Fundamentals of the Physics of Solids / 15-Elementary Excitations in Magnetic Systems

15.3 Antiferromagnetic Magnons

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At finite temperature, the sublattice magnetization is

MA =	1	gμB	SAz i =		1	gμB	[S − ai†ai ]
	V				V
			i			i	(15.3.22)
	1
=		gμB 21 N S −		k	ak† ak .
	V

Changing to magnon creation and annihilation operators anew, the formula for the temperature dependence of sublattice magnetization becomes

1			uk2	αk†	αk	+ vk2 βk† βk	.	(15.3.23)
MA(T ) = MA(0) − gμB V k

Since the two types of magnons possess the same energy, by using the Bose– Einstein statistics for the occupation numbers, and inserting the expressions for u2k and vk2 from (15.3.12), we have

1			1	−				1
MA(T ) = MA(0) − gμB								−		.	(15.3.24)
	V		eβ ωk		1
		k							k

To determine the leading-order correction, the linearity of the dispersion relation is assumed at low temperatures, ωk = Dk. Introduction of the new variable x = Dk/kBT and the customary replacement of the sum by an integral leads to

MA(T ) = MA(0) − gμB	|Dπ| 2		D	2	xmax	ex − 1 ,	(15.3.25)
					0
	J zS		kBT			x dx

where xmax = Dkmax/kBT , while kmax is determined by

N	=	1		4π	k3	,	(15.3.26)
2V		(2π)3
				3 max

in analogy to the Debye wave number. At low temperatures, where one may extend the integration to infinity, the sublattice magnetization varies as the square of the temperature. The specific heat contribution of magnons – similarly to that of phonons, since the two dispersion relations are identical – is found to be proportional to T 3.

15.3.4 Excitations in Anisotropic Antiferromagnets

If we had determined the energy of excited states in isotropic antiferromagnets in the presence of a magnetic field, but with the constraint that in the two sublattices the spins are respectively lined up parallel and antiparallel to the applied field, we would have arrived at the simple result

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"

ωk = 2|J|zS 1 − γk2 ± gμBB . (15.3.27)

The fact that in the vicinity of k = 0 the excitation energies are negative in one branch signals immediately that our assumption is flawed. In isotropic models the state in which the sublattice magnetization is parallel to the magnetic field direction is not stable. As we saw in the mean-field-theoretical treatment, the magnetic moments in an isotropic antiferromagnet are not aligned with the applied magnetic field but are slightly turned with respect to the direction perpendicular to it. A uniaxial anisotropy, however, may render the alignment of the sublattice magnetization along the applied field stable. If we write the Hamiltonian as

	Jij SixSjx + SiySjy + ΔSiz Sjz	!
H = −		,	(15.3.28)

i,j

like in (14.3.6), the excitation spectrum can be determined applying methods similar to those used in the isotropic case. After a Holstein–Primako transformation from spin to boson operators, the Hamiltonian may be diagonalized by a Bogoliubov transformation. In two-sublattice antiferromagnets, if exchange takes place only between nearest neighbors located in di erent sublattices, the spin-wave excitation energies are found to be

"

ωk = 2|J|zS 2 − γk2 ± gμBB . (15.3.29)

In the absence of an applied magnetic field there is a finite gap in the magnon spectrum for > 1. There is no contradiction with Goldstone’s theorem, for in this easy-axis anisotropic situation the ordered antiferromagnetic ground state does not break any continuous symmetry. The twofold degeneracy in the excitation spectrum is lifted by an external magnetic field. The energy in one branch decreases for increasing field intensities, and vanishes at a critical value of the magnetic field. It is at this critical field strength that the alignment of the sublattice magnetization along the easy axis of magnetization becomes unstable and that the spin-flop transition seen in the previous chapter occurs.

15.3.5 Magnons in Ferrimagnets

The magnetic excitations in ferrimagnets are examined in a bipartite lattice in the special case when the two sublattices are built up of spins of magnitude SA and SB, respectively, and each atom located in either sublattice is surrounded by the same z number of nearest neighbors in the other sublattice. Employing a straightforward generalization of the method used for antiferromagnets, the excitations are found to have two branches, with energies

"

ωk = |J|z (SA − SB)2 + 4SASB(1 − γk2 ) ± (SA − SB) . (15.3.30)

15.4 Experimental Study of Magnetic Excitations

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For SA = SB the spectrum of antiferromagnets is recovered. When the two spins are di erent, one excitation branch still starts o at zero, in accordance with Goldstone’s theorem, while the energy values in the other branch are always finite. Just like for phonons, the gapless modes are called acoustic magnons, while the others are termed optical magnons.

The structure of most ferrimagnets is more complex than the one presented above. There might be more than two sublattices, with di erent coordination numbers and exchange integrals for each of them. In magnetite, e.g., there are six sublattices, and, accordingly, six magnon branches, of which five are optical. In the magnon spectrum of yttrium–iron garnet (YIG), 20 excitation branches are found, of which 19 are optical. In those rare-earth garnets, in which the rare-earth ions are magnetic, 32 modes are present, of which 31 are optical.

15.4 Experimental Study of Magnetic Excitations

In Chapter 13 we gave a detailed account of the experimental methods used to study lattice vibrations. We saw that by measuring the wave vector and energy of the particles (photons, neutrons etc.) in the incoming and scattered beams in scattering experiments, one can determine the spectrum of lattice vibrations created or annihilated in the scattering process. The general considerations presented there are also valid for magnetic excitations, as long as the particles in scattered beam can flip the spins of the magnetically ordered system, creating or absorbing magnetic excitations. Earlier we had also seen that neutrons, via their magnetic moments, can interact with magnetic moments localized to atoms – hence elastic neutron scattering is the most adequate method of magnetic structure determination. Therefore it comes as no surprise that inelastic neutron scattering is the method of choice for determining the dispersion relation of magnetic excitations.

According to the Van Hove formula presented in Appendix E, the double di erential cross section of inelastic scattering in a vibrating lattice can be expressed in terms of the spatial and temporal correlation function of the position of atoms – or, more precisely, is proportional to its Fourier transform with respect to space and time variables. Likewise, the magnetic scattering cross section is proportional to the Fourier transform of the spatial and temporal correlation function of magnetic moments. Because of the vector character of magnetic moments, the orientation of the moments relative to the polarization direction of neutrons will be essential. We shall skip the details here – the Van Hove formula of magnetic scattering will be given in Appendix E –, and content ourselves with the result: the cross section formula contains the spatial and temporal Fourier transform of the correlation function

Siα(t)Sjβ (0) ,

(15.4.1)

as

548 15 Elementary Excitations in Magnetic Systems

		∞
ij	e−iK·(Ri −Rj )		dt eiεt/ Siα(t)Sjβ (0) ,	(15.4.2)
		−∞

where K = k −k is the change of the neutron propagation vector, also known as the scattering vector, and ε = Ef − Ei is the energy transfer in the process.

These correlation functions can be easily determined for systems described by the Heisenberg model if the leading term of the Holstein–Primako transformation is used for the spin operators. In the magnetically ordered phase the term Siz Sjz gives rise dominantly to elastic scattering (di raction), from which the static magnetic structure can be determined. Inelastic processes arise from the terms Si+Sj− and Si−Sj+ . Considering, for the sake of simplicity, the ferromagnetic case, it is easily seen that, using the bosonic creation and annihilation operators, the terms appearing in the inelastic scattering cross section can be written as

d2σ		ij		q	e−i(K+q)·(Ri −Rj )			∞	dt eiεt/ aq† (t)aq (0)
dΩ dε
						−∞				(15.4.3)
								∞
	+		ij	q		e−i(K−q)·(Ri −Rj )			dt eiεt/ aq (t)aq† (0) .
								−∞

This expression takes a simple form when the interaction between magnons is neglected. Writing out explicitly the time dependence of the operators, which is easily obtained for free magnons, the Fourier transform of the spin–spin correlation function gives a set of Dirac delta peaks at pairs of K and ε that correspond to the creation or annihilation of magnons:

d2σ

dΩ dε A nq δ(K +q) δ(ε+ ωq)+B 1+ nq δ(K −q) δ(ε− ωq) . (15.4.4)

The first term – which is proportional to the magnon occupation number and has a sharp peak at k = k + q, Ef = Ei − ωq – corresponds to processes in which a magnon is absorbed by the scattered particle (neutron). The term proportional to 1 + nq arises from processes in which a magnon is created, and therefore k = k − q, Ef = Ei + ωq .

By measuring the peaks as a function of the transferred energy and the scattering angle, the dispersion relation for magnons can be recovered, much in the same manner as for phonons. Interactions between magnons will broaden these peaks. Since the resulting line width is related to the decay rate of magnons, the magnon lifetime can be determined from scattering experiments.

15.5 Low-Dimensional Magnetic Systems

A lot of attention has been recently devoted to the study of magnetic systems in which the exchange interaction between atoms carrying magnetic moments

15.5 Low-Dimensional Magnetic Systems

549

is appreciable only in one or two directions – while in other directions the energy of this type of interaction is negligibly small compared to the thermal energy. As far as magnetic properties are concerned, such systems should be considered as oneor two-dimensional, even if they behave as truly threedimensional crystals in other respects.

15.5.1 Destruction of Magnetic Order by Thermal and Quantum Fluctuations

When studying the temperature dependence of magnetization in ferromagnets we found in (15.2.32) that the deviation from the saturation value is given by the integral

M = gμB	(2π)3	eβ ωk	− 1 .		(15.5.1)
	dk	1

Obviously, in the more general case of d dimensions, one has to evaluate the integral

M = gμB	ddk	1		.	(15.5.2)
	(2π)d		eβ ωk − 1

To provide an estimate, suppose that the dispersion relation is strictly quadratic. At low temperatures, where only low-energy magnons are excited, this is a good approximation. Just as in the method employed for phonons, we shall integrate over a d-dimensional sphere of the same volume as the Brillouin zone. This leads to

	ddk	1			kmax	kd−1
M = gμB					0
				= gμBKd			dk , (15.5.3)
	(2π)d		eβDk2 − 1			eβDk2 − 1

where Kd = 1/(πd/22d−1Γ (d/2)). In terms of the new variable x = βDk2 the integral becomes

			kBT	d/2	xmax	x(d−2)/2
	1				0
M =	2 gμBKd						dx .	(15.5.4)
			D			ex − 1

In d > 2 dimensions the integral is convergent. Its value can be determined approximately by choosing the upper limit as infinity, and using the formulas given in Appendix C. For d ≤ 2, on the other hand, it is readily seen from the expansion of the integrand about x = 0 that the integral

∞	x(d 2)/2	∞	x(d−2)/2
0	−	dx 0		dx	(15.5.5)
	ex − 1		x

blows up at the lower limit. This indicates that the spin-wave approximation cannot be applied to such systems. An even stronger statement can also be

550 15 Elementary Excitations in Magnetic Systems

made: long-range ferromagnetic order cannot exist at any finite temperature in the isotropic Heisenberg model for d ≤ 2. The ordered ferromagnetic state can appear only as the ground state at T = 0, as at arbitrarily low but finite temperatures thermal fluctuations destroy the order, and spin–spin correlation functions decay exponentially.

For antiferromagnets a somewhat di erent calculation has to be performed. According to (15.3.23), the sublattice magnetization has to be determined from the expression

1			uk2	αk†	αk	+ vk2 βk† βk .	(15.5.6)
MA(T ) = MA(0) − gμB V k

Using the Bose–Einstein statistics for the magnon occupation number, and exploiting the fact that the thermal correction is governed mostly by longwavelength magnons, the sum in the above expression can be approximated by the integral

1		1						1			ddk 1				1			.	(15.5.7)
V k		eβ ωk		1							(2π)d		eβDk		1		k
			−				1	−	γk2					−

For any finite temperature there is a region close to the lower limit of integration where βDk < 1. To determine the contribution of this region, one has to evaluate

Kd	kd−1 dk	(15.5.8)
	βDk2 .

At the lower limit (k = 0) the last integral is finite for d = 3, while it is logarithmically divergent for d = 2. The divergence is even stronger for d < 2. This shows that antiferromagnetic order is also destroyed by thermal fluctuations in low-dimensional systems. This is in agreement with the Mermin–Wagner theorem, which has already been mentioned in connection with thermal disordering in two-dimensional lattices. The theorem, which was originally formulated for models described by the Heisenberg Hamiltonian, declares that in d ≤ 2 dimensions no long-range ordered state may exist at any finite temperature that breaks a continuous symmetry of the Hamiltonian. This clearly applies to the isotropic Heisenberg model, where the continuous symmetry is the rotational symmetry of the spins.

The problem of the ground state is even more interesting. Of course, an ordered ferromagnetic ground state may exist at T = 0 in arbitrary dimension because it is an eigenstate of the Hamiltonian (in other words: because the order parameter of the ferromagnetic state, magnetization, is conserved). The situation is di erent in antiferromagnets, where the Néel state is not an eigenstate. Long-range order may exist at T = 0 in two-dimensional systems, but it is destroyed by quantum fluctuations even at T = 0 for d < 2.

To demonstrate this, consider the zero-point spin contraction that characterizes the correction to the Néel state. According to (15.3.21), the correction to the average value of the spin due to spin waves is

15.5 Low-Dimensional Magnetic Systems

551

Sz = N

&

1 γ2

− 1' N

(2π)d k .

(15.5.9)

1

1

V

ddk 1

k

−

k

For a two-dimensional square lattice, the correction is about 0.2. Thus, a spin S = 1/2 is contracted to approximately 0.3, nonetheless the Néel-type order is preserved. For d < 2, and in particular for d = 1 the contribution at the lower limit is divergent, indicating that Néel-type antiferromagnetic order cannot exist in the ground state of one-dimensional systems. As we shall see, the ground state of the one-dimensional isotropic antiferromagnetic Heisenberg chain can be regarded as a singlet spin liquid.

15.5.2 Vortices in the Two-Dimensional Planar Model

The foregoing considerations were concerned with systems described by a Heisenberg model that is isotropic in spin space. As it was shown, such systems are on a borderline when the spatial dimension of the lattice of spins is two. In higher dimensions, e.g. in three-dimensional systems an ordered magnetic state – characterized by some order parameter – can emerge at low temperatures, and transition to the disordered phase takes place at a finite critical point. The correlation length becomes infinite in this point, and the spin–spin correlation function exhibits a power-law decay. At all other temperatures the correlation function decays exponentially.

On the other hand, we have seen in two dimensions that an ordered ground state may emerge for antiferromagnetic and ferromagnetic couplings alike, however, in accordance with the Mermin–Wagner theorem, there does not exist any state at finite temperature that breaks the continuous rotational symmetry of the Heisenberg model and features long-range order, since thermal fluctuations destroy any such order. The critical (Curie or Néel) temperature of the isotropic Heisenberg model is Tc = 0 for d = 2. The spin–spin correlation function decays exponentially at any finite temperature.

The Mermin–Wagner theorem does not apply to the Ising model, for the latter does not possess continuous rotational symmetry in spin space. The transition between ordered and disordered phases is well known to occur at fi- nite temperature in the two-dimensional Ising model. What about the planar, or XY model that falls between the Ising model and the isotropic Heisenberg model with respect to the dimensionality of the allowed spin space? In this model, by definition, spins lie in the (x, y) plane. As we shall see, this model exhibits an unusual phase transition. It takes place at a finite temperature, hence quantum e ects may be ignored. Regarding the spins as classical vectors, it will be assumed that these vectors are of unit length, located at lattice sites Ri, that they lie in the (x, y) plane, and their orientation is characterized by a polar angle θi:

Six = cos θi ,

Siy = sin θi .

(15.5.10)

Assuming that only nearest-neighbor spins interact, the Hamiltonian of the XY model reads

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	cos θi cos θj + sin θi sin θj = −J
H = −J		cos(θi − θj ), (15.5.11)
<ij>		<ij>

where ij denotes adjacent lattice sites. Since the model has a continuous rotation symmetry in the (x, y) plane, at finite temperatures there cannot exist a phase with long-range order that breaks this symmetry. If, nevertheless, there exists an “ordered” phase at low temperatures, it cannot be truly ordered – however the correlation in the orientation of the classical vectors may be stronger than in a usual disordered state. The phase transition manifests itself in a di erent analytic form of the correlation functions at low and high temperatures.

At finite temperatures the free energy of the system is obtained from the partition function

Z =	?	dθi e−H/kBT ,	(15.5.12)
	i

which is the sum (or integral) over all possible configurations of the angles θi. Some of the configurations correspond to spin-wave-like excitations. To study their e ect, we shall assume that the spins of nearest neighbors are only slightly rotated with respect to one another, and thus it is su cient to keep the leading-order term in the expansion of the cosine:

H = E0 + 21 J (θi − θj )2 ,	(15.5.13)

<ij>

where E0 is the energy of the completely ordered state. Replacing the variable θi defined at discrete lattice sites by the smooth function θ(r), we have

H = E0 + 12 J

In terms of the Fourier components:

H = E0 + 12 J

dr θ(r) 2 .	(15.5.14)
k2|θˆ(k)|2 .	(15.5.15)

k

However, we may encounter other configurations in which – apart from some singular points – θi varies slightly between adjacent lattice sites, nevertheless the sum of the di erences θi along a closed path C encircling a singular point will not be zero but an integral multiple of 2π,

θi = 2πq .

(15.5.16)

C

Such a configuration, featuring a singular point, is obtained, e.g., for

	θi = θ0	+ qϕi = θ0	+ q arctan	yi	,	(15.5.17)
				xi

15.5 Low-Dimensional Magnetic Systems

553

that is when the inclination of a spin is q times the polar angle of its lattice site ri. Needless to say, θi may just as well be a more complicated function of the polar angle. Such configurations, with quantum number q = ±1 and q = ±2 are presented in Figs. 15.4 and 15.5, respectively. Because of their vortex-like character, such configurations are called vortices and the quantum number is called vorticity.

q

q

q

Fig. 15.4. Vortex configurations of quantum number q = ±1 in the two-dimensional planar model

q

q

Fig. 15.5. Vortex configurations of quantum number q = ±2 in the two-dimensional planar model

Following the procedure used to show the topological character of dislocations it is easy to show that the integer q is a topological quantum number.

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In this classical spin model the variable θ(r) varies on the unit circle, thus the order parameter space is the unit circle. Going counter-clockwise around the core of a vortex of vorticity q = 1, the variable θ(r) goes around the unit circle once. If the spin orientation is distorted continuously, i.e., θ(r) is varied continuously locally, then every time when one goes around the vortex along the same circuit, a deformed path is found in the order parameter space, but its deformation is also continuous, and since it has to remain on the unit circle, it is not possible to reduce it to zero or to reach configurations of another quantum number.

Even when studying the role of vortex configurations it is admissible – apart from the immediate neighborhood of the core of the vortex – to approximate θi defined at lattice points only by a continuous function θ(r). Then, going around the vortex of vorticity q on a closed path C, we have, instead of (15.5.16),

C0	θ(r)
	ddr	dl = 2πq .	(15.5.18)

Assuming that the rate of change of the angle is constant on a circle of radius r around the core,

| θ(r)| =		2πq		.	(15.5.19)
		2πr
In polar coordinates	q
θ(r) =					(15.5.20)
			eϕ	,
	r

where eϕ is the tangential unit vector drawn to the point r. Using the radial unit vector er and the unit vector ez perpendicular to the plane,

q
θ(r) = −r er × ez = −q × (ez ln r) .	(15.5.21)

To estimate the energy of such a configuration, we shall further assume that this form is valid all the way from the lattice constant a to a radius R characteristic of the size of the system. Then

E = 21 J							R
		θ(r)		2 dr =	21 J				θ(r)	2	2πr dr
R							a				(15.5.22)
		q2				R
= πJ a
			dr = πJq2 ln				.
		r				a

Since this energy diverges as the size of the system is increased indefinitely, no such configuration is excited thermally at low temperatures.

However, such configurations may appear at high temperatures, since they may lower the free energy through the −T S term. As the vortex core can be at any lattice site, the number of vortex configurations is R2/a2, and the entropy of the vortex is