# Theory of Spin Waves

In this chapter, we study the properties of the spin waves excited in a system of interacting spins. We suppose that the spins are localized on the lattice sites. The exchange interaction between spins makes them rotate about the quantization axis in a collective manner: These collective excitations are called "spin waves” or "magnons” when quantized. Such collective excitations or elementary excitations are common phenomena observed in systems of interacting particles or quasi-particles in condensed matter. One can mention phonons (waves of atomic motions around the atoms’ equilibrium positions in a system of interacting atoms) and plasmons (waves of the charge density). Note that there is no spin wave in systems of Ising spins because Ising spins cannot be deviated from their axis to create a wave.

Spin waves propagate in systems with a translation invariance. In systems with a broken translation, spin waves can be localized or damped such as in the presence of an impurity or a surface. Spin waves cannot be excited in disordered systems due to their wave nature since a propagating wave should have a constant amplitude and a constant phase shift in space. The same is observed with a system of interacting atoms: Phonons can be excited only at low temperatures when the system has a crystalline structure.

*Physics of Magnetic Thin Films: Theory and Simulation *Hung T. Diep

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com

At low temperatures, spin waves yield the main thermal behavior of a system of ordered spins such as in a ferromagnet, as seen below. This chapter presents the spin wave theory as applied to ferromagnets, antiferromagnets and, to a lesser extent, ferrimagnets. It gives a necessary background for understanding the applications in part II of the book.

## Spin Waves in Ferromagnets

### Classical Treatment

We consider a ferromagnet where the spins are parallel, aligned along the *Oz* axis in the ground state. Each spin is considered as a vector of modulus S with three components. This is the classical Heisenberg spin model. As the temperature increases, the spins rotate around their *z* axis in a collective manner as shown in Fig. 3.1. The energy provided by the temperature is carried by the excited spin wave.

In the following, we calculate the spin wave energy for the classical Heisenberg spin model on a lattice. The spins are supposed to interact with each other via a nearest-neighbor exchange interaction *J.* The interaction of the spin S/ with its nearest neighbors is written as

Figure 3.1 Side view and top view of a spin wave.

where R is the vector connecting the spin *S _{t}* with a nearest neighbor. The sum is performed over all nearest neighbors. Equation (3.1) can be rewritten as

where

and

*g* and *ixв* are the Lande factor and Bohr magneton. М/ and H_{ex} are, respectively, the magnetic moment of spin S/ and the field of the nearest neighbors acting on S/.

We write the equation of motion of the kinetic moment *hSi* as follows:

If S/ is parallel to its neighbors then the right-hand side of Eq. (3.5) is zero, one has = 0, i.e., S/ is equal to a constant vector, thus there is no spin wave.

In an excited state due to the spin rotation around the *z* axis, one can decompose S/ as shown in Fig. 3.2:

where *8Si* represents the deviation and S_{0} the z spin component. For a homogenous system and for a given spin wave mode, it is natural to suppose that S_{0} is space- and time-independent. Equation (3.5) becomes

where *8Si* and 8Si_{+r} are supposed to be small so that we neglect their second-order terms. This hypothesis is justified at low temperatures.

Figure 3.2 Decomposition of a spin.

We define now the spin components by: S_{0} = *Sfk = S _{0}k,* (<5S/)

^{X}= S* and (SS/)-*' =

*Sf, к*being the unit vector on the

*z*axis (see Fig. 3.3). We rewrite Eq. (3.7) as

Figure 3.3 Spin components.

The last equation indicates that *Sf* is a constant of motion. One now looks for the solutions of S* and *Sf* of the form

where к and 1 are the wave vector and the position of S/, respectively. Replacing (3.11)-(3.12) in (3.8)-(3.9), one obtains

where *Z* is the coordination number (number of nearest neighbors). The non-trivial solutions of *U* and *V* verify

from which one has

where

Replacing (3.15)-(3.16) in the above coupled equations of *U* and *V, *one finds *V =* -/(/.This relation indicates that * U* | = |K|; therefore the spin rotation has a circular precession around the *z* axis.

The relation (3.15) is called "dispersion relation” of spin waves in ferromagnets.

Example: In the case of a chain of spins, one has

Figure 3.4 Spin wave dispersion relation of a ferromagnet in one dimension.

where *a* is the lattice constant. Equation (3.15) becomes

Figure 3.4 shows *hw _{k}* versus

*к*in the first Brillouin zone. When

*ka*

*w _{k}* is thus proportional to

*k*for small

^{2}*к*(long wavelength). This behavior is also true for other dimensions. However, for antiferromagnets we will see below that

*w*Since macroscopic physical properties are calculated by averaging over the spin wave excitations, we will see the difference between ferromagnets and antiferromagnets. This difference is not observed in the mean-field theory.

_{k}oc k.Remark: Effect of the crystal symmetiy is contained in the factor yu. Here are a few examples:

(1) Square lattice:

(2) Simple cubic lattice:

(3) Centered cubic lattice:

(4) Face-centered cubic lattice:

### Quantum Spin Wave Theory: Holstein–Primakoff Approximation

We consider now the case of quantum spins. The spin S/ at the lattice site / can be decomposed into the following spin operators: *Sf* and Sf = S* ± *iSf.* These spin operators obey the following commutation relations:

The Holdstein-Primakoff method consists in introducing the operators *a* and *a ^{+}* as follows:

where

The transformations (3.26)—(3.28) are called "Holstein-Primakoff transformations.” We note that *a ^{+}a* in Eq. (3.26) corresponds to the diminution of S due to the excited spin wave.

*a*is called therefore "spin wave number operator.”

^{+}aLet us calculate the spin wave dispersion relation using the Holstein-Primakoff transformations (3.26)-(3.28). We consider the following Heisenberg Hamiltonian

where for simplicity we suppose that interactions are limited to pairs of nearest neighbors (/, m) with a ferromagnetic exchange integral *J >* 0. *H* is the amplitude of a magnetic field applied along the *z* direction, *g* and *x _{B}* are, respectively, the Lande factor and Bohr magneton.

Using *S ^{±}* = 5

^{X}±

*iS*for

^{y}*Si*and S

_{m}, one rewrites

*H*as

Replacing 5* and *S ^{z}* by (3.26)-(3.28) while keeping position indices / and

*m*of operators

*a*and o

^{+}one obtains

It is impossible to find a solution of this equation because of non-linear terms such as a_{;}^{+}a/0^o_{m}, //(5) and f„(S). In a first approximation, one can assume that the number of excited spin waves *n* is small with respect to 25 (namely *a ^{+}a* <£ 25) so that one can expand //(5) and /,„(5) as follows:

Equation (3.32) becomes, to the quadratic order in *a* and a^{+ }
where one has used the following relation

R being the vector connecting the spin at / to one of its nearest neighbors, Z the coordination number and *N* the total number of spins.

The first term of (3.35) is the energy of the ground state where all spins are parallel and the second is a constant which will be omitted in the following. One introduces next the following Fourier transformations

One can show that a_{k} and obey the boson commutation relations just as real-space operators й/ and a_{;}^{+} (see Problem 9 in Section 3.6). Putting (3.38) in (3.35) one finds

where

One sees that in the case where *H* = 0 one recovers the magnon dispersion relation (3.15) e_{k} = *hw _{k},* obtained by the classical treatment. The Holstein-Primakoff method allows, however, to go further by taking into account terms of order higher than quadratic in

*a*and

^{+}*a.*By using expansions (3.33)-(3.34), one obtains terms of four operators, six operators, ... which represent interactions between spin waves. These terms play an important role when the temperature increases.

### Properties at Low Temperatures

One studies here some low-temperature properties of spin waves using the dispersion relation (3.40).

#### Magnetization

One has seen above that *a* and *a ^{+}* are boson operators. The number of spin waves (or magnons) of к mode at temperature

*T*is therefore given by the Bose-Einstein distribution

where *f)* = *[квТ] ^{-1}.* The magnetization defined as the magnetic moment per unit volume is given for the volume £2 = 1:

where the sum is performed over all spins.

With (3.38), Eq. (3.42) becomes

where < n_{k} > is given by (3.41).

One shows here how to calculate *M* in the case of a simple cubic lattice and *H* = 0. The sum in (3.43) reads

Using yu of (3.21), one has

where one used an expansion for small к because at low temperatures (large /1) the main contribution to the integral (3.44) comes from the region of small k. With £2 = 1, *Z =* 6 (simple cubic lattice) and (3.45), Eq. (3.44) becomes

where *m = 1 +* 1. Note that the upper limit of the integral which is the border of the first Brillouin zone has been replaced by oc. This is justified by the fact that important contributions are due to small *k. *Changing the variable *x* = *mf52J S[ka) ^{2},* one obtains

One notes that the integral of the right-hand side is equal to and that the sum on *m* is the Riemann's series <■ (3/2). Finally, one arrives at

The magnetization decreases with increasing Г by a term proportional to T^{3/2}. This is called the Bloch's law.

As *T* increases further one has to take into account higher-order terms in (3.45). In doing so, one obtains

where *t* = *$fjj.* This result, exact at low temperatures, has been confirmed by experiments at least up to *T ^{s/2}.*

Note: <(3/2) = 2.612, <(5/2) = 1.341, <(7/2) = 1.127

#### Energy and heat capacity

The energy of a ferromagnet is calculated in the same manner. From Eq. (3.39), one writes

where the first term is the ground-state energy and the second term the energy of excited magnons at the quadratic order (free magnons).

The magnetic heat capacity *C™* is thus

We see that at low temperatures Cj/ is proportional to *T*^{3/2} while the heat capacity of an electron gas *C ^{e}v* is proportional to

*T*. It is also different from that of phonons where Cf a

*T*The dependence of

^{3}.*C™*on

*T*has been experimentally confirmed.