Appendix a Properties of Yttrium–Iron–Garnet (YIG)

Appendix A Properties of Yttrium–Iron–Garnet (YIG)

Structure and Physical Properties

Chemical formula Y3Fe5O12 Crystal structure Cubic Number of formula units per unit cell 8 Lattice constant (25 ◦C) 12.376 A˚ Mass density (25 ◦C) 5172 kg/m3 Octahedral Sublattice 27 −3 Site density Na 8.441 × 10 m Angular momentum Ja 5/2 Land´e g factor ga 2 Magnetization at 0 K Ma (0 K) 391.5 kA/m Tetrahedral Sublattice 27 −3 Site density Nd 12.66 × 10 m Angular momentum Jd 5/2 Land´e g factor gd 2 Magnetization at 0 K Md (0 K) 587.2 kA/m Molecular Field Constants [1]

Octahedral λaa 735.84 Tetrahedral λdd 344.59 Nearest-neighbor λad 1100.3 Phenomenological Exchange Constant [2] 2 −16 2 Deﬁned by hex = λex∇ m (3.104) λex 3 × 10 m Macroscopic Magnetic Properties

Magnetization at 0 K Mtot (0 K) 196 kA/m Magnetization at 298 K Mtot (298 K) 140 kA/m Curie temperature Tc 559 K

333 334 References

Anisotropy Constants 3 First-order cubic (4.2 K) [3] Kc1 (4.2 K) −2480 J/m 3 First-order cubic (295 K) [3] Kc1 (295 K) −610 J/m 3 Second-order cubic (4.2 K) [4] Kc2 (4.2 K) −118.0J/m 3 Second-order cubic (273 K) [4] Kc2 (273 K) −26.0J/m Optical Properties [5–7] Refractive index (1.2 μm) n 2.2 Faraday rotation (1.2 μm) φF 240 deg/cm Absorption coeﬃcient (1.2 μm) α 0.069 cm−1 μ 2 − × −4 Cotton–Mouton tensor elements (1–3 m) g44MS 1.14 10 2 × −5 ΔgMS 5.73 10

Cotton–Mouton rotation (1.14 μm) ΦCM −160 deg/cm

References

[1] E. E. Anderson, “Molecular field model and magnetization of YIG,” Phys. Rev., vol. 134, p. A1581, 1964. [2] E. H. Turner, “Interaction of phonons and spin waves in yttrium iron garnet,” Phys. Rev. Lett., vol. 5, p. 100, 1960. [3] P. Hansen, “Anisotropy and magnetostriction of gallium-substituted yttrium iron garnet,” J. Appl. Phys., vol. 45, p. 3638, 1974. [4] P. Escudier, “L’anisotropie de l’aimantation; un paramétre important de l’étude de l’anisotropie magnétocrystalline,” Ann. Phys. (Paris), vol. 9, p. 125, 1975. [5] M. J. Weber, Ed., CRC Handbook of Laser Science and Technology, Vol. IV Optical Materials Part 2: Properties. Boca Raton, FL: CRC Press, 1986, p. 291. [6] R. V. Pisarev, I. G. Sinii, N. N. Kolpakova, and Y. M. Yakovlev, “Magnetic birefringence of light in iron garnets,” Sov. Phys. JETP, vol. 33, p. 1175, 1971. [7] G. A. Smolensky, R. V. Pisarev, I. G. Sinii, and N. N. Kolpakova, “Cotton- Mouton birefringence of ferrimagnetic garnets,” J. Physique Colloque, vol. C1 32, pp. c1–1048, 1971. Appendix B Currents in Quantum Mechanics

B.1 Density of States

The valence electrons in a metal can be modeled as free electrons contained in a box the size of the sample. Because of the boundary conditions at the faces of the box, the allowed values of wave number k are quantized. The actual conditions at the surface of a metal may be very complicated, but much of solid-state physics depends primarily on the fact that the modes are quantized, rather than on exactly what the boundary conditions of a particular sample happen to be. Consequently, it is common to make the simple assumption of periodic boundary conditions. In this assumption, the wave function of an electron at x = L should equal to that at x = 0, where L is the length of the sample along the x-direction. For this to be true, we have exp(ikL) = exp(0) = 1, which gives the discrete values of wave number k =2πn/L, where n is an integer. In k space, the total number of states N in a sphere of radius |k| is given by the volume of the sphere divided by the volume occupied by one state. Thus

4 1 4πk3L3 N =2 πk3 =2 , (B.1) 3 (2π/L)3 3(2π)3 where we have assumed the sample is a cube with length L on each side so that 3 the minimum volume in k space is ΔkxΔkyΔkz =(2π/L) . The factor of 2 accounts for the degeneracy in the energy of spin-up and spin-down electrons.1 In the free electron approximation, the Hamiltonian H is merely the kinetic energy operator (cf. Table 1.1)

p2 2 H = = − ∇2 , (B.2) 2m 2m 1 When we consider ferromagnetic metals, this degeneracy is lifted by the interaction with the internal exchange ﬁeld causing a splitting of the energy band diagram.

335 336 Appendix B Currents in Quantum Mechanics and when we assume an electron wavefunction of the form ψ ∼ eik·r,the Schr¨odinger equation Hψ = Eψ yields

2k2 = E. (B.3) 2m Combining (B.1) and (B.3), the number of states with energy less than E can be written 4πL3 2m 3/2 N =2 E3/2. (B.4) 3(2π)3 2 The density of states is deﬁned as the number of states per unit energy, or dN g(E)= . (B.5) dE Thus, the density of states is given by 4πL3 2m 3/2 3 g(E)=2 E1/2 (B.6) 3(2π)3 2 2 L3 2m 3/2 √ = E. (B.7) 2π2 2

The Fermi energy is deﬁned as the highest occupied energy state by an electron, at absolute zero temperature. Interactions at higher temperatures 2 2 typically occur near the Fermi energy, EF = kF/(2m), since electrons at the Fermi energy have nearby empty states. Consequently, the current is often calculated at the Fermi energy while assuming the parabolic band approximation g(E) ∝ E1/2. Multiplying and dividing (B.7) by E and comparing the result with (B.4), we see that the relationship between the Fermi energy and the density of states can be expressed 3 N g(EF)= . (B.8) 2 EF At ﬁnite temperature, some of the electrons will be excited to states above the Fermi level. In this case, the density of states is related to the number of particles by ∞ N = g(E)f(E)dE , (B.9)

0 where − − 1 f(E)= 1+e(E EF)/(kBT ) (B.10) is the Fermi–Dirac distribution describing the probability of ﬁnding an elec- −23 tron with energy E at temperature T and kB =1.38 × 10 J/Kisthe Boltzmann constant. B.2 Electric and Spin Current Densities 337

Note that the above expressions assume degenerate spin-up and spin-down subbands. In a ferromagnetic metal, the molecular field interacting with the electron spin splits the bands as illustrated in Figure 10.1. If the magnetization is oriented upwards, then the Hamiltonian with the effects of the molecular field is (cf. Eq. (2.30)) −2 d 2 H = + μ μ σH , (B.11) 2m dx 0 B m where Hm is the molecular field and σ = ±1. Defining Em = μ0μBσHm,the density of states for the spin subbands can be expressed 1 g (E)= g (E − σE ) , (B.12) σ 2 m and the total density of states is given by g(E)= gσ(E) (B.13) σ = g↑(E)+g↓(E) (B.14) 1 = [g (E − E )+g (E + E )] . (B.15) 2 m m (B.16)

B.2 Electric and Spin Current Densities

Consider ψ to be a solution of the 1D Schr¨odinger equation

∂ψ 2 i = − ∇2ψ. (B.17) ∂t 2m Multiplying (B.17) by ψ∗ and subtracting the complex conjugate of the re- sulting equation, we obtain ∂ −i (ψ∗ψ)+ ∇·[ψ∗∇ψ − ψ∇ψ∗]=0. (B.18) ∂t 2m In obtaining (B.18), we have used the vector identity

∇·(f∇g)=∇f ·∇g + f∇2g. (B.19)

If we define the probability density of finding an electron as ψ∗ψ,wecan express the charge density as ρ = qψ∗ψ. We can also define a current density j that satisfies the conservation equation ∂ρ + ∇·j =0. (B.20) ∂t 338 Appendix B Currents in Quantum Mechanics

Multiplying (B.18) by the electronic charge q and comparing with (B.20), we make the identiﬁcation −iq j = [ψ∗∇ψ − ψ∇ψ∗] 2m (B.21) q = Im(ψ∗∇ψ). m √ Choosing ψ =1/ keikx for the 1D case ensures the normalization dψ Im ψ∗ =1. (B.22) dx Under the two-band approximation, we treat the spin-up and spin-down carriers as being independent. Under this two-channel approximation, we can borrow standard techniques from quantum mechanics to solve for the probability of carrier transmission across an interface, or even through a barrier with multiple interfaces.

B.3 Reﬂection and Transmission at a Boundary

Consider an electron that is incident on a planar interface between two media as illustrated in Figure B.1. The electron will be reﬂected with probability

Medium 1 Medium 2

t r x

Fig. B.1. Geometry for calculating the reﬂection and transmission coeﬃcients of a particle at a boundary between two materials. amplitude r, and transmitted with probability amplitude t. In medium 1, the wave function is

ψ1 = ψinc + ψref (B.23) 1 − = √ eik1x + re ik1x , (B.24) k1 (B.25) while in medium 2, the wave function is B.4 Tunneling Through a Barrier 339 t ik2x ψ2 = ψtr = √ e . (B.26) k2 The reﬂection and transmission coeﬃcients can be found by requiring both the wave function and its derivative to be continuous at the boundary x =0. − + Requiring ψ1(0 )=ψ2(0 ) gives 1 1 √ (1 + r)=√ t. (B.27) k1 k2 Similarly, requiring dψ/dx to be continuous at x = 0 gives k1 (1 − r)=t k2. (B.28)

Solving (B.27) and (B.28) simultaneously gives the desired reflection and transmission amplitudes: k − k r = 1 2 , (B.29) k + k √1 2 2 k k t = 1 2 . (B.30) k1 + k2 It is straightforward to show that the probabilities of reflection and transmission sum to 1: r2 + t2 =1. (B.31) These equations can be readily applied to the transmission and reflection of spin-up and spin-down electrons at an interface between a normal metal and a ferromagnetic metal. To do this, we make the identifications k1 = kN and k2 = kσ.Thus,wehave

kN − kσ rσ = , (B.32) kN + kσ √ 2 kN kσ tσ = , (B.33) kN + kσ and 2 2 rσ + tσ =1. (B.34)

B.4 Tunneling Through a Barrier2

Consider the quantum mechanical tunneling of particles with energy E through a potential barrier ) V −a/2

Fig. B.2. Tunneling through a potential barrier. A particle incident from the left has a ﬁnite probability of tunneling through the barrier, even though the particle energy E is less than the barrier height V . where V>E(Figure B.2). Assuming a free particle solution to the Schr¨odinger equation Hψ = Eψ, the wavefunction will have the form (for 1D problem)

ψ = Ae+ikx + Be−ikx, (B.35) where the two terms represent plane waves of amplitude A and B traveling along the +xˆ and −xˆ directions, respectively. The wavenumber k is a measure of the oscillatory nature of ψ and can be obtained by substituting (B.35) into the Schr¨odinger equation 2 − ∇2ψ + U(x)=Eψ (B.36) 2m to yield 1 − k(x)= 2m(E U(x)). (B.37) We write the wavefunction for the particle in the three regions as3 ⎧ ⎨ Aeikx + Be−ikx x<−a/2, −κx κx − ψ(x)=⎩ Ce + De a/2

Since U(x) is piecewise constant, we can define 1 √ k = 2mE (B.39) outside the barrier, where U(x) = 0, and within the barrier U(x)=V and we define the real quantity κ: 1 − κ = 2m(V E). (B.40) 2 We rely heavily on the mathematics in [1]. 3 The normalization is not important in this case, since we are interested in the ratio of the transmitted to incident amplitudes, and the initial and final media both have the same k. References 341

We interpret 1/κ as an exponential decay length characteristic of tunneling through a barrier. The wavefunction in each region is determined by requiring that both ψ and dψ/dx be continuous across the interfaces x = −a/2and x = a/2. The boundary conditions at x = −a/2 yield the equations

Ae−ika/2 + Beika/2 = Ceκa/2 + De−κa/2, (B.41) ikA e−ika/2 − ikB eika/2 = −κC eκa/2 + κD e−κa/2, (B.42) while the boundary conditions at x = a/2 give

Ce−κa/2 + Deκa/2 = Feika/2, (B.43) −κC e−κa/2 + κD eκa/2 = ikF eika/2. (B.44)

Normalizing the equations by A, we have four equations in the four unknown normalized coeﬃcients. The ratio of the transmitted and incident wave amplitudes can thus be uniquely determined as [1]

F e−ika = , (B.45) A cosh κa + i(ε/2) sinh κa where κ k ε = − . (B.46) k κ We are usually interested in quantities such as the transmissivity of the barrier deﬁned as 2 F T = . (B.47) A For a high and wide barrier (κa 1), we obtain κk 2 T ≈ 16 e−2κa . (B.48) k2 + κ2

Note, however, that when we apply a bias between the two ferromagnetic electrodes, the potential U(x) will contain a linear term. Consequently, the solutions for ψ(x) within the barrier will no longer have the exponential form e±κx and will instead be replaced by Airy functions.

References

[1] E. Merzbacher, Quantum Mechanics, 3rd ed. New York: John Wiley & Sons, 1998. Appendix C Characteristics of Spin Wave Modes

This appendix provides a summary of the characteristics of spin wave modes, as described in Chapters 5 and 6. We identify

• H0 as the total internal field, • MS as the saturation magnetization, • HDC as the applied DC bias field, • g as the Lande’ g factor (2 for spin), • q and m as the charge (negative) and effective mass, respectively, of an electron, and • d as the film thickness • w as the width of the microstrip transducer • s as the separation between the transducer and the surface of a magnetic film and define

ω0 = −γμ0H0,

ωM = −γμ0MS, gq γ = (which is negative). 2m

The coordinate system is generally deﬁned so that H0 and HDC are along the zˆ direction. For simplicity, magnetocrystalline anisotropy is neglected in the following equations.

C.1 Constitutive Tensors

C.1.1 Polder Susceptibility Tensor

χ −iκ χ¯ = , iκ χ

343 344 Appendix C Characteristics of Spin Wave Modes where ω ω ωω χ = 0 M and κ = M . 2 − 2 2 − 2 ω0 ω ω0 ω

C.1.2 Permeability Tensor ⎡ ⎤ 1+χ −iκ 0 ⎣ ⎦ μ = μ0 iκ 1+χ 0 . 001

C.2 Uniform Precession Mode Frequencies

• Normally magnetized ﬁlm with H0 = HDC − MS,

ω = ω0 = −γμ0(HDC − MS).

• Tangentially magnetized ﬁlm with H0 = HDC, 1/2 ω =[ω0(ω0 + ωM)] . • Sphere, ω = −γμ0HDC.

C.3 Spin Wave Resonance Frequencies Assuming pinned boundary conditions:

• Normally magnetized ﬁlm with H0 = HDC − MS, nπ 2 ωn = ω0 + ωMλex d ,n=1, 2, 3,...

• Tangentially magnetized ﬁlm with H0 = HDC, nπ 2 nπ 2 1/2 ω = ω + ω λ ω + ω λ + ω , n 0 M ex d 0 M ex d M n =1, 2, 3,...

C.4 General Magnetostatic Field Relations

h = −∇ψ, m = χ¯ · h,

b = μ0 (1 + χ¯) · h, ∇×e = iωb, kk · m h = − , k2 ωμ e = − 0 k × m. k2 C.5 Forward Volume Spin Waves 345 C.5 Forward Volume Spin Waves

Normally magnetized ﬁlm

HDC

H0 = HDC − MS

Dispersion relation: − − ktd 2 1 e ω = ω0 ω0 + ωM 1 − . ktd

Group velocity: 1 χκ 2 = − ktd . vg (1 + χ)ωMd χ

For lowest order n = 0 mode, 1 4 = . v ω d g ktd=0 M Loss: 2 2 2 2 1 ω0 + ω ω0 + ω = |γ|μ0ΔH = α . Tk 4ω0ω 2ω0

Potential function: ⎧ · − ⎪ψ ektd/2 cos( −(1 + χ) k d/2) eikt r ktz,z>d/2, ⎨ 0 t (e) ikt·r ψ (r)= ψ0 cos( −(1 + χ) ktz) e , |z|≤d/2, ⎪ ⎩ ktd/2 ikt·r+ktz ψ0 e cos( −(1 + χ) ktd/2) e ,z<−d/2.

Normalization: For a mode power of P , mW/mm, 2 4P ψ0 = . −(1 + χ)ωμ0kd 346 Appendix C Characteristics of Spin Wave Modes

Radiation resistance of mth mode with wavenumber km: ⎧ 2 ⎨ ωμ0 2 kmd F − cos −(1 + χ) ,m=0, 2, 4,..., r = [ (1+χ)]kmd 2 I rm ⎩ ωμ 2 k d 2 0 sin m −(1 + χ) F ,m=1, 3, 5,..., [−(1+χ)]kmd 2 I where F − sin(kmw/2) = e kms . I kmw/2

C.6 Backward Volume Spin Waves

Tangentially magnetized ﬁlm

HDC

H0 = HDC Dispersion relation: − − kz d 2 1 e ω = ω0 ω0 + ωM . kzd

Group velocity: 1 χκ 2 k d = + z . vg ωMd χ 1+χ

For lowest order n = 1 mode, 1 4 ω0(ω0 + ωM) = − . v ω d ω g kz d=0 M 0

Loss: 2 2 2 2 1 ω0 + ω ω0 + ω = |γ|μ0ΔH = α . Tk 4ω0ω 2ω0 Potential function: ⎧ ⎪ kz d/2 iνkz z−kz y ⎨⎪ψ0 e sin(kyd/2)e ,y>d/2, (o) iνkz z ψ (r)= ψ0 sin(kyy)e , |y|≤d/2, ⎪ ⎩ kz d/2 iνkz z+kz y −ψ0 e sin(kyd/2)e ,y<−d/2. C.7 Surface Spin Waves 347

Normalization: For a mode power of P , mW/mm, 2 4P ψ0 = . ωμ0kzd

Radiation resistance of mth mode with wavenumber km: ⎧ ⎪ ωμ 2 k d 2 ⎨ 0 sin √ m F ,m=1, 3, 5,..., kmd 2 −(1+χ) I rrm = ⎪ ωμ k d 2 ⎩ 0 cos2 √ m F ,m=2, 4, 6,..., kmd 2 −(1+χ) I

where F − sin(km w/2) =e kms . I kmw/2

C.7 Surface Spin Waves

Tangentially magnetized ﬁlm

HDC

H0 = HDC Dispersion relation (k>0):

ω2 ω2 = ω (ω + ω )+ M 1 − e−2kd . 0 0 M 4 Or equivalently, − 1 4 − 2 k = ln 1+ 2 ω0(ω0 + ωM) ω . 2d ωM 348 Appendix C Characteristics of Spin Wave Modes

Group velocity: 1 4ω = e2kd v ω2 d g M 1 4 ω0(ω0 + ωM) = . v ω d ω g kz d=0 M M Loss: 1 |γ|μ0ΔH = (2ω0 + ωM)=α(ω0 + ωM/2). Tk 4ω Potential function: ⎧ ⎪ψ ekd + p(ν) e−ky+iνkx,y>d/2, ⎨ 0 ky −ky iνkx ψν (r)= ψ e + p(ν)e e , |y|≤d/2, ⎪ 0 ⎩ kd ky+iνkx ψ0 1+p(ν)e e ,y<−d/2, ψ − χ − νκ where p(ν) ≡ 0 = e−kd, ψ0+ χ +2+νκ χ +2− νκ p(ν)= ekd. χ + νκ

Normalization: For a mode power of P , mW/mm, 2 P ψ0ν = . −(1 + χ)ωμ0p(ν)kd

Radiation resistance: 2 (ν) μ0ω 1+χ F r = ,ν= ±1. r 2kd (1 + νκ)2 − (1 + χ)2 I

where, for a SW mode with wavenumber k,

F sin(kw/2) = e−ks . I kw/2 Appendix D Mathematical Relations

D.1 Trigonometric Identities

2cosα cos β =cos(α + β)+cos(α − β) 2sinα sin β =cos(α − β) − cos(α + β) 2sinα cos β =sin(α + β)+sin(α − β) tan(θ − π/2) = − cot θ sin 2θ =2sinθ cos θ cos 2θ =cos2 θ − sin2 θ cosh2(x) − sinh2(x)=1 tanh2(x)+sech2(x)=1 coth2(x) − cosh2(x)=1

D.2 Vector Identities and Deﬁnitions

∇·(∇×A)=0 ∇×∇ϕ =0 ∇·∇ϕ = ∇2ϕ A · (B × C)=C · (A × B)=B · (C × A) ∇·(A × B)=B · (∇×A) − A · (∇×B) ∇×(A × B)=A∇·B − B∇·A +(B ·∇)A − (A ·∇)B ∇·(fG)=G · (∇f)+f(∇·G) ∇·(f∇g)=∇f ·∇g + f∇2g (M ×∇) × r = −2M

349 350 Appendix D Mathemati cal Relations

k · k = kk − k2I (kk) = k k ⎡ ij i j ⎤ 0 −kz ky ⎣ ⎦ k ≡ k × I = kz 0 −kx −ky kx 0 k × A = k · A

Divergence Theorem: A · ds = ∇·A dv. S V Stokes Theorem: (∇×A) · ds = A · dl.

S C

D.3 Fourier Transform Deﬁnitions

∞ ∞ dω f(t)= e−iωtf(ω),f(ω)= dt eiωtf(t), 2π −∞ −∞ ∞ ∞ d3k f(r)= eik·rf(k),f(k)= d3re−ik·rf(r). (2π)3 −∞ −∞ Index

Angular momentum, 6–12, 37 steady state, 295 addition of, 20 Birefringence orbital, 10, 11, 16 circular, 223 Anisotropic Bragg diffraction, 253 linear, 223 Anisotropic medium, 113 Bistability, 267, 268 Anisotropy Bogoliubov transformation, 276, 278, cubic, 86 303 energy density, 84 Bohr magneton, 23, 70 field, 85, 86, 90, 107 Boltzmann factor, 70 tensor, 87 Bose-Einstein statistics, 43 uniaxial, 303, 323, 325 Bosons, 43, 57 Annihilation operator, 57 Boundary conditions, 206, 208 Anomalous dispersion, 289 Bragg diffraction, 253 Antiferromagnetism, 4 Brillouin function, 72, 73, 77, 79, 107 Array factor, 191 Brillouin zone, 49 Asteroid, 103 Canonical equations, 302 Auto-oscillations, 295 Canonical variables, 35, 54, 271, 272, fingers, 266 275, 276, 278, 302, 304 Charge density, 112 Backward volume waves, 158, 160, 161, Circular polarization, 123 328 Commutation relation, 15, 16, 21, 54, dispersion relation, 159, 167 59, 272 group velocity, 160 Commutator, 15–18 insertion loss, 197 Compensation point, 78 normalization, 182 Conservation of charge, 121 orthogonality relation, 181 Constitutive relations, 112, 113, 139 potential functions, 160 Continuity equation, 115, 121 radiation resistance, 195, 196, 200 Conversion length, 251, 260 return loss, 196 Cotton-Mouton effect, 223, 252, 253, solitons, 291 334 stationary formulas, 213 Coulomb gauge, 34, 37 Bifurcation Coulomb potential, 24 Hopf, 295 Coupled-mode equations, 238

351 352 Index

Coupling coefficient, 247, 251 Energy velocity, 127, 169, 170 Creation operator, 57 Equipartition theorem, 322 Cubic anisotropy, 86 Euler-Lagrange equation, 35, 205, 208 Curie Law, 72 EuO, 75, 76 Curie temperature, 5, 73, 82 Exchange interaction, 3, 33, 76, 141, Curie-Weiss law, 73 311 energy, 81, 82 Damping, 94 field, 81, 82, 93 Decay instability, 280 Pauli spin exchange operator, 45, 64 Demagnetizing field, 145, 318 YIG, 142 Demagnetizing tensor, 146, 149, 150, Exchange spin waves, 141 319 Density of states, 310 False nearest neighbor, 299 Diamagnetism, 3, 11, 67 Faraday rotation, 135, 223, 236, 240 susceptibility, 106 Faraday’s law, 34, 68, 225, 259 Dielectric waveguide, 224 Fermi energy, 310, 336 Dipolar spin waves, 134, 142 Fermi-Dirac distribution, 336 Dipole gap, 265, 290, 293, 298, 300–302 Fermi-Dirac statistics, 43, 310 Dispersion energy density, 211 Fermions, 43 Dispersion relation, 49, 55, 64, 122 Ferrimagnetism, 4 backward volume waves, 159, 167 Ferromagnetic resonance, 98, 108 dielectric waveguide, 227, 228, 257, foldover, 267 258 line width, 199 ferrite, 128, 130 Ferromagnetism, 3 forward volume waves, 166 Field displacement non-reciprocity, 165, isotropic medium, 122 197 magnetostatic wave, 140, 142 Foldover, 267, 269 surface waves, 163, 167 Forward volume waves, 155, 157 Dispersive medium, 113 bistability, 269 Divergence theorem, 114, 229 dispersion relation, 166 Domains, 3–5 group velocity, 155 Doppler effect, 309, 325, 327 insertion loss, 192, 193 Dyadic product, 122 normalization, 180 Dynamical interaction, 62 orthogonality relation, 179 potential function, 156 Effective magneton number, 72 radiation resistance, 191, 192, 200 Eigenfunction, 14–17 return loss, 192 Eigenvalue, 14, 17, 18, 21–23, 29 stationary formulas, 212, 214 Electric energy density, 119 Fourier transform, 58, 96, 124, 125, 191, Electric field intensity, 111 242, 276, 289, 322, 323 Electric flux density, 111 Functional, 204, 208 Electric volume current density, 111 Functional derivative, 205 Electromagnetic wave, 142 extraordinary, 135 Gadolinium gallium garnet, 184 ordinary, 135 Gilbert damping, 95 Elliptical polarization, 123 Group velocity, 126, 171 Energy density backward volume waves, 160 electric, 119 forward volume waves, 155 magnetic, 119 surface waves, 163 Index 353

Gyromagnetic ratio, 9–12, 69 Lorentz reciprocity theorem, 229 Lowering operator, 53 Hamiltonian, 14, 36–38, 42, 46, 47, 54, Lumped element model, 185 58, 63 Harmonic oscillator, 50, 62 Magnetic circular birefringence, 223 classical, 50, 271 Magnetic energy density, 119, 211 eigenfunction, 51, 52 Magnetic field intensity, 111 operators, 56, 57 Magnetic flux density, 112 quantum mechanical, 50 Magnetic linear birefringence, 223 Heisenberg Hamiltonian, 39, 44, 46, 64 Magnetic susceptibility tensor, 67 Heisenberg uncertainty principle, 13, Magnetization, 69 16, 23 Magnetocrystalline anisotropy, 84 Hermite polynomials, 51 Magnetoquasistatics, 139 Hermitian matrices, 117 Magnetostatic approximation, adjoint, 55 132, 169 conjugate, 63 Magnetostatic limit, 132 Hilbert transforms, 187 Magnetostatic modes, 140 Holstein-Primakoff transformation, 57, Magnetostatic scalar potential, 140 275 Magnetostatic waves, 142 Hund’s rules, 26, 28 Magnons, 50, 57, 62, 63 ground state, 74, 75 Maxwell’s equations, 34, 111, 112, 120–122, 129, 132, 151, 209, 225, Impedance, 185, 186 231, 237, 243, 258 Information dimension, 300 Micromagnetics, 104, 322, 323 Inner product, 46 mid-point rule, 105 Insertion loss, 187 Microstrip, 183, 184 Interaction bandwidth, 256 Modulational instability, 286 Iron group, 28 Molecular field, 73, 75, 76 Isotropic medium, 113, 122 Néel temperature, 5, 79 Kerr effect, 252, 324 Néel theory of ferrimagnetism, 76 Kinematical interaction, 62 Nonlinear Schrödinger equation, 304 Kramers-Kronig relations, 187 Kronecker delta, 88 Orbital, 24 Orbital angular momentum, 68 Lagrange multipliers, 219, 220 Orthogonality relation, 178 Lagrangian, 35, 206, 210, 212, 220, 263, backward volume waves, 180 271, 272 forward volume waves, 178 Landé g factor, 12, 22, 30, 72 surface waves, 182 Landau-Lifshitz equation, 91, 95, 104, TE modes, 230 108, 151, 282, 311 TM modes, 230 Langevin diamagnetism, 69 Larmor precession, 11, 38, 104 Paramagnetic susceptibility, 72, Left-hand polarization, 123 78, 107 Legendre equation, 147 Paramagnetism, 3, 70 polynomials, 148 Pauli exclusion principle, 42, 57 Lenz’s law, 3, 67 Pauli spin matrices, 29 Limit cycle, 295, 296 Permeability, 112, 113 Linear polarization, 123 tensor, 139 354 Index

Permittivity, 112, 113, 236, 237, 258, pulse train, 291 259 spin wave bullet, 291 tensor, 135 Spin current, 311–314 Phase matching, 240, 253 density, 312, 314, 337 Phase velocity, 120 Spin torque, 310, 312, 316, 317 Polarization, 122 Spin transfer torque, 319 circular, 123 Spin wave manifold, 64, 151 density, 112 Spin wave resonance, 142, 143, 166 elliptic, 123 Spin waves, 33, 134, 141 left-hand, 123 Spin-orbit coupling, 26, 84 linear, 123 Spinors, 30 right-hand, 123 Stationary formulas, 41, 212 Polder susceptibility tensor, 92 Stoke’s theorem, 233 Poynting theorem, 115, 117 Stoner-Wohlfarth model, 102, 108 Poynting vector, 115, 126, 170, 326 Subshell, 24–26 Precession frequency, 7, 9–11, 29, 166, Subsidiary absorption, 278 318, 319, 324 Suhl processes, 278 Propagation loss, 176 first instability, 292 Pseudo-energy density, 210, 211 first order, 278 second instability, 280, 296 Quantum numbers, 24 second order, 279 Quasi-particle number density, 212 subsidiary absorption, 278 Surface waves, 158, 162, 164, 166, 328 Radiation reactance, 187 dispersion relation, 163, 167 Radiation resistance, 188 group velocity, 163 backward volume waves, 195, 196, insertion loss, 199 200 normalization, 183 forward volume waves, 191, 192, 200 orthogonality relation, 182 surface waves, 196, 198 potential function, 165 Raising operator, 53 radiation resistance, 196, 198 Rare earths, 28 return loss, 198 Reciprocal lattice vector, 49 stationary formulas, 212 Reflection coefficient, 187, 339 Susceptibility, 73 Relaxation time, 171, 172 antiferromagnet, 79 Residue theorem, 97 diamagnetism, 69, 106 Resonance frequency, 93 EuO, 75 line width, 101, 108 paramagnetism, 72, 78, 107 Return loss, 187 tensor, 67, 91, 92 Right-hand polarization, 123 Russell-Saunders coupling, 84 TE modes, 224 Thermal average, 71 Saturation magnetization, 85 Torque, 6, 68, 106, 311, 317 Schrödinger equation, 47, 50, 51, 285 current loop, 8 time dependent, 36 Torque equation, 92, 93, 145, 150, 151, time independent, 14, 33 210, 283, 318 Screening, 24 Total angular momentum, 21 Shell, 24, 26, 27 Transducer, 184, 185 Soliton, 286, 287, 304 Transduction loss, 187 Lighthill criterion, 289 Transition elements, 28 Index 355

Transmission line, 185 Wave function, 13, 24 Transverse susceptibility, 94 Wave impedance, 134 Wave number, 120 Uniaxial anisotropy, 84 Wave packet, 126 Uniform plane waves, 120 Wave vector diagram, 253 Uniform precession mode, 144, 166, 171 Wavelength, 120 Weiss theory of ferromagnetism, 73 Variational derivative, 205 Yttrium iron garnet, 5, 23, 28, 78, 333 Walker’s equation, 139, 140, 158, 162, exchange constant, 142 220 Wave equations, 122 Zeeman energy, 37, 70, 82, 102, 210, 220