Introduction to antiferromagnetic magnons

Cite as: J. Appl. Phys. 126, 151101 (2019); https://doi.org/10.1063/1.5109132 Submitted: 06 May 2019 . Accepted: 22 September 2019 . Published Online: 15 October 2019

Sergio M. Rezende, Antonio Azevedo, and Roberto L. Rodríguez-Suárez

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J. Appl. Phys. 126, 151101 (2019); https://doi.org/10.1063/1.5109132 126, 151101

© 2019 Author(s). Journal of Applied Physics TUTORIAL scitation.org/journal/jap

Introduction to antiferromagnetic magnons

Cite as: J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132

Submitted: 6 May 2019 · Accepted: 22 September 2019 · View Online Export Citation CrossMark Published Online: 15 October 2019

Sergio M. Rezende,1,a) Antonio Azevedo,1 and Roberto L. Rodríguez-Suárez2

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1Departamento de Física, Universidade Federal de Pernambuco, 50670-901, Recife, Pernambuco, Brazil 2Facultad de Física, Pontifícia Universidad Católica de Chile, Casilla 306, Santiago, Chile

a)Author to whom correspondence should be addressed: [email protected]

ABSTRACT The elementary spin excitations in strongly magnetic materials are collective spin deviations, or spin waves, whose quanta are called magnons. Interest in the experimental and theoretical investigation of magnons attracted many groups worldwide about 4–6 decades ago and then waned for some time. In recent years, with the advent of the field of spintronics, the area of magnonics has gained renewed atten- tion. New phenomena have been discovered experimentally, and others have been predicted theoretically. In this tutorial, we briefly review the basic concepts of magnons in antiferromagnetic (AF) materials. Initially, we present a semiclassical view of the equilibrium spin configu- rations and of the antiferromagnetic resonance in AF materials with two types of magnetic anisotropy, easy-axis and easy-plane. Then, we present a quantum theory of magnons for these materials and apply the results to two important AF insulators, MnF2 and NiO. Finally, we introduce the concept of antiferromagnetic magnonic spin current that plays a key role in several phenomena in antiferromagnetic spintronics.

Published under license by AIP Publishing. https://doi.org/10.1063/1.5109132

I. INTRODUCTION In his 1970 Nobel lecture, Louis Néel2 stated that “Antiferromagnetic materials are extremely interesting from the Antiferromagnets are magnetic materials that have no net ” macroscopic magnetization and, therefore, are almost insensitive to theoretical viewpoint, but do not seem to have any application. external magnetic fields. The existence of antiferromagnetism was This pessimistic view of antiferromagnets would change dramati- proposed by Louis Néel1 to account for the magnetic behavior of cally a few decades later. Today, these materials have practical some salts with complicated lattice structures. Néel first assumed applications and promise to deliver several more. The debut of that the ordered magnetic arrangements could be described in antiferromagnets in technology was made possible with the discov- “ ” ery of the giant magnetoresistance effect (GMR) in 1988 by Albert terms of spin sublattices, each having all spins aligned in the 3 4 same direction. Then, he considered that the interaction between Fert and Peter Grünberg. They showed that in magnetic struc- the spins in different sublattices could be negative so that they tures with nanometer thick multilayers, the electron transport would be aligned antiparallel. Now we know that the coupling could be controlled by the spin instead of the electric charge as in between spins originates mainly in the exchange interaction, with conventional electronics. This discovery triggered research in mag- magnitude represented by an exchange parameter J. The intra- netic multilayers and gave birth to the field of spintronics, which sublattice J is positive because it is due to direct exchange, while has revolutionized magnetic recording technologies and promises the intersublattice J is negative because it arises from the super- new functionalities to electronic devices. exchange that is mediated by ligand ions between the magnetic Antiferromagnetic materials proved to be essential in GMR ions. If the sublattices have different magnetizations, the materials sensors for pinning the magnetization of a reference ferromagnetic – are called “ferrimagnets,” which have spontaneous net magnetiza- layer by means of the exchange bias phenomenon.5 7 Since the late tion below the ordering temperature Tc and in many respects 1990s, disk-drive magnetic read-heads are based on GMR sensors behave like ferromagnets. Materials with J , 0 that have sublattices that are much more sensitive to changes in magnetic fields than the with the same magnetization are called “antiferromagnets.” In the traditional induction heads. Although important, antiferromagnetic ordered phase, below a critical temperature called Néel temperature (AF) materials have had a passive role in spintronic devices; the TN , they are magnetic but have no net macroscopic magnetization. active role is played by ferromagnetic materials. However, in recent

J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132 126, 151101-1 Published under license by AIP Publishing. Journal of Applied Physics TUTORIAL scitation.org/journal/jap

years, this scenario began to change with new experimental and constant. The important exchange interactions are illustrated in theoretical results showing that antiferromagnets have several Fig. 1(a). In MnF2, the exchange parameters determined by inelas- 19 advantages over ferromagnets in spintronic phenomena. One of tic neutron scattering measurements are J1 ¼ 0:028 meV, them is the fact that AF materials are insensitive to external mag- J2 ¼0:152 meV, and J3 ¼0:004 meV. Thus, the negative inter- netic perturbations. Another one is the ultrafast dynamics of anti- sublattice J2 is the dominant interaction and determines the antifer- ferromagnets that promise device operation in the terahertz frequency romagnetic arrangement of the sublattices. range. These late developments gave rise to the field of antiferro- In order to find the spin equilibrium configuration, we use a 8–14 ~ ~ magnetic spintronics. macrospin approximation and associate to Si and Sj the uniform In this tutorial, we review the basic concepts and properties of sublattice magnetizations in sublattices 1 and 2, respectively, given ~ ~ the collective spin excitations in antiferromagnets, the spin waves, by M1,2 ¼ γ hN Si,j, where N is the number of spins per unit whose quanta are called magnons. Antiferromagnets are known to volume. For T ¼ 0, the static sublattice magnetizations have the exist with a great variety of crystalline structures and physical proper- same value, M1 ¼ M2 ¼ M. Considering a magnetic field H0 ties, and new materials continue to be discovered.13 Here, we will applied along the c axis, the z-direction of the system shown in restrict attention to simple AF insulators with only two sublattices, Fig. 1, from Eq. (1), we have the energy per unit volume one class with uniaxial magnetic anisotropy, also called easy-axis anisotropy, such as the fluorides MnF and FeF , and the other with H H 2 2 E ¼H (M þ M ) þ E M~ M~ A (M2 þ M2 ), (2) two anisotropy axes, or easy-plane anisotropy, such as the oxides 0 1z 2z M 1 2 2M 1z 2z NiO and MnO. where HE and HA are the effective exchange and anisotropy fields II. SPIN CONFIGURATIONS IN SIMPLE defined by ANTIFERROMAGNETS ¼ =γ ¼ =γ We consider initially two simple antiferromagnets with uniax- HE 2SzJ2 h, HA 2SD h, (3) ial, or easy-axis, anisotropy, MnF2 and FeF2. Both have the rutile crystal structure, a body-centered tetragonal lattice with the mag- where we have considered only intersublattice exchange interaction netic ions occupying the corner and body-centered positions, as between the z nearest neighbors. Due to the symmetry, the sublat- 15 tice magnetizations and the applied field are in the same plane so shown in Fig. 1(a). Below the Néel temperature, 66.5 K for MnF2 that the energy in Eq. (2) becomes and 78.4 K for FeF2, and in the absence of an external magnetic field, the spins are arranged in two oppositely directed sublattices, θ θ pointing along the easy anisotropy direction (c axis). The magnetic E( 1, 2) ¼H0(cos θ1 þ cos θ2) þ HE cos(θ1 þ θ2) properties are described very well by a Hamiltonian consisting of M contributions from Zeeman, exchange, and magnetic anisotropy HA 2 2 – (cos θ þ cos θ ), (4) energies in the form16 19 2 1 2 X X X ~ ~ ~ z 2 θ θ ¼γ ~ þ 0 0 where 1 and 2 are the polar angles of the corresponding sublattice H h H0 Si 2Jii Si Si D (Si ) , (1) i i;i0=i i magnetizations. The conditions @E=@θ1 ¼ @E=@θ2 ¼ 0 give two possible equilibrium configurations. As illustrated in Fig. 1(b), θ1 ¼ γ ¼ μ = θ ¼ π where g B h is the gyromagnetic ratio, g is the spectroscopic 0, 2 corresponds to the antiferromagnetic (AF) phase, while splitting factor, μ is the Bohr magneton, h is the reduced Plank θ ¼ θ ¼ cos1[H =(2H H )] corresponds to the spin-flop ~ B ~ 1 2 0 E A constant, Si is the spin (in units of h) at a generic lattice site i, H0 (SF) phase. Substitution of these angles in Eq. (4) shows that the AF is the applied magnetic field, Jii0 is the exchange constant of the phase has lower energy for fields H0 , HSF, while the SF phase has 1=2 ~ ~ 0 . ¼ 2 interaction between spins Si and Si and D is the uniaxial anisotropy lower energy for H0 HSF, where HSF (2HEHA HA) .For

FIG. 1. (a) Crystal structure of the easy-axis rutile antifer- 2+ romagnets MnF2 and FeF2. The ions M represent Mn or Fe2+. (b) Spin configurations in the antiferromagnetic (AF) and spin flop (SF) phases with the external field applied along the easy axis.

J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132 126, 151101-2 Published under license by AIP Publishing. Journal of Applied Physics TUTORIAL scitation.org/journal/jap

small fields applied in the z-direction, the spins are along the c axis in the AF phase. As the field intensity increases and reaches HSF, the system undergoes a first-order phase transition to the SF phase. The magnetic interactions in MnF2 and FeF2 are dominated by nearest-neighbor exchange, having comparable intersublattice 15 exchange fields, respectively, HE ¼ 526 kOe and HE ¼ 540 kOe. fi 2+ In MnF2, the ground state con guration of the magnetic Mn ions 5 6 is 3d ( S5/2), which has no orbital angular momentum so that it has very small single-ion anisotropy. The origin of the magnetic anisotropy of MnF2 relies mainly in the dipolar interaction, with a 19,20 relatively small effective anisotropy field HA ¼ 8:2 kOe so that 1=2 HSF (2HEHA) ¼ 93 kOe. The temperature dependence of the spin-flop field has been measured by ultrasonic techniques,21 anti- 22 ff 23 ferromagnetic resonance, and spin Seebeck e ect. In FeF2, the fi 2+ 5 5 ground state con guration of the magnetic Fe ions is 3d ( D4), which has a finite orbital angular momentum, resulting in a large effective anisotropy field HA ¼ 200 kOe arising from the spin-orbit 24,25 coupling. Thus, the spin-flop field is large, HSF ¼ 505 kOe, and can be attained only with pulsed magnetic fields.26 FIG. 2. Crystal structure and spin arrangements in the easy-plane antiferromag- The other material we will investigate here is NiO, which, 2− net NiO. The small yellow circles represent O ions and the large circles repre- due to its simple structure and spin interactions, is considered a sent the Ni2+ ions. prototypical room-temperature antiferromagnetic insulator. Its magnetic structure and spin interactions are similar to those in MnO.27,28 This material has been extensively used in experimental Considering the external field applied along one of the – – 0 investigations of many phenomena, such as exchange bias,5 7,29 31 z -directions in the {111} plane, with Eq. (5), we obtain for the inelastic light scattering,32,33 and magnetic response at terahertz energy per unit volume – frequencies.34 36 More recently, it has been shown that a thin HE ~ ~ HAx 2 2 layer of NiO in spintronic devices can be used to transport a mag- E ¼H0(M1z0 þ M2z0 ) þ M1 M2 þ (M þ M ) – 1x 2x nonic spin current while blocking charge current.37 42 Also, NiO M 2M HAz 2 2 has the potential to generate radiation with terahertz frequency in 0 þ 0 (M1z M2z ), (6) spin-torque nano-oscillators,43 and under a thermal gradient it 2M ff 44 exhibits the spin Seebeck e ect at room temperature. where, with the relevant intersublattice exchange constant, the In the paramagnetic phase, NiO has the face-centered cubic effective exchange field is given by the same expression as in structure of sodium chloride. Below the Néel temperature fi 2+ Eq. (3), but now we have two anisotropy elds TN 523 K, the Ni spins are ordered ferromagnetically in {111} h i planes, lying along 112 axes, with adjacent planes oppositely HAx ¼ 2SDx=γ h, HAz ¼ 2SDz=γ h: (7) magnetized due to a super-exchange AF interaction, as illustrated in Fig. 2. It is characterized by two distinct anisotropies, a negative In order to simplify calculations, we assume a monodomain one (hard) along h111i axes that forces the spins into the {111} sample with the field applied along an easy direction. In this case, h i planes, and a positive (easy) in-plane one along 112 axes. The M1x ¼ M2x ¼ 0 and Eq. (6) reduces to Eq. (2) so that the result spin Hamiltonian with Zeeman energy, exchange interaction previously obtained is valid here. In NiO, the effective fields are ¼ ¼ : ¼ : fi energy, and out-of-plane (x) and in-plane (z) anisotropy energies HE 9 684 kOe, HAx 6 35 kOe,pHffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAz 0 11 kOe so that the eld 28 47 can be written as for the spin-flop transition HSF ¼ 2HEHAz is 46.3 kOe. X X X 0 2 III. ANTIFERROMAGNETIC RESONANCE: THE k =0 H ¼γ h H~ ~S þ 2J ~S ~S þ D (Sx )2 D (Sz ) 0 i,j ij i j x i,j z i,j : MAGNONS i,j i,j i,j (5) We now derive the resonance frequencies of the uniform pre- cession modes of the sublattice magnetizations, which correspond The difference to Eq. (1) is that the Hamiltonian now contains to the magnons with zero wave vector. The equations of motion for the magnetization components are obtained from the Landau– two anisotropy terms, with anisotropy constants Dx and Dz. Both are taken to be positive so that the signs in Eq. (5) imply that x is a Lifshitz equation hard direction along a h111i axis, while z0 are easy directions along ~ dM1,2 ~ ~ h112i axes. Since there are three equivalent directions in the easy- ¼ γM1,2 Heff 1,2, (8) dt plane and four {111} planes, in bulk NiO crystals there are AF ~ domains with spins in 12 different directions, which complicates where Heff 1,2 represent the effective fields that act on the sublattice the interpretation of some magnetic measurements.45,46 magnetizations, given by

J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132 126, 151101-3 Published under license by AIP Publishing. Journal of Applied Physics TUTORIAL scitation.org/journal/jap

γ þ þ þ þ ω γ ¼ ~ ¼∇ ~ : m1x(H0 HE HAx HAz) i m1y HEm2x 0, (10b) Heff 1,2 M~1,2[E(M1,2)] (9)

Since the uniaxial antiferromagnet is a particular case of the γ þ ω þ γ ¼ biaxial AF, we will consider the energy in Eq. (6), with the static HEm1y i m2x m2y (H0 HE HAz) 0, (10c) field applied along a direction of equilibrium so that the result can be used for both easy-axis and easy-plane AFs. Writing for the ~ ¼ ^ þ ^ þ ^ iω t magnetizations M1,2 zM1,2z (xm1,2x ym1,2y) e , we obtain þγ HEm1x γ m2x(H0 HE HAx HAz) þ iω m2y ¼ 0, (10d) with Eqs. (8) and (9) the linearized equations of motion for the transverse components of the sublattice magnetizations where we have considered M1z ¼ M, M2z ¼M. The antiferro- magnetic resonance (AFMR) frequencies are the eigenvalues of the ω þ γ þ þ þ γ ¼ i m1x m1y(H0 HE HAz) HEm2y 0, (10a) resonance matrix [A], given by

2 3 00γH (A C) B 6 0 7 6 00B γH0 þ (A C) 7 [A] ¼ 4 5, (11) γH0 (A þ C) B 00 B γH0 þ (A þ C)0 0

where the parameters are ωα,β ¼ γ(Hc + H0), (16)

A ¼ γ (HE þ HAx=2 þ HAz), B ¼ γ HE, C ¼ γ HAx=2: (12) where

Equations (10) can be written as an eigenvalue equation in 1=2 H ¼ (2H H þ H2 ) , H ¼ H : (17) matrix form c E A A A Az Thus, the two modes are degenerate in the absence of an [A][S] ¼ [S][ω], (13) external magnetic field. The application of an external field H0 in where the columns of [S] are eigenvectors of [A] that represent the the direction of the anisotropy lifts the degeneracy. Equation (16) T shows that the frequency of mode α increases with an increasing four normal modes with components (m) ¼ (m1x, m2x, im1y, im2y) , ω field, while the frequency of mode β decreases with an increasing and [ ] is the diagonal matrix of the eigenvalues, which are given fi by the roots of eld. In order to characterize the behavior of the sublattice magne- tizations in the AFMR, one has to solve the full eigenvalue equa- ¼ fi det{[A][ω]} ¼ 0: (14) tion. For H0 0, it can be shown that Eq. (13) is satis ed by the eigenvector matrix16 Note that the problem can also be formulated with the equations of 2 3 ~ motion for the Néel vector l ¼ m~ m~ and the total small-signal ηα ηα ηβ ηβ 1 2 6 7 magnetization m~ ¼ m~ þm~ .43 6 11117 1 2 [S] ¼ 4 5, (18) ηα ηα ηβ ηβ 1111 A. Easy-axis antiferromagnets where Initially, we apply the results for the case of easy-axis AFs by ¼ ¼ setting HAx 0. The solution of Eq. (14) with C 0 gives for the þ þ þ 16 HE HA Hc HE HA Hc AFMR frequencies (eigenvalues) ηα ¼ , ηβ ¼ : (19) HE HE pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ω1 ¼ω2 ¼ γH0 þ A B , (15a) Hence, the components of the sublattice magnetizations in the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi two modes are 2 2 ω3 ¼ω4 ¼ γH0 A B : (15b) 0 1 0 1 0 1 0 1 m1x ηα m1x ηβ B C B C B C B C B m2x C B 1 C B m2x C B 1 C We shall use α to denote modes 1 and 2, and β for modes 3 @ A ¼ Aα@ A, @ A ¼ Aβ@ A, (20) im ηα im ηβ and 4. From Eqs. (12) and (15), the positive frequencies for the two 1y 1y im im modes are 2y α 1 2y β 1

J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132 126, 151101-4 Published under license by AIP Publishing. Journal of Applied Physics TUTORIAL scitation.org/journal/jap

where Aα and Aβ are amplitude factors that depend on the driving experiments in MnF2 at microwave frequencies to measure field. Equations (20) show that the two sublattice magnetizations the absorption lineshapes and obtain information about the are nearly opposite to each other, and they precess circularly in the damping mechanisms.49,50 This technique was also used in same sense, counterclockwise in mode α and clockwise in mode β. Ref. 22 to measure the limit of stability of the AF phase in α In the -mode, the ratio between the precession amplitudes of the MnF2. As shown in Fig. 4(a), measurement of the AFMR two sublattices is m1x=m2x ¼ηα, which is larger than unity in frequency vs field and extrapolation to zero-frequency yields a absolute value. Thus, the angle of M~ with the z axis is larger than precise value for the field H (T). ~ 1 c the angle of M2, as illustrated in Fig. 3 (a). On the other hand, in The complete picture of the field dependence of the two β = ¼η the -mode, m1x m2x β, which is smaller than unity in abso- AFMR frequencies in MnF2 at low temperatures was measured in lute value, showing that the angle of M~ with the z axis is smaller Ref. 51 using a combination of microwave millimeter and sub- ~ 1 than the angle of M2,asinFig. 3(a). millimeter sources. Figure 3(b) shows the data of Ref. 51 and the Note that the effect of the anisotropy field is enhanced by results of the calculations described here for the three phases of the interplay with exchange, and this gives rise to the gap in spin arrangement in MnF2. In the AF phase, according to Eq. (16), the spectrum. The opposite directions of the anisotropy fields as the field increases, the frequency of the mode with counterclock- for the two sublattice magnetizations make it impossible for wise precession (α) increases, as in a ferromagnet. On the other them to precess in the same direction, and this brings the hand, the frequency of the clockwise precession mode (β) decreases exchange field into play. The interplay between the anisotropy with an increasing field. As we will show in Sec. IV B, in the and the large exchange interaction results in a magnetization spin-flop (SF) phase, the frequency of one mode is zero and the ω ¼ γ 2 2 1=2 precession with frequencies that are much larger in antiferro- other is 0 (H0 Hc ) , as in the data of Fig. 3(b). Note that magnets than in ferromagnets. This fast magnetization dynam- if the field is applied perpendicularly to the c axis, the equilibrium ics constitutes one of the motivations for pushing research in positions of the sublattice magnetizations are at some angle with – antiferromagnetic spintronics.9 14 the field, characterizing a canted phase. In this case, the two modes Equation (16) shows that the frequency of the β-mode are degenerate and there is no spin-flop transition, as will be shown decreases with an increasing field and vanishes for H0 ¼ Hc, the in Sec. IV C. Figure 4(b) shows a comparison of the phase dia- fi fi eld value that de nes the limit of stability of the AF phase. Note grams for MnF2 measured by various techniques, as presented in that this value is larger than the one obtained before for the ther- Ref. 23. The AFMR has also been investigated in detail in FeF2 modynamic transition, but for small anisotropy, the two expres- using a conventional far-infrared monochromator24 and also with 1=2 sions are approximately the same HSF Hc (2HEHA) . With transmission measurements employing the radiation of molecular 52,53 no applied field, the AFMR frequency is ω0 ¼ γHc. Using for far-infrared lasers. ¼ γ ¼ : = ¼ MnF2 g 2, 2 8 GHz kOe, and Hc 93 kOe, this gives for ¼ : the AFMR frequency 260 GHz. Using for FeF2 g 2 25 and Hc ¼ 505 kOe, we obtain 1.59 THz. B. Easy-plane antiferromagnets The AFMR frequencies have been measured in MnF2 and fi In the case of the biaxial, or easy-plane, antiferromagnet, FeF2 with several techniques. The rst measurements in MnF2, Eqs. (11) and (14) yield for the two AFMR frequencies made in zero field by Johnson and Nethercot48 with a millime- ter microwave spectrometer, showed very broad lineshapes, but 2 2 2 2 2 2 ωα β ¼ (A þ γ H ) (C þ B ) served to give the temperature dependence of the AFMR fre- , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 quency. By applying a large magnetic field along the c axis, + γ2 2 2 2 þ 2 2: Jaccarino and coauthors managed to carry out AFMR 2 H0 (A B ) B C (21)

FIG. 3. (a) Illustration of the preces- sions of the sublattice magnetizations of the two modes of antiferromagnetic resonance in an easy-axis antiferro- magnet. (b) Symbols represent the measured magnetic field dependence of the AFMR frequencies in MnF2 at 1.8–5.0 K, with the field applied parallel and perpendicular to the c-axis.51 The solid lines represent the field depen- dencies of the frequencies calculated with Eqs. (16) and (17) for the AF phase, with Eqs. (61) for the SF phase, and with Eqs. (59) and (67) for ~ ~ the canted phase (H0?c).

J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132 126, 151101-5 Published under license by AIP Publishing. Journal of Applied Physics TUTORIAL scitation.org/journal/jap

22 FIG. 4. Experimental data for MnF2. (a) Measurements of the AFMR frequency vs applied field at two temperature values. Reprinted with permission from S. M. Rezende et al., Phys. Rev. B 16, 1126 (1977). Copyright 1977 American Physical Society. (b) Phase diagram for MnF2 measured with various techniques: ultrasonic attenuation (US);21 differential magnetization (M);21 antiferromagnetic resonance (AFMR);22 and spin Seebeck effect (SSE).23 Reprinted with permission from S. M. Wu et al., Phys. Rev. Lett. 116, 097204 (2016). Copyright 2016 American Physical Society.

Using the definition of the parameters in Eqs. (12), for In order to obtain the eigenstates of the two uniform magnon HE HAx, HAz,wehave modes in the AF phase of easy-plane antiferromagnets, we consider H0 ¼ 0 and use for the eigenvalues (frequencies) ω1 ¼ω2 ¼ ωα ω2 ffi γ2 þ þ 2 and ω3 ¼ ω4 ¼ ωβ, given by Eq. (21). The solution of the eigen- α,β {HE(HAx 2HAz) H0 value Eq. (13) gives for the eigenvector matrix + 2 þ þ 2 2 1=2 : [4H0 HE(HAx 2HAz) HEHAx] } (22) 0 1 εα εα εβ εβ B C Considering HAx HAz, appropriate for NiO, the two fre- εα εα εβ εβ [S] ¼ B C, (25) quencies become simply @ 111 1 A 1111 ω2 γ2 þ 2 ω2 γ2 2 : α0 (2HEHAx 3H0 ), β0 (2HEHAz H0 ) (23) where On the other hand, for H0 ¼ 0, regardless of the relative fi ωα ωβ values of the anisotropy elds, Eq. (22) gives εα ¼ , εβ ¼ , (26) A (B C) A þ (B þ C) ω2 ¼ γ2 þ ω2 ¼ γ2 : α0 2HE(HAz HAx), β0 2HEHAz (24) are the ellipticities, defined as ε ¼ mx=imy. Using Eqs. (12) and (24), and considering H H H , appropriate for NiO, Clearly, in the absence of the hard-axis anisotropy, in zero E Ax Az Eqs. (26) give for the ellipticities field, the two modes are degenerate, as seen earlier for the uniaxial AF. The presence of the hard-axis anisotropy lifts the degeneracy   1=2 1=2 of the two modes. This feature gives NiO the property of transport- ε 2HE ε HAz : 54 α , β (27) ing a magnonic spin current even with no applied field. HAx 2HE Equations (23) show that as the field increases, the frequency of the α-mode increases, while the one for the β-mode decreases. The Using the parameters for NiO, Eqs. (27) give for the elliptici- 1=2 β-mode frequency goes to zero at a field HSF (2HEHAz) , which ties εα 55 and εβ 1=419. From Eqs. (25) and (27), we see that represents the limit of stability of the AF phase. Using the values of in the α-mode M~ precesses counterclockwise about the z axis, ~ 1 the effective fields for NiO, HE ¼ 9 684 kOe, HAx ¼ 6:35 kOe, while M2 precesses clockwise, with very elliptical trajectories, with HAz ¼ 0:11 kOe, and g = 2.18, we find for the zero-field frequencies the major axis oriented along the x axis (hard anisotropy axis). The approximately 1.07 THz and 0.140 THz. The AFMR of the higher precessions in opposite directions give rise to an oscillating magne- frequency mode, called optical mode, has been measured with tization component in the y-direction, illustrated in Fig. 5. On the 34,36 ~ several techniques. other hand, in the β-mode, M1 precesses clockwise about the z

J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132 126, 151101-6 Published under license by AIP Publishing. Journal of Applied Physics TUTORIAL scitation.org/journal/jap

18,56,57 ai that destroys a spin deviation ! y 1=2 þ = a a S ¼(2S)1 2 1 i i a , (29a) 1i 2S i

! y 1=2 = y a a S ¼(2S)1 2 a 1 i i , (29b) 1i i 2S

z ¼ y : S1i S ai ai (29c)

For the down-spin sublattice, the raising spin operator is related to the operator that creates a spin deviation, and we have ! y 1=2 þ = y b bj FIG. 5. Illustration of the precessions of the sublattice magnetizations of the S ¼(2S)1 2b 1 j , (30a) two modes of antiferromagnetic resonance in an easy-plane (or hard-axis) 2j j 2S antiferromagnet. ! y 1=2 b b ¼ 1=2 j j ~ S2j (2S) 1 bj, (30b) axis, while M2 precesses counterclockwise, with very elliptical tra- 2S jectories, with the major axis oriented along the y axis, giving rise to an oscillating magnetization component in the x-direction, as in z ¼ þ y Fig 5. In Sec. IV D, we shall return to the discussion of the eigen- S2j S bj bj, (30c) modes in easy-plane AFs in connection with the spin waves in y y these materials. where ai , ai,andbj , bj, are the creation and destruction operators for spin deviations at sites i,j of sublattices 1 and 2, which satisfy the y y ¼ δ 0 0 ¼ ¼ δ 0 IV. ANTIFERROMAGNETIC MAGNONS boson commutation rules [ai, ai0 ] ii ,[ai, ai ] 0, [bj, b j0 ] jj 0 ¼ and [bj, b j ] 0. The next step consists of introducing a transforma- In this section, we present a quantum formulation of spin tion from the localized field operators to collective boson operators waves in antiferromagnets. We treat the quantized excitations of y that satisfy the commutation rules, [ak, a 0 ] ¼ δ 0 ,[ak, a 0 ] ¼ 0, the magnetic system with the approach of Holstein–Primakoff y k kk k [bk, b 0 ] ¼ δ 0 ,[bk, b 0 ] ¼ 0, (HP),55 which consists of transformations expressing the spin oper- k kk k X ~ X ~ ators in terms of boson operators that create or annihilate quanta 1=2 ik:~r 1=2 ik:~r a ¼ N e i a , b ¼ N e j b , (31) of spin waves, or magnons. This approach was developed for ferro- i k j k k k magnets, but it also works well for antiferromagnets.56 We treat ~ separately the cases of the easy-axis AFs, such as MnF2 and FeF2, where N is the number of spins in each sublattice and k is a wave and that of the easy-plane AF, such as NiO and MnO. vector, and we have the orthonormality condition

X 0 ~~ :~ 1 i(k k ) ri ¼ δ 0 : A. Easy-axis antiferromagnets: AF phase N e k,k (32) i Here we consider that the spins of the two sublattices point in the direction +z of the symmetry axis, as in Fig. 1(b), and write Introducing in Eq. (28) the transformations (29) and (30) the Hamiltonian (1) for the easy-axis AF in terms of raising and with the binomial expansions of the square roots, and using lowering spin operators Eqs. (31) and (32), we obtain a Hamiltonian with the form (2) (4) (6) X X X H ¼ E0 þ H þ H þ H þ, where each term contains ¼γ ~ ~ þ þ þ þþ z z z 2 H h H0 Si,j Jij(Si Sj Si Sj 2Si Sj ) D (Si,j) , an even number of boson operators. The quadratic part of the i,j i=j i,j Hamiltonian is (28) X (2) ¼ γ þ þ y þ þ y H h (HE HA H0)akak (HE HA H0)bkbk where the subscripts i and j refer to the sites in sublattices 1 and 2, k respectively. In the first HP transformation, the components of the þγ þ y y kHE(akbk akbk), (33) local spin operators are related to the creation and annihilation P γ fi γ ¼ = ~ ~δ operators of spin deviations. For the up-spin sublattice, the lower- where k is a structure factor de ned by k (1 z) δ exp(i k ), y ~δ ing spin operator is related to the operator ai that creates a spin are the vectors connecting the z nearest neighbors in opposite sub- deviation, while the raising spin operator is related to the operator lattices, and we have considered only intersublattice exchange with

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the effective fields defined as in Eq. (3). The next step consists of per- forming canonical transformations from the collective boson opera- y y tors ak, ak, bk, bk into magnon creation and annihilation operators αy α βy β 18 k, k, k, k. This is done with the Bogoliubov transformation

¼ α βy ak uk k vk k,(34a)

y ¼ α þ βy : bk vk k uk k (34b)

Substituting these expressions in Eq. (33) and imposing that the Hamiltonian be cast in the diagonal form X (2) ¼ ω αyα þ ω βyβ H h( αk k k βk k k), (35) k

FIG. 6. Magnon dispersion relations for MnF2 with no applied field with the ωα ωβ where k and k are the frequencies of the two magnon modes, c fi ffi wave vector along the -axis. Symbols represent inelastic neutron scattering one can nd the frequencies and the transformation coe cients. data at three temperatures,58 and the solid lines represent the dispersions calcu- These are given by lated for the same temperatures with Eqs. (36) and (38) and with the magnon energy renormalization in Eqs. (40) and (41).

ωα ¼ ω þ γ ωβ ¼ ω γ k k H0, k k H0, (36a)

2 2 2 1=2 The decrease in frequency with increasing temperature is due ωk ¼ γ[2HEHA þ H þ H (1 γ )] (36b) A E k to the magnon energy renormalization. This was calculated using the terms of the Hamiltonian with four magnon operators, and obtained by substitution of the transformations (29)–(31) and (37)  in Eq. (28), and retaining only those that have pairs of creation- γ þ γ þ ω 1=2 annihilation operators for the same mode, in any order, such as HE HA k y y y y y y uk ¼ , (37a) α α α α α α β β β β β β 2ω k1 k2 k3 k4, k1 k2 k3 k4, and k1 k2 k3 k4. Then, we use a k random-phase approximation57 to replace one of the pairs by  its thermal average such that the term has the same form 1=2 α αy βy β Δ ~ þ~ ~ ~ γH þ γH ω as in Eq. (35). For example, k1 k4 (k1 k4 k2 k3)is ¼ E A k : y y k2y k3 y vk (37b) replaced by hα α iβ β ¼ (nα þ 1)β β þ nβ α α , where nα β is 2ωk q q k k q k k q k k q, q the thermal occupation number for the α-orβ-mode with ~ – ffi wave vector q, given by the Bose Einstein distribution Notice that the transformation coe cients satisfy the ortho- ωα β = nα β ¼ 1=(eh q, q kBT 1). The averages of other pairs of operators 2 2 ¼ q, q y y normality condition uk vk 1. Using for the body-centered α α β β α α β β such as k1 k2 k3 k4,or k1 k3 k2 k4, vanish or do not contribute tetragonal structure of MnF2 and FeF2, the vectors connecting ~δ ¼ +^ = + ^ = + ^ = to the energy of either mode. Applying this approximation to each nearest neighbors x(a 2) y(a 2) z(c 2), one can show four-magnon term, the Hamiltonian reduces to a quadratic form that the geometric structure factor is X (4) y y γ ¼ = = = : H ) h Δωα α α þ Δωβ β β , (39) k cos(kxa 2) cos(kya 2) cos(kzc 2) (38) k k k k k k k Note that for magnons with k ¼ 0, γ ¼ 1, and Eqs. (36) are k Δω Δω the same as Eq. (16) for the AFMR frequencies. Figure 6 shows the where h αk and h βk represent the renormalization of the magnon dispersion relations for MnF , with no applied field, with energies of the two modes. Thus, clearly the total magnon 2 ω the wave vector along the c axis, measured by inelastic neutron energies become temperature dependent, given by h αk,βk(T) ¼ ω þ Δω scattering at three temperatures.58 The curves in Fig. 6 were calcu- h αk,βk(0) h αk,βk(T). Using the contributions from the lated with Eqs. (36) and (38) and with the magnon energy exchange and anisotropy interactions, the total 4-magnon contribu- tion to the α-mode energy becomes47,59 renormalization, to be described next, using HE ¼ 570 kOe, HA ¼ 8:2 kOe, and g = 2.0. The value of the exchange field was adjusted to fit the data at T = 3 K and is about 7% higher than the 2 2 2 hΔωα (T) ¼ 2zJ[(u þ v 2u v γ )C þ (u v γ v )E value given before, because Eq. (36) does not include the effect of k k k k k k q k k k k q þ (u v γ u2)F ] 4D(u2E þ v2F ), (40) the intrasublattice interaction J1 (see Fig. 1). k k k k q k q k q

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where X ¼ 1 γ þ þ Cq uqvq q (nαq nβq 1), (41a) NS q

X ¼ 1 2 þ 2 þ Eq uq nαq vq (nβq 1), (41b) NS q

X ¼ 1 2 þ 2 þ : Fq uq nβq vq (nαq 1) (41c) NS q

The correction for the frequency of the β-mode is given by α $ β the same expressions as (40) and (41) with the interchange . FIG. 7. Coordinate systems used to represent the components of the spin oper- It is important to note that even at T = 0 there is a correction to the ators for the two sublattices in the spin flop phase. magnon energies due to the magnon interactions. This zero-point correction is a characteristic feature of antiferromagnets that was first noted by Oguchi several decades ago.60 The temperature dependence of the magnon energy renormal- Eqs. (29),wehave ization for MnF due to 4-magnon interactions was calculated 2 pffiffiffiffiffi pffiffiffiffiffi numerically with Eqs. (40) and (41) replacing the sum over wave þ 0 y0 þ 00 y00 S ¼Sx þ iS ¼ 2S a , S ¼Sx þ iS ¼ 2S b , (43a) vectors by an integral over the Brillouin zone i i i i j j j j X ð Ω 0 0 pffiffiffiffiffi 00 00 pffiffiffiffiffi 1 ~ y y y y ! d k, (42) S ¼Sx iS ¼ 2S a , S ¼Sx iS ¼ 2S b , (43b) π 3 i i i i j j j j N k (2 )

~ 0 00 where N is the number of allowed k values in the first Brillouin z ¼ y z ¼ y : Si S ai ai, Sj S bj bj (43c) zone and Ω is the volume of the unit cell. For each temperature, the integrals in (41) were evaluated numerically by discrete sums in ~ k space. In the first cycle of the evaluation of Δωαk and Δωβk, the Using the transformation to collective boson operators in frequencies and the Bose factors are calculated for each point k in Eq. (31) and the relation (32), one can show that the Hamiltonian the Brillouin zone without renormalization. In the following cycles, (28) leads to the following quadratic form: the new Bose factors are calculated with the magnon frequencies X y y y y hωα β (T) ¼ hωα β (0) þ hΔωα β (T), using the renormalized (2) ¼ þ þ þ k, k k, k k, k H h Ak(akak bkbk) Bk(akbk akbk) energies of the previous cycle. The process is repeated until the k change in frequency at all points is smaller than 0.1%. Figure 5 1 y y þ Ck(akak þ bkbk þ H: c:) þ Dk(akb þ a bk), (44) shows that the calculated dispersion relations in MnF2 for three 2 k k temperature values agree quite well with the experimental data. where B. Easy-axis antiferromagnets: SF phase  As mentioned earlier, if the applied field H0 exceeds the criti- 2 1 2 Ak ¼ γ H0cos θ HEcos 2θ þ HA cos θ sin θ , (45a) cal value Hc, the energy of the β-mode at k ¼ 0, given by Eq. (16), 2 becomes negative and the AF phase is no longer stable. Then the spins flip to the configuration illustrated in Fig. 1(b), at an angle θ ¼ θ ¼ θ fi ¼ γγ 2 θ ¼ 1 γ 2 θ ¼ γγ 2 θ 1 2 with the eld and in a plane determined by the small Bk kHE sin , Ck HA sin , Dk kHE cos , anisotropy that exists in the plane normal to the symmetry axis. 2 We shall study the spin wave excitations in the SF phase using two (45b) different coordinate systems, one for each sublattice, as shown in 1 Fig. 7. In each system, the z axis is chosen to point along the spin where the angle is given θ ¼ cos [H0=(2HE HA)]. The next equilibrium direction in that sublattice, the y-axes are in the direc- step consists of performing canonical transformations from the y y tion of the y axis of the crystal, as in Fig. 1, and the x-axes are collective boson operators ak, ak, bk, bk into magnon creation and ^ ¼ ^ ^ αy α βy β determined by x y z. annihilation operators k, k, k, k such that the quadratic As in Sec. IV A, we express the spin components in each sub- Hamiltonian is cast in the diagonal form (35). lattice in terms of spin deviation operators. Considering only the Here, we follow the method developed by White et al.61 that first order terms in the binomial expansions of the square roots in generalizes the Bogoliubov transformation for diagonalizing

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quadratic Hamiltonians and write Eq. (44) in the matrix form and with (50)–(52) we obtain the eigenvalue equation

X y [H][Q] ¼ [g] 1[Q][g]h[ω]: (54) H ¼ h Hk,Hk ¼ (X) [H](X), (46) k.0 The solution of this equation gives the eigenfrequencies and the elements of the transformation matrix [Q]. Due to the symme- where the matrices are try of the 4 × 4 matrices, the matrix equations can be reduced to 0 1 0 1 relations between 2 × 2 matrices. This is done by first noticing that ak Ak Dk Ck Bk the Hamiltonian (47) can be written as B bk C B C ¼ B y C ¼ B Dk Ak Bk Ck C:    (X) @ a A, [H] h@ A (47) k Ck Bk Ak Dk [H1][H2] Ak Dk Ck Bk y [H] ¼ ,[H1] ¼ h ,[H2] ¼ h : Bk Ck Dk Ak bk [H2][H1] Dk Ak Bk Ck (55) The next step consists of introducing a linear transformation to new operators Also, note that the transformations of the operators a and b y y k k are the complex conjugates of the ones for a and b so that the 0 1 k k matrix [Q] can also be written in terms of submatrices αk B β C B k C   ¼ ¼ B y C [Q ][Q ] Q Q Q Q (X) [Q](Z), (Z) @ α A (48) ¼ 1 2 ¼ 11 12 ¼ 13 14 : k [Q] * * ,[Q1] ,[Q2] y [Q2][Q1] Q21 Q22 Q23 Q24 β k (56)

such that the Hamiltonian in Eq. (46) can be written as The matrix [g]in(52) can also be written in the form X  ¼ y ω I 0 H h (Z) [ ](Z), (49) [g] ¼ , (57) k 0 I

where [ω] is a diagonal eigenvalue matrix. In order to find the where I is a 2 × 2 diagonal unit matrix. Using Eqs. (55)–(57) in transformation matrix [Q], one needs a few relations. The first (47), we obtain the following relations between the 2 × 2 matrices follows from the introduction of (48) in (46) and comparison with (49). This leads to þ * ¼ ω [H1][Q1] [H2][Q2] h [Q1][ ], (58a)

y [Q] [H][Q] ¼ h[ω]: (50) þ * ¼ ω : [H1][Q2] [H2][Q1] h[Q2][ ] (58b)

Another relation is obtained from the boson commutation The solution of Eqs. (58) yields the diagonal Hamiltonian in rules. They can be written in matrix form as Eq. (35), with the eigenfrequencies given by

y y * T T ω2 ¼ 2 2 [X, X ] ¼ X(X) (X X ) ¼ [g], (51a) αk (Ak Dk) (Bk Ck) , (59a)

ω2 ¼ þ 2 þ 2 y y T βk (Ak Dk) (Bk Ck) (59b) [Z, Z ] ¼ Z(Z) (Z*ZT ) ¼ [g], (51b) and the elements of the transformation matrix given by where  2 3 þ ω 1=2 þ þ ω 1=2 ¼ (Ak Dk) αk ¼ (Ak Dk) βk) 10 0 0 Q11 , Q12 , 4ωα 4ωβ 6 01 0 07 k k [g] ¼ 6 7: (52) 4 00105 (60a)  00 0 1 1=2 1=2 (Ak Dk) ωαk (Ak þ Dk) ωβk) Q13 ¼ , Q14 ¼ , 4ωαk 4ωβk Using Eqs. (47) and (48) in (51a) and in (51b), we obtain an (60b) orthonormality relation for the transformation matrix ¼ ¼ ¼ ¼ y and Q21 Q11, Q22 Q12, Q23 Q13, Q24 Q14. These results ¼  [Q][g][Q] [g], (53) greatly simplify if H0, HA HE, which is the case of MnF2. Using

J. Appl. Phys. 126, 151101 (2019); doi: 10.1063/1.5109132 126, 151101-10 Published under license by AIP Publishing. Journal of Applied Physics TUTORIAL scitation.org/journal/jap

 γ 2= α ¼jα j f β for ka 1, 1 k (ka) 8, the two magnon frequencies become where k k exp(i αk). For the k mode, we have

1 x0 1=2 ~ ωαk γHE ak, (61a) hS i¼(2S=N) (Q þ Q )jβ jcos(k ~r ωβ þ fβ ), (63a) 2 i 14 12 k i k k

0 1=2 y 1=2 ~ 2 2 1 2 2 2 hS i¼(2S=N) (Q Q )jβ jsin(k ~r ωβ þ fβ ), (63b) ωβ γ(H H þ H a k ) : (61b) i 14 12 k i k k k 0 c 4 E h x00 i¼ = 1=2 þ jβ j ~~ ω þ f As we saw in Sec. IV B, in the AF phase, the frequency of Sj (2S N) (Q14 Q12) k cos(k rj βk βk), (63c) the αk mode increases linearly with magnetic field intensity for 00 any wave vector. However, in the SF phase, the frequency h y i¼ = 1=2 jβ j ~~ ω þ f S (2S N) (Q14 Q12) k sin(k rj βk βk), (63d) becomes independent of the applied field and, moreover, it van- j ishes for the k ¼ 0 magnon, as can be seen in Eq. (61a).This 51 β ¼jβ j f fi behavior has been clearly observed experimentally in MnF2, as where k k exp(i βk). With Eqs. (62) and (63), we can nd the shown in Fig. 3(b). The frequency increases linearly with field, ellipticities of the spin precessions. Considering HE H0, HA in ffi α but as the field approaches Hc, it falls abruptly to a very small the coe cients (60), the ellipticity in both sublattices for the k value. This is so because this mode corresponds to the rotation mode is of the spins about the symmetry axis at no cost of energy. In h x0 i þ γ þ ω 1=2 þ γ ω 1=2 fact, in crystals there is always a small anisotropy in the plane Si max Q11 Q13 ( HE αk) ( HE αk) α ¼ ¼ perpendicular to the symmetry axis that can be represented by e y0 1=2 1=2 , h i Q Q γ þ ωα γ ωα Si max 11 13 ( HE k) ( HE k) an effective field HAp. In this case, the frequency of the k ¼ 0, αk 1=2 (64) magnon is ωα0 ¼ γ(2HEHAp) , which is still independent of fi β the applied eld, but not zero. In regard to the k mode, in the β AF phase, its frequency decreases linearly with field and, for and similarly for the k mode, it is ¼ ¼ . the k 0 magnon, it goes to zero at H0 Hc.ForH0 Hc, 0 h x i þ γ þ ω 1=2 γ ω 1=2 according to Eq. (61b), the frequency increases with field as Si max Q14 Q12 ( HE βk) ( HE βk) = β ¼ ¼ : 2 2 1 2 e y0 1=2 1=2 ωβk ¼ γ(H H ) , in agreement with the experimental data h i Q14 Q12 γ þ ωβ þ γ ωβ 0 c Si max ( HE k) ( HE k) shown in Fig. 3(b). (65) We can find the behavior of the sublattice spins of the magnon modes in the SF phase, by calculating the expectation 62,63 These equations show that the ellipticity of the spin precession values of the spin components in coherent magnon states. For varies strongly with the wave number. Near the center of the the αk mode, we obtain Brillouin zone, Eqs. (61) show that the frequencies are ωαk, ωβk  γHE so that the ellipticities become eα ¼ 8γHE=ωαk and 0 = h x i¼ = 1 2 þ jα j ~~ ω þ f eβ ¼ ωβ =(8γH ). In this case, the spin precessions are highly ellip- Si (2S N) (Q11 Q13) k cos(k ri αk αk), (62a) k E tical, with the major axis much larger than the minor. Figure 8 illus- 0 y 1=2 ~ trates the precessions of the spins in the two sublattices for both hS i¼(2S=N) (Q Q )jα jsin(k ~r ωα þ fα ), (62b) i 11 13 k i k k modes, with k = 0. In the field-independent αk mode, the spins precess with the major axes along the x-axes of the two sublattices, h x00 i¼ = 1=2 þ jα j ~~ ω þ f with a net spin component along the external field. Thus, this mode Sj (2S N) (Q11 Q13) k cos(k rj αk αk), (62c) can be driven by a rf field parallel to the static field. On the other fi β 00 hand, in the eld dependent k mode, the major axes of the ellipti- h y i¼ = 1=2 jα j ~~ ω þ f Sj (2S N) (Q11 Q13) k sin(k rj αk αk), (62d) cal precessions are along the y-direction for each sublattice, with a

FIG. 8. Illustration of the elliptical pre- cessions of the sublattice spins in the k = 0 magnon modes in the spin-flop phase. (a) αk mode. (b) βk mode.

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net spin component along y so that it can be driven by a rf field par- to the easy axis, calculated with Eqs. (59) and (67),using allel to this direction. However, for magnons at the Brillouin zone HE ¼ 570 kOe, HA ¼ 8:2kOe, g = 2.0, and a geometric structure ω ω γ γ ¼ π = boundaries, αk βk HE, and Eqs. (64) and (65) show that factor for a spherical Brillouin zone, k cos( k 2kmax). For the eα eβ 1 so that the spin precessions are circular. zero field, the two modes are degenerate and the dispersion relations are the same as in Fig. 6. Application of the field results in a behavior fl α C. Easy-axis antiferromagnets: Canted phase similar to the spin- op phase, the frequency of the -mode becomes independent of the field, while the frequency of the β-mode increases fi We now consider the con guration where the external magnetic with the field. This is illustrated further in Fig. 9(b) showing the field fi eld is applied in a direction perpendicular to the easy axis, the c axis dependence of the frequencies for the k = 0 modes. The frequency of fi in MnF2 and FeF2. In this case, in the equilibrium con guration, the the β-mode increases continuously with field, as observed experi- two sublattice spins lie in the same plane along directions at an angle mentally,51 and shown in Fig. 3(b). Note that the frequency of the θ fi fl with the eld, similar to the spin- op phase, shown in Fig. 1(b). α-mode is not shown in Fig. 3(b) because the experiments of Ref. 51 θ ¼ = þ Minimization of the energy gives cos H0 (2HE HA). In this were carried out with fixed frequency and scanning field so that the phase, the Hamiltonian (28) expressed in terms of boson operators α-mode could not be detected. introduced by the transformations in Eqs. (43) and (31) becomes, with only the quadratic terms D. Easy-plane antiferromagnets in the AF phase X ¼ y þ y þ þ y y H h Ak(akak bkbk) Bk(akbk akbk) We now consider easy-plane, or hard-axis, antiferromagnets, k initially with no applied field. As in the previous cases, the first 1 y y step consists in writing the Hamiltonian (5), with H0 ¼ 0, in terms þ C (a a þ b b þ H: c:) þ D (a b þ a b ), (66) 2 k k k k k k k k k k of raising and lowering spin operators where X X ¼γ þ þ þþ z z þ x 2 z 2 H h Jij(Si Sj Si Sj 2Si Sj ) Dx(Si,j) Dz(Si,j) : i,j i,j A ¼ γ [H cos θ H cos 2θ þ H (2 sin2θ cos2θ)=2], (67a) k 0 E A (68)

¼ γγ 2 θ Bk kHE sin ,(67b)Next, we use Eqs. (29)–(32) for the transformation from the spin operators into boson collective spin deviation operators and 2 Ck ¼ γ HA cos θ=2, (67c) obtain the quadratic Hamiltonian X y y y y ¼ γγ þ θ = : ¼ þ þ þ Dk kHE(1 cos 2 ) 2 (67d) H h Ak(akak bkbk) Bk(akb k akbk) k The Hamiltonian (66) has the same form as Eq. (44) for the 1 þ Ck(akak þ bkbk þ H:c:), (69) spin-flop state so that the frequencies and the transformation 2 coefficients are given by the same expressions as in Eqs. (59) where the parameters are related to the effective fields by and (60), with the parameters defined by Eqs. (67). Figure 9(a) shows the magnon dispersion relations for MnF2 H with a magnetic field of H ¼ 200 kOe applied perpendicularly A ¼ γ(H þ H =2 þ H ), B ¼ γγ H , C ¼ γ Ax , (70) 0 k E Ax Az k k E k 2

FIG. 9. Magnon frequencies calculated with Eqs. (59) and (67) for MnF2, with the external field applied perpendicu- larly to the easy axis, in the spin- canted configuration. (a) Dispersion relations for H0 ¼ 200 kOe. (b) Field dependence of the k = 0 magnon frequencies.

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γ ω = π ¼ : where k is the geometric structure factor. The Hamiltonian (69) the value β0 2 0 140 THz measured by Brillouin light scat- 32 has the same form as Eq. (44) for the spin flop state so that the tering and ωα0=2π ¼ 1:07 THz obtained from magnetization frequencies are given by the same expressions as in Eqs. (59a) oscillations in the far infrared.36 With these values in Eqs. (73), and (59b), with Dk ¼ 0. Thus, the magnon frequencies of the two we determine the effective anisotropy fields, HAx ¼ 6:35 kOe and modes are HAz ¼ 0:11 kOe. Figure 10(a) shows the dispersion relations calcu- lated with Eqs. (72) assuming a spherical Brillouin zone and using 2 2 2 ωα β ¼ A (Ck + Bk) : (71) γ ¼ π = ¼ π= , k for the structure factor k cos( k 2km), where km al, al being the lattice parameter, as well as the data of Ref. 28. The two Using the expressions for the parameters in Eq. (70), one can magnon modes are nearly degenerate for large wave vectors, but the write the frequencies of the two magnon modes in terms of the frequency separation is evident in the blow up near the Brillouin effective fields zone center as shown in Fig. 10(b). In the case where an external field is applied in the easy plane, ω2 ¼ γ2 2 þ þ þ þ γ þ 2 γ2 in a direction along the easy axis, it can be shown47 that the αk [HAz 2HEHAz HAxHAz HEHAx(1 k) HE(1 k)], unrenormalized magnon frequencies are (72a) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 ¼ 2 þ γ2 2 2 þ 2 + γ2 2 2 2 þ 2 2: ω2 ¼ γ2 2 þ þ þ γ þ 2 γ2 : α,β (Ak H0 ) (Ck Bk) 2 H0 (Ak Bk) BkCk βk [HAz 2HEHAz HAxHAz HEHAx(1 k) HE(1 k)] (72b) (74)

The dashed lines in Fig. 10(b) represent the dispersion rela- For NiO, considering HE HAx, HAz, the frequencies of the fi ¼ γ ¼ tions in NiO calculated for a eld H0 40 kOe. As one can see, for Brillouin-zone center k =0( k 1) magnons are k = 0, the frequency of the αk-mode increases with an increasing fi β 1=2 eld, while the frequency of the k-mode decreases. In fact, the ωα0 ¼ γ[2HE(HAx þ HAz)] , (73a) β fi softening of the k-mode with an increasing eld determines the spin-flop transition in NiO.47 ω ¼ γ 1=2 β0 (2HEHAz) , (73b) Similar to the case of the spin-flop phase, studied in Sec. IV B, we can calculate the expectation values of the spin components in which agree with Eq. (24) obtained with the semi-classical treat- coherent magnon states in the AF phase of the easy-plane antifer- ment. These equations show that, regardless of the relative romagnets. In both modes, the spin precessions in the two sublatti- ¼h x i =h y i values of the anisotropy fields, the frequency of the α-mode ces are characterized by an ellipticity e Si,j max Si,j max. Using – is always larger than for the β-mode.InthecaseofNiO,HAx the transformations (30) (32) and (48), one can express the spin 1=2 ffi HAz so that ωα0 γ(2HAxHE) ,inagreementwithEq.(24) for components in terms of the coe cients in Eqs. (60) and show that α β the α-mode and H0 ¼ 0. As shown in Ref. 54,thevaluesofthe the ellipticities for modes k and k are parameters for NiO can be obtained by fitting Eqs. (72) to three sets of experimental data. Fitting to the neutron scattering mea- Q11 þ Q13 Q14 jQ12j 28 eα ¼ , eβ ¼ : (75) surements of Hutchings and Samuelsen gives the value of the Q11 Q13 Q14 þjQ12j exchange field HE ¼ 9 684 kOe, considering for the g-factor g =2.18. Since the neutron data do not have sufficient resolution As in the spin-flop phase, the ellipticities of the spin precession to determine the frequencies of the zone-center magnons, we use vary strongly with the wave number. Near the center of the

FIG. 10. Spin wave dispersion rela- tions in antiferromagnetic NiO at T = 300 K. (a) Solid curves show the magnon frequencies calculated with Eqs. (72). Symbols represent the neutron scattering data of Ref. 28. (b) Blow up of the Brillouin zone center showing the separation of the frequen- cies of the α (upper blue curve) and β (lower red curve) magnon modes. The dashed curves are calculated with Eq. (74) for a field of H0 ¼ 40 kOe applied along the easy axis.

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Using the transformation to the magnon operators given by Eqs. (31) and keeping only terms with magnon number operators, we have X z ¼ αyα þ βyβ : S ( k k k k) (78) k

The opposite signs in the angular momenta of the two modes are consistent with the semiclassical picture of the spins precessing in opposite directions. Considering for each mode μ the group ^ velocity ~vμk ¼ k @ωμ=@k, the spin current density operator is

X ~z ¼ h ~ αyα þ~ βyβ : JS [ vαk k k vβk k k] (79) V k

In this equation, which is valid for both easy-axis and αyα βyβ FIG. 11. Wave number dependence of the ellipticities of the two magnon hard-axis antiferromagnets, k k and k k represent the number modes in the easy-plane antiferromagnet NiO. operators of magnons of each mode. In easy axis antiferromagnets, such as MnF2 and FeF2, in the absence of an applied magnetic field, the two magnon modes are degenerate and thus have equal occupation numbers in thermal equilibrium. Since they also have γ ω ω Brillouin zone, for HE αk, βk,Eqs.(75) give the same results equal group velocities, in easy-axis AFs, the spin current created by as in Eqs. (27), showing that the spin precessions are highly elliptical, thermal gradients vanishes at zero field. as in Fig. 5. However, for magnons near the Brillouin zone boundar- In order to illustrate a method to calculate the magnonic spin ω ω γ ! ies, αk βk HE, Q13, Q14 0, and one can see from Eq. (75) current in spintronics, we consider the spin Seebeck effect, which that the ellipticities approach unity, corresponding to circular spin consists of the generation of spin currents by a thermal gradient – precessions. The dependencies of the ellipticities on the wave across a thin layer of a magnetic material.67 69 In this case, the number, calculated for NiO with Eq. (75), are shown in Fig. 11. magnonic spin current can be calculated with the Boltzmann and diffusion equations for the magnon accumulation, defined for AF V. MAGNONIC SPIN CURRENT IN materials by extending the concept introduced for ferromagnets.65 ANTIFERROMAGNETS Denoting by nμk the number of magnons in the μ ¼ α, β mode A key concept in spintronics is the spin current, which with wave number k in the whole volume V of the AF layer, and by 0 – expresses the flow of spin angular momentum in a material. In nμk the number in thermal equilibrium, given by the Bose Einstein δ ~ ¼ ~ 0 metals, the spin current is carried by the spins of the conduction distribution, the quantity nμk(r) nμk(r) nμk is the number in electrons, whereas in magnetic insulators, the spin current is trans- excess of equilibrium. Since the contributions of the two modes to the spin current have opposite signs, we define the magnon accu- ported by magnons. In a simple ferromagnetic insulator with one δ ~ 54,66,70 spin per unit cell so that there is only one magnon mode, consider- mulation nm(r) in antiferromagnets as ing z the equilibrium direction of the spins, the spin-current ð ~ 1 3 0 0 density with polarization z carried by magnons with wave vector k δ n (~r) ¼ d k[(nα n ) (nβ n )], (80) 64–66 m π 3 k αk k βk and energy εk ¼ hωk is (2 ) ð h ~Jz ¼ d3k~v [n (~r) n0]: (76) so that the magnon spin-current density in Eq. (76) becomes S π 3 k k k (2 ) ð ~z h 3 0 0 ~ ~ J ¼ d k [~vα (nα (~r) n ) ~vβ (nβ (~r) n )]: (81) Here, vk is the magnon velocity, nk(r) is the number of S π 3 k k αk k k βk ~ ~ 0 (2 ) magnons with wave vector k at a position r, and nk is the number – in thermal equilibrium, given by the Bose Einstein distribution fl with zero chemical potential. In a two-sublattice antiferromagnet, The distribution of the magnon number under the in uence of a the spin current is transported by the two magnon modes. In the thermal gradient can be calculated with the Boltzmann transport equation. In the absence of external forces and in the relaxation AF phase, the total z-component ofP the spin angular momentum z ¼ z þ z approximation, in steady state, we obtain for each magnon mode carried by magnons is given by S i,j (Si Sj ). With Eqs. (29c) and (30c), one can write the z-component of the spin angular ~ 0 ¼τ ~ ∇ 0 τ ~ ∇ ~ 0 momentum as nμk(r) nμk μkvμk nμk μkvμk [nμk(r) nμk], (82) X z ¼ y þ y : τ μ S akak bkbk (77) where μk is the k-magnon relaxation time. Using Eq. (82) in k Eq. (81), one can show that the spin current is the sum of two

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~z ¼~z þ~z In regard to the excitation of magnons in antiferromagnets, parts, JS JS∇T JSδn, where we have discussed only one method, namely, the spin Seebeck ð   @ 0 @n0 @ effect, by which magnonic spin currents are generated by thermal ~z ¼ h 3 τ 2 nαk þ τ 2 βk T J ∇ d k αkvα βkvβ (83) gradients. Actually, magnons can be excited by several processes, S T (2π)3 ky @T ky @T @y such as spin current injection,73 ultrafast optical excitation,35,36 fl current driven dynamics produced by spin-transfer torque,43 is the contribution of the ow (convection) of magnons due to the 8–14 temperature gradient, assumed to be in the y direction, and among others. Thus, the experimental and theoretical study of magnonic phenomena in the wide variety of antiferromagnetic ð materials, known and yet to be discovered, constitutes a challenging ~z ¼ h 3 τ ~ ~ ∇δ τ ~ ~ ∇δ J δ d k[ αkvαkvαk nαk βkvβkvβk nβk] (84) and exciting field of research in magnetism and spintronics. S n (2π)3

is due to the spatial variation of the magnon accumulation, which ACKNOWLEDGMENTS is governed by the diffusion equation.65,66 With the temperature This research was supported in Brazil by the Conselho gradient normal to the film plane, Eq. (83) gives for the spin Nacional de Desenvolvimento Científico e Tecnológico (CNPq), current in the y-direction66 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Financiadora de Estudos e Projetos (FINEP), and z ¼ z∇ JS SS T, (85) Fundação de Amparo à Ciência e Tecnologia do Estado de "#Pernambuco (FACEPE), and in Chile by Fondo Nacional de ð ω = ω = 2 h βk kBT ω 2 h αk kBT ω 2 fi h e βkvβ e αkvα Desarrollo Cientí co y Tecnológico (FONDECYT) No. 1170723. Sz ¼ dk k2 ky ky , S π2 2 ω = 2 ω = 2 6 kBT ηβ (eh βk kBT 1) ηα (eh αk kBT 1) k k REFERENCES (86) 1 L. Néel, “Influence des fluctuations des champs moléculaires sur les propriétés magnétiques des corps,” Ann. 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