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Submitted on 1 Jan 1971
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FERROMAGNETIC-LIKE MAGNON DISPERSION IN A FACE-CENTERED CUBIC WITH TYPE-I
ANTIFERROMAGNETIC ORDER
J. Kouvel
To cite this version:
J. Kouvel. FERROMAGNETIC-LIKE MAGNON DISPERSION IN A FACE-CENTERED CUBIC WITH TYPE-I ANTIFERROMAGNETIC ORDER. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-695-C1-696. �10.1051/jphyscol:19711244�. �jpa-00214073�
JOURNAL DE PHYSIQUE Colloque C I , supplbment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1
-
695FERROMAGNETIC-LIKE MAGNON Dl SPER SION IN A FACE-CENTERED CUBIC
WITH TYPE-I ANTIFERROIMAGNETIC ORDER
J. S. KOUVEL
Department of Physics, University of Illinois, Chicago, Illinois 60680, U. S. A.
Rbsum6. - Bien que le spectre magnon d'un antiferromagnkte Type-I est normal (dispersion lineaire, energie d'in- tervalle dependant de 1'Cchange et de l'anisotropie) prks k = 0, celui-la se trouve anormalement semblable a un ferroma- gnete (dispersion quadratique, energie d'intervalle dependant seulement de l'anisotropie) prks de certains points de syme- trie aux bornes de la zone magnetique Brillouin. Cette f a ~ o n d'agir, dkrivant d'une exacte equivalence en energie d'echange dans une famille de variantes non colineaires d'arrangement Type-I, pourrait prkdominer en determinant plusieurs propriktes thermodynamiques.
Abstract. - Although the magnon spectrum for a Type-I antiferromagnet is normal (linear dispersion, gap energy dependent on exchange and anisotropy) near k = 0, it is found to be anomalously ferromagnetic-like (quadratic disper- sion, gap energy dependent only on anisotropy) near certain symmetry points on the magnetic Brillouin zone boundaries.
The latter behavior, arising from an exact equivalence in exchange energy among a family of non-collinear variants of Type-I ordering, could predominate in determining various thermodynamic properties.
As is well known, atomic moments on a f. c. c, cell remains exactly cubic, so that all interactions of lattice can form a stable antiferromagnetic configura- the same range can be considered equal. Since we are tion only if nearest-neighbor exchange interactions are not immediately concerned about effects that are accompanied by interactions of longer range. In fact, specifically quantum mechanical, we resort to the the longer range interactions determine the type of spin-vector equation of motion,
ordering that can take place [I]. Less known, however, -
A
is that Type-I ordering (shown in Fig. 1) has a whole ti - -
as^,
at - 2 ~ J , , ! ? i x ~ $ j n + 2 ~ , ? i x ~ n jn S ( 1 )
-
FIG. 1. - Type-I antiferromagnetic structure.
family of non-colinear equivalent structures [2, 31, which could have a profound effect on the magnon spectrum. A recent experimental study of the magnon dispersion in y-Mn [4], a metallic Type-I antiferroma- gnet, has given us an immediate incentive for carrying out the spin-wave analysis described below.
In this analysis, the spin axis of the colinear Type-I structure will be taken to be parallel to the unique tetragonal axis, i. e., the z axis in figure 1. However, the choice of spin axis has no effect on our final results.
A more significant assumption is that the chemical
involving isotropic exchange interactions of different range (n) and an anisotropy. For Type-I ordering to be the stable magnetic state, it is necessary and sufficient that Jl < 0 and J, > 0 for the nearest and next- nearest neighbor interactions, respectively. At low temperatures, the two sublattice spin vectors shown in figure 1 can each be expressed by small plane-wave spin deviations in the x-y plane and a z-component that is static to first order in the deviations, so that
and similarly for
S^(2)
with the z-component reversed.Inserting these expressions into eq. (1) and neglecting terms quadratic in the spin deviations, we obtain a set of linear equations whose solution yields the following dispersion relationship :
( A W ) ~ = AZ - B~ (3) A - 8 J l S ( l
+ 8,
By)+
-I- 24 J 2 S [ 1 - +(8:
+ +
p:)]+
2 KB - 8 JI
s(Bx +
By)PI
where
p,
= cos & a, k,, etc., a, being the cubic lattice parameter. The solution also gives the following relationship between the spin deviations :sy'/s'l'
=- s?'/sF'
= + - 1( 4 ) S;"/S?) =
+
i [ ( A _+ B)/(A T B)]%.
Hence, for any value of k, the sublattice spins precess in identical elliptical orbits but in opposite senses
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711244
C 1
-
696 J. S. KOWEL (since the effective anisotropy fields on the two sublat-tices are antiparallel). The sign changes in eq. (4)
Y
merely interchange the x and y axes and/or the senses
of precession.
s "Q
At the reciprocal lattice points for this tetragonal magnetic structure, i. e., at (hko with h
+
k even,eq. (3) reduces to
and near these points it gives for K z 0,
hw z S(J? cos2 cp - J1 J2)% Sao k (5b) where Q is the anale between and
2.
The form of eq. (5aj for the gap energy and the linear dispersion described by eq. (5b) are the familiar magnon proper- ties of a typical antiferromagnet. However, at or near (100) and other cubic points located on the magnetic Brillouin zone boundaries, we find that the analoguesof eqs. (5a) and (5b) derived from eq. [3) are FIG. 2. -Spin precession in basal (x-y) plane at (100) and h o = 2 K (6a) equivalent points in k-space.
and
where $ is measured from these points, 8 and (g being the angles it makes with
x^
and2
respectively. Here, since the gap energy depends only on K and the dispersion is quadratic, the magnon behavior resembles that of a ferromagnet near k = 0. This anomaly can be traced to the pattern of neighboring spin deviations for these special values of k, which is shown in figure 2 as derived from eq. (4). We can readily deduce that the net exchange field at -any atom is exactly parallel to the precessing spin and that its magnitude is inde- pendent of the amplitude of spin deviation, which is what normally obtains in a ferromagnet at k = 0.Indeed, this situation reflects the exact equivalence in total exchange energy between the collinear Type-I structure and all the non-collinear structures in which the static spin canting is as described in figure 2 (with the spin deviations frozen in time). This family
of equivalent antiferromagnetic structures was first noted by Kasper and myself [2] and later rediscovered by Hastings et al. [3].
The existence of exchange interactions of range beyond next-nearest neighbor atoms alters the effective values of J , and J, but otherwise leaves the above equations unchanged. Quite generally, therefore, for any material having Type-I antiferromagnetic order and an anisotropy (including that arising from any crystallographic distortion) much weaker than the effective exchange, the magnon energies related to the ferromagnetic-like part of the spectrum may be considerably lower (with a higher density of states) than the energies associated with the normal magnon behavior near k = 0. These anomalous low-lying states may therefore predominate in determining various thermodynamic properties, such as the varia- tion of the sublattice magnetization and specific heat at low temperatures.
Helpful discussions with Dr R. D. Lowde and his associates at Harwell [4] are gratefully acknowledged References
[l] ANDERSON (P. W.), Phys. Rev., 1950, 79, 705. 141 HAYWOOD (B. C . G.), LOWE (R. D.), STRINGFELLOW [2] KOUVEL (J. S.) and KASPER (J. S.), J. Phys. Chem. (M. W.), and WAEBER (W. B.), Proceedings of
Solids, 1963, 24, 529. this Conference, 1971.
[3] HASTINGS (J. M.), CORLISS (L. M.), BLUME (M.), and PASTERNAK (M.), Phys. Rev., B, 1970, 1, 3209.
