and the YIG film is very thin, an assumption of one-dimensionaltransport is used to describe the magnon movement from the injector to detector.

2.5.2 Magnon chemical potential and spin diffusion equation

Because each magnon carries a spin of ∼ h, transport of magnon quasiparticles resultsin a flow of spin current. In a mesoscopic system with the mean free path smallerthan the size of the transport channel, the spin diffusion equation may be used tosolve the resulting one-dimensional transport problem, quite similar to the methodsused to describe electron spin transport in metals [57].

One assumes in the following that the thermalization of magnons is very fast,much faster than the decay of their number. The non-equilibrium state can then be

2


described by the Bose-Einstein distribution function

f(E) =1

exp(E−µmkBTm

) − 1, (2.30)

where µm is the magnon chemical potential, which parametrizes the deviation of themagnon system from the equilibrium (µm = 0 in thermal equilibrium). The relationbetween magnon chemical potential and magnon spin current is the same as thatbetween chemical potential for electron spin accumulation and electron spin current[20, 57]: In the spin diffusion approach, the magnon chemical potential is assumedto depend slowly on position and obey the relation

∇2µm =

µm

λ2m, (2.31)

while a gradient induces the diffusion current

jm =D∇µm (2.32)

where λm is the magnon diffusion length characterizing the relaxation length of thenon-zero magnon chemical potential, and D is the diffusion constant governed bythe magnon spin conductivity σm. These formulas reflect the fact that the quasi-equilibrium state for magnons is reached quickly and magnons start to diffuse in themagnet. Simultaneously, magnon energy relaxes into the crystal lattice via intrinsicGilbert damping or via extrinsic processes such as scattering with defects. This issimilar to the electron spin current in metals: In both cases, angular momentum isnot conserved and energy may be converted into lattice vibrations. Therefore, itis natural that the magnon spin transport shares the diffusion-relaxation equationswith the same form as those of the electron spin current (Eq. 2.31 and Eq. 2.32).

Transport can occur in two regimes depending on the length of the transportchannel, i.e. the distance between the magnon injector and detector d in the non-local setup, compared to the magnon spin diffusion length λm. When d is smallerthan λm, the diffusive transport manifests itself by showing Ohmic behavior, mean-ing that the nonlocal signal is proportional to 1/d. However, the relaxation processstarts to dominate after d exceeds λm, which gives rise to an exponential decay of thenonlocal signals as a function of d. The two distance dependent transport regimeshave been observed in the Pt∣YIG∣Pt nonlocal devices as described in section 2.5.1.From the slope of the exponential decay, a magnon diffusion length of around 10µmis extracted for magnon spin transport at room temperature.

The interface spin resistance between the heavy metal and YIG also affects thedistance dependence of spin transport [29, 42, 57, 58]: When the interface is relativelytransparent, i.e. a spin mixing conductance that is large compared with the magnon

2

2.5. Magnon transport theory 35

spin conductance of the bulk transport channel, both ”1/d”-decay and exponentialdecay regimes can be observed. By contrast, when the interface is opaque, i.e. with asmall spin mixing conductance, only exponential decay can be observed. The formercase corresponds to high-quality Pt∣YIG interfaces, while the latter is observed for aTa∣YIG interface as discussed in Chapter 5.

2.5.3 Energy-dependent magnon transport

The electrical nonlocal magnon spin transport experiment is not energy-resolved.Similarly, in the magnon chemical potential theory approach, after the establishmentof the quasi-equilibrium magnon state, the diffusion and relaxation processes are notintegrated over frequency. In other words, a single magnon diffusion length doesnot capture the feature that magnons with different energy have different transportproperties, such as group velocity, effective mass and relaxation rate. This lack ofenergy resolution may have more profound consequences for the magnon (boson)spin transport than for the electron (fermion) spin transport: In the case of bosons thewhole spectrum contributes to the transport, while for fermions, only the particlesat the Fermi surface play a role in transport. In Chapter 4 of this thesis, anisotropicproperties of the magnon transport are observed, which indicates the contributionfrom magnons in the low energy regime where dipolar energy dominates. Unlikethe magnon Hall effect observed in strong spin-orbit-effect magnets [59], YIG hasweak spin-orbit coupling as discussed in section 1.4. The origin of the anisotropicmagnetotransport of magnons in YIG is likely due to the contribution from dipolarmagnons. Besides, it is found in Chapter 6 that there are circumstances in which thecontribution of the magnons at the band minimum is very significant.

2.5.4 From the linear to nonlinear response regime

If the magnon numbers keep increasing, the response of the magnon system to eitherthe electrical current (Chapter 7) or the microwave power (Chapter 6) will no longerbe linear anymore, meaning that the magnons are in the nonlinear regime. In thiscase, magnon-magnon interaction has to be taken into account [44, 60–65]. In Chap-ter 6, the magnet responds nonlinearly to the high microwave power, whereas inChapter 7 on a three terminal magnon transistor, the nonlocal magnon transport sig-nals from injector to detector respond nonlinearly to the high dc current through themodulator. When the magnon number crosses a critical value, macroscopic conden-sation (magnon Bose-Einstein condensation) can happen, which has been observedfor microwave-generated magnons [66–68] and has been theoretically predicted forelectrically excited thermal magnons [18, 69].

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2.5.5 Summary

To sum up, the magnon spin transport in the linear response regime can be describedby a diffusion equation: Magnon spin current is driven by a gradient of the magnonchemical potential. This theory is in good agreement with the results of the nonlo-cal electrical magnon transport experiment on 210 nm thick YIG film [38]. Also, thisshows the analogy between the magnon spin transport in insulators and the electronspin transport in metals [25, 26]. On the other hand, the difference between themlies in the statistical properties of two (quasi)particles: Electrons are fermions andthe ones propagating in metals have the Fermi energy, whereas magnons are bosonsand particles from the whole spectrum can contribute to magnon propagation. Theelectrical nonlocal magnon transport experiment is not energy-resolved. Differenttechniques such as Brillouin light scattering [45, 67, 68, 70] or NV center nanomag-netometry [71], can be combined to test the roles of magnons with different energy.Moreover, by increasing the strength of excitation sources, the linear response of themagnon system becomes nonlinear. The resulting large effects in the nonlinear re-sponse regime (Chapters 6 and 7) can be interesting for applications.

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Bibliography 37

Bibliography

[1] C. Kittel and P. McEuen, Introduction to solid state physics, John Wiley & Sons, Inc., 8th ed., 2005.[2] D. D. Stancil and A. Prabhakar, Spin waves, Springer, 2009.[3] F. Bloch, “Zur Theorie des Ferromagnetismus,” Zeitschrift fur Physik 61(3-4), pp. 206–219, 1930.[4] 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.[5] A. A. Serga, A. V. Chumak, and B. Hillebrands, “YIG magnonics,” Journal of Physics D: Applied

Physics 43(26), p. 264002, 2010.[6] C. Herring and C. Kittel, “On the theory of spin waves in ferromagnetic media,” Physical Re-

view 81(5), p. 869, 1951.[7] B. A. Kalinikos and A. N. Slavin, “Theory of dipole-exchange spin wave spectrum for ferromagnetic

films with mixed exchange boundary conditions,” Journal of Physics C: Solid State Physics 19(35),p. 7013, 1986.

[8] J. Liu, F. Feringa, B. Flebus, L. J. Cornelissen, J. C. Leutenantsmeyer, R. A. Duine, and B. J. van Wees,“Microwave control of thermal-magnon spin transport,” Physical Review B 99(5), p. 054420, 2019.

[9] J. Barker and G. E. W. Bauer, “Quantum thermodynamics of complex ferrimagnets,” arXiv preprintarXiv:1902.00449 , 2019.

[10] J. S. Plant, “Spinwave dispersion curves for yttrium iron garnet,” Journal of Physics C: Solid StatePhysics 10(23), p. 4805, 1977.

[11] J. Barker and G. E. W. Bauer, “Thermal spin dynamics of yttrium iron garnet,” Physical Review Let-ters 117(21), p. 217201, 2016.

[12] A. J. Princep, R. A. Ewings, S. Ward, S. Toth, C. Dubs, D. Prabhakaran, and A. T. Boothroyd, “Thefull magnon spectrum of yttrium iron garnet,” npj Quantum Materials 2(1), p. 63, 2017.

[13] J. C. Slonczewski, “Current-driven excitation of magnetic multilayers,” Journal of Magnetism and Mag-netic Materials 159(1-2), pp. L1–L7, 1996.

[14] L. Berger, “Emission of spin waves by a magnetic multilayer traversed by a current,” Physical ReviewB 54(13), p. 9353, 1996.

[15] L. D. Landau and E. M. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferro-magnetic bodies,” in Perspectives in Theoretical Physics, pp. 51–65, Elsevier, 1992.

[16] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross,A. Kamra, J. Xiao, et al., “Experimental test of the spin mixing interface conductivity concept,” Phys-ical Review Letters 111(17), p. 176601, 2013.

[17] F. K. Dejene, N. Vlietstra, D. Luc, X. Waintal, J. Ben Youssef, and B. J. van Wees, “Control of spin cur-rent by a magnetic yig substrate in nife/al nonlocal spin valves,” Physical Review B 91(10), p. 100404,2015.

[18] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, “Electronic pumping of quasiequilibrium Bose-Einstein-condensed magnons,” Physical Review Letters 108(24), p. 246601, 2012.

[19] S. S.-L. Zhang and S. Zhang, “Magnon mediated electric current drag across a ferromagnetic insula-tor layer,” Physical Review Letters 109(9), p. 096603, 2012.

[20] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, “Magnon spintransport driven by the magnon chemical potential in a magnetic insulator,” Physical Review B 94(1),p. 014412, 2016.

[21] Y. Kajiwara, K. Harii, S. Takahashi, J.-i. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai,K. Ando, K. Takanashi, et al., “Transmission of electrical signals by spin-wave interconversion in amagnetic insulator,” Nature 464(7286), p. 262, 2010.

[22] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, “Spin-torque ferromagnetic resonance inducedby the spin Hall effect,” Physical Review Letters 106(3), p. 036601, 2011.

2


[23] V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and S. O.Demokritov, “Magnetic nano-oscillator driven by pure spin current,” Nature Materials 11(12), p. 1028,2012.

[24] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, “Electric manipulationof spin relaxation using the spin Hall effect,” Physical Review Letters 101(3), p. 036601, 2008.

[25] M. Johnson and R. H. Silsbee, “Interfacial charge-spin coupling: Injection and detection of spin mag-netization in metals,” Physical Review Letters 55(17), p. 1790, 1985.

[26] F. J. Jedema, A. T. Filip, and B. J. van Wees, “Electrical spin injection and accumulation at roomtemperature in an all-metal mesoscopic spin valve,” Nature 410(6826), p. 345, 2001.

[27] X. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J. Zhang, K. S. M. Reddy, S. D. Flexner,C. J. Palmstrøm, and P. A. Crowell, “Electrical detection of spin transport in lateral ferromagnet–semiconductor devices,” Nature Physics 3(3), p. 197, 2007.

[28] M. Schreier, G. E. W. Bauer, V. I. Vasyuchka, J. Flipse, K.-i. Uchida, J. Lotze, V. Lauer, A. V. Chumak,A. A. Serga, S. Daimon, et al., “Sign of inverse spin Hall voltages generated by ferromagnetic res-onance and temperature gradients in yttrium iron garnet platinum bilayers,” Journal of Physics D:Applied Physics 48(2), p. 025001, 2014.

[29] J. Liu, “Investigation of magnon transport in yttrium iron garnet using platinum and tantalum spininjection/detection electrodes,” Master’s thesis, Rijksuniversiteit Groningen, 2015.

[30] V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van Wees, “Platinum thickness dependence of theinverse spin-hall voltage from spin pumping in a hybrid yttrium iron garnet/platinum system,”Applied Physics Letters 101(13), p. 132414, 2012.

[31] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprags,M. Opel, S. Takahashi, et al., “Spin hall magnetoresistance induced by a nonequilibrium proximityeffect,” Physical Review Letters 110(20), p. 206601, 2013.

[32] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, “Spin-hall magnetoresistancein platinum on yttrium iron garnet: Dependence on platinum thickness and in-plane/out-of-planemagnetization,” Physical Review B 87(18), p. 184421, 2013.

[33] N. Vlietstra, Spin transport and dynamics in magnetic insulator/metal systems. PhD thesis, Rijksuniver-siteit Groningen, 2016.

[34] N. Vlietstra, J. Shan, B. J. van Wees, M. Isasa, F. Casanova, and J. Ben Youssef, “Simultaneous de-tection of the spin-Hall magnetoresistance and the spin-Seebeck effect in platinum and tantalum onyttrium iron garnet,” Physical Review B 90(17), p. 174436, 2014.

[35] A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. W. Bauer, B. Noheda, B. J. van Wees, and T. T. M. Palstra,“Spin-Hall magnetoresistance and spin Seebeck effect in spin-spiral and paramagnetic phases ofmultiferroic CoCr2O4 films,” Physical Review B 92(22), p. 224410, 2015.

[36] G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. T. M. Palstra, and B. J. van Wees, “Negative spin Hallmagnetoresistance of Pt on the bulk easy-plane antiferromagnet NiO,” Applied Physics Letters 111(5),p. 052409, 2017.

[37] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. Goennenwein, E. Saitoh, and G. E. W.Bauer, “Theory of spin Hall magnetoresistance,” Physical Review B 87(14), p. 144411, 2013.

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

[39] J. Shan, Coupled Charge, Spin and Heat Transport in Metal-insulator Hybrid Systems. PhD thesis, Rijk-suniversiteit Groningen, 2018.

[40] L. J. Cornelissen, Magnon spin transport in magnetic insulators. PhD thesis, Rijksuniversiteit Gronin-gen, 2018.

[41] K. Uchida, J. Xiao, H. Adachi, J.-i. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa,

2

Bibliography 39

H. Kawai, et al., “Spin Seebeck insulator,” Nature materials 9(11), p. 894, 2010.[42] J. Shan, L. J. Cornelissen, N. Vlietstra, J. Ben Youssef, T. Kuschel, R. A. Duine, and B. J. van Wees, “In-

fluence of yttrium iron garnet thickness and heater opacity on the nonlocal transport of electricallyand thermally excited magnons,” Physical Review B 94(17), p. 174437, 2016.

[43] J. Shan, L. J. Cornelissen, J. Liu, J. Ben Youssef, L. Liang, and B. J. van Wees, “Criteria for accurate de-termination of the magnon relaxation length from the nonlocal spin Seebeck effect,” Physical ReviewB 96(18), p. 184427, 2017.

[44] M. G. Cottam, Linear and nonlinear spin waves in magnetic films and superlattices, World Scientific, 1994.[45] D. Bozhko, Spin transport in magnon and magnon-phonon gases and condensates. PhD thesis, Technische

Universitat Kaiserslauten, 2017.[46] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, “Enhanced Gilbert damping in thin ferromagnetic

films,” Physical Review Letters 88(11), p. 117601, 2002.[47] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, “Electrical detec-

tion of spin pumping due to the precessing magnetization of a single ferromagnet,” Physical ReviewLetters 97(21), p. 216603, 2006.

[48] C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. Ben Youssef, “Detection of mi-crowave spin pumping using the inverse spin Hall effect,” Physical Review Letters 111(21), p. 217204,2013.

[49] C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. Ben Youssef, “Erratum: Detection ofMicrowave Spin Pumping Using the Inverse Spin Hall Effect [Phys. Rev. Lett. 111, 217204 (2013)],”Physical Review Letters 112(17), p. 179901, 2014.

[50] A. Kapelrud and A. Brataas, “Spin pumping and enhanced gilbert damping in thin magnetic insula-tor films,” Physical Review Letters 111(9), p. 097602, 2013.

[51] K. Harii, T. An, Y. Kajiwara, K. Ando, H. Nakayama, T. Yoshino, and E. Saitoh, “Frequency depen-dence of spin pumping in Pt/Y3Fe5O12 film,” 2011.

[52] R. Takemasa, Y. Tateno, and K. Ando, “Spatial mapping of spin pumping from magnetic insulator,”Applied Physics Letters 110(4), p. 042404, 2017.

[53] V. Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, “Yttrium iron garnet thickness andfrequency dependence of the spin-charge current conversion in yig/pt systems,” Physical ReviewB 90(21), p. 214434, 2014.

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

[55] B. Flebus, P. Upadhyaya, R. A. Duine, and Y. Tserkovnyak, “Local thermomagnonic torques in two-fluid spin dynamics,” Physical Review B 94(21), p. 214428, 2016.

[56] K. S. Das, Controlling spins in nanodevices via spin-orbit interaction, magnons and heat. PhD thesis,Rijksuniversiteit Groningen, 2019.

[57] S. Takahashi and S. Maekawa, “Spin injection and detection in magnetic nanostructures,” PhysicalReview B 67(5), p. 052409, 2003.

[58] J. Liu, L. J. Cornelissen, J. Shan, B. J. van Wees, and T. Kuschel, “Nonlocal magnon spin transportin yttrium iron garnet with tantalum and platinum spin injection/detection electrodes,” Journal ofPhysics D: Applied Physics 51(22), p. 224005, 2018.

[59] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, “Observation of the magnonHall effect,” Science 329(5989), pp. 297–299, 2010.

[60] P. W. Anderson and H. Suhl, “Instability in the motion of ferromagnets at high microwave powerlevels,” Physical Review 100(6), p. 1788, 1955.

[61] H. Suhl, “The theory of ferromagnetic resonance at high signal powers,” Journal of Physics and Chem-istry of Solids 1(4), pp. 209–227, 1957.

[62] H. Suhl, “Origin and use of instabilities in ferromagnetic resonance,” Journal of Applied Physics 29(3),

2


pp. 416–421, 1958.[63] M. T. Weiss, “Microwave and low-frequency oscillation due to resonance instabilities in ferrites,”

Physical Review Letters 1(7), p. 239, 1958.[64] I. D. Mayergoyz, G. Bertotti, and C. Serpico, Nonlinear magnetization dynamics in nanosystems, Elsevier,

2009.[65] S. M. Rezende and F. M. de Aguiar, “Spin-wave instabilities, auto-oscillations, and chaos in yttrium-

iron-garnet,” Proceedings of the IEEE 78(6), pp. 893–908, 1990.[66] 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 underpumping,” Nature 443(7110), p. 430, 2006.

[67] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S.Lvov, and B. Hillebrands, “Supercurrent in a room-temperature Bose–Einstein magnon condensate,”Nature Physics 12(11), p. 1057, 2016.

[68] A. J. E. Kreil, D. A. Bozhko, H. Y. Musiienko-Shmarova, V. I. Vasyuchka, V. S. Lvov, A. Pomyalov,B. Hillebrands, and A. A. Serga, “From kinetic instability to Bose-Einstein condensation and magnonsupercurrents,” Physical Review Letters 121(7), p. 077203, 2018.

[69] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A. Duine, “Two-fluid theory for spin superfluidity inmagnetic insulators,” Physical Review Letters 116(11), p. 117201, 2016.

[70] A. A. Serga, C. W. Sandweg, V. I. Vasyuchka, M. B. Jungfleisch, B. Hillebrands, A. Kreisel, P. Kopi-etz, and M. P. Kostylev, “Brillouin light scattering spectroscopy of parametrically excited dipole-exchange magnons,” Physical Review B 86(13), p. 134403, 2012.

[71] C. Du, T. van der Sar, T. X. Zhou, P. Upadhyaya, F. Casola, H. Zhang, M. C. Onbasli, C. A. Ross, R. L.Walsworth, Y. Tserkovnyak, et al., “Control and local measurement of the spin chemical potential ina magnetic insulator,” Science 357(6347), pp. 195–198, 2017.

university of groningen controlled magnon spin transport

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