On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet
Macroscopic magnetoelastic domain structure of the defectless layered antiferromagnet of CoCl₂-type with the "easy-plane" magnetic anisotropy is studied theoretically in the framework of phenomenological approach. In assumption of mobile domain walls, the finite-size effects are shown to r...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
1999
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| Цитувати: | On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet / E.V. Gomonaj, V.M. Loktev // Физика низких температур. — 1999. — Т. 25, № 7. — С. 699-707. — Бібліогр.: 16 назв. — англ. |
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Gomonaj, E.V. Loktev, V.M. 2018-06-17T16:55:51Z 2018-06-17T16:55:51Z 1999 On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet / E.V. Gomonaj, V.M. Loktev // Физика низких температур. — 1999. — Т. 25, № 7. — С. 699-707. — Бібліогр.: 16 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/137858 Macroscopic magnetoelastic domain structure of the defectless layered antiferromagnet of CoCl₂-type with the "easy-plane" magnetic anisotropy is studied theoretically in the framework of phenomenological approach. In assumption of mobile domain walls, the finite-size effects are shown to result in the formation of a stable domain structure that changes reversibly under the action of the external magnetic field and can be treated as equilibrium. It is found that in antiferromagnets, where (in contrast to ferromagnets) long-range forces of magnetic origin are absent, the domain structure and its collective behavior are governed by elasticity. Field dependence of a domain structure, magnetostriction and low-frequency AFMR of poly- and monodomain samples are calculated, the external magnetic field being directed perpendicular to the main symmetry axis of the crystal. The results obtained are in qualitative agreement with the available experimental data. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низкотемпеpатуpный магнетизм On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet Article published earlier |
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On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet |
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On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet Gomonaj, E.V. Loktev, V.M. Низкотемпеpатуpный магнетизм |
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On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet |
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On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet |
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On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet |
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On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet |
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on the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet |
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Gomonaj, E.V. Loktev, V.M. |
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Gomonaj, E.V. Loktev, V.M. |
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Низкотемпеpатуpный магнетизм |
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Низкотемпеpатуpный магнетизм |
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1999 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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Macroscopic magnetoelastic domain structure of the defectless layered antiferromagnet of CoCl₂-type with the "easy-plane" magnetic anisotropy is studied theoretically in the framework of phenomenological approach. In assumption of mobile domain walls, the finite-size effects are shown to result in the formation of a stable domain structure that changes reversibly under the action of the external magnetic field and can be treated as equilibrium. It is found that in antiferromagnets, where (in contrast to ferromagnets) long-range forces of magnetic origin are absent, the domain structure and its collective behavior are governed by elasticity. Field dependence of a domain structure, magnetostriction and low-frequency AFMR of poly- and monodomain samples are calculated, the external magnetic field being directed perpendicular to the main symmetry axis of the crystal. The results obtained are in qualitative agreement with the available experimental data.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/137858 |
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On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet / E.V. Gomonaj, V.M. Loktev // Физика низких температур. — 1999. — Т. 25, № 7. — С. 699-707. — Бібліогр.: 16 назв. — англ. |
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2025-11-25T22:54:40Z |
| last_indexed |
2025-11-25T22:54:40Z |
| _version_ |
1850576167082393600 |
| fulltext |
Fizika Nizkikh Temperatur, 1999, v. 25, No 7, p. 699–707Gomonaj E. V. and Loktev V. M.On the theory of equilibrium magnetoelastic domain structure in easy-plane
antiferromagnetGomonaj E. V. and Loktev V. M.On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet
On the theory of equilibrium magnetoelastic domain
structure in easy-plane antiferromagnet
E. V. Gomonaj1 and V. M. Loktev1,2
1 National Technical University of Ukraine KPI, 37, Ave. Peremogy, Kiev, 252056, Ukraine
2 Bogolyubov’ Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 14-b,
Metrologichna str., Kiev, 252143, Ukraine
E-mail: malyshen@ukrpack.net
vloktev@bitp.kiev.ua
Received March 5, 1999
Macroscopic magnetoelastic domain structure of the defectless layered antiferromagnet of CoCl
2
-type
with the «easy-plane» magnetic anisotropy is studied theoretically in the framework of phenomenologi-
cal approach. In assumption of mobile domain walls, the finite-size effects are shown to result in the
formation of a stable domain structure that changes reversibly under the action of the external magnetic
field and can be treated as equilibrium. It is found that in antiferromagnets, where (in contrast to
ferromagnets) long-range forces of magnetic origin are absent, the domain structure and its collective
behavior are governed by elasticity. Field dependence of a domain structure, magnetostriction and
low-frequency AFMR of poly- and monodomain samples are calculated, the external magnetic field being
directed perpendicular to the main symmetry axis of the crystal. The results obtained are in qualitative
agreement with the available experimental data.
PACS: 75.50.Ee, 75.60.Ch
E. V. Gomonaj and V. M. Loktev
Introduction
The origin of equilibrium domain structure (DS)
in antiferromagnetic (AFM) insulators is studied
for a long time [1,2], but the question is still
obscure, despite a well developed theory for the
close vicinity of the magnetic 1-st order phase
transitions (in particular, for the field-induced spin-
flop transitions in the easy-axis antiferromagnets
(see the review [3] and recent paper [4] where this
theory has been generalized for the case of the
hexagonal AFM with the easy-plane magnetic an-
isotropy)). The DS of the pure antiferromagnets is
usually treated as the result of structural imperfec-
tions (such as dislocations, twins, impurities, etc.)
that cause the so-called sprout AFM domains (in-
cluding 180° domains). Sometimes, the origin of the
DS is attributed to the entropy factor, which de-
creases the free energy of the sample in the vicinity
of the critical temperature in the case of spatially
inhomogeneous ordering.
Both mentioned and some other possibilities
were analyzed in a recent paper [5] where numerous
experimental evidences of the equilibrium (almost
insensitive to growth conditions) domain structure
in dihaloids of transition metals MX2 (M = Mn,
Co, Ni; X = Cl, Br) were given. According to obser-
vations, the domain structure changes almost re-
versibly under the action of the external magnetic
field; it gradually disappears when the field is
switched on and recovers after it is switched off.
Such a behavior evidently points to the equilibrium
nature of the DS observed in these compounds*.
The authors [5] also suggested that magneto-
elastic interactions play a dominant role in forma-
tion of the equilibrium DS in the layered easy-plane
antiferromagnets of CoCl2-type. However, the only
condition pointed out is not sufficient and there is
an additional requirement necessary for DS forma-
tion, namely, the account of the sample surface. The
finite size effects bring about the appearance of the
DS during ferromagnetic and ferroelastic phase
© E. V. Gomonaj and V. M. Loktev, 1999
* We cannot exclude another possibility, when, for example, due to the high defect concentration, the internal stresses govern the
local equilibrium orientation of AFM vector. After the magnetic field is removed, this vector reverts to the initial state or to the
nearest easy direction. On the other hand, observations [5] point to the regular DS rather than to the stochastic.
transitions as well as the magneto-elastic interac-
tions.
These effects should play an important role in
the AFM, and particularly, in dihaloids of transi-
tion metals, where the antiferromagnetic domains
bear a magnetostrictive character and are of rather
small size. In fact, the width of the antifer-
romagnetic domain wall can be evaluated as
δ0 ∼ a(2HE/HA)1/2 ∼ a(2HE/Hsf ) ∼ 15−20a, whe-
re a is interatomic distance, HE , HA and Hsf are
the exchange field, the interplane anisotropy field
and the spin-flop field, correspondingly (for CoCl2
Hsf = 2 kOe [6] and spin-flip field 2HE = 32 kOe
[7]). Small thickness and low energy of the domain
walls (2⋅10−4 mJ/m2, compared with the typical
value 4 mJ/m2 for a ferromagnet) give grounds for
expecting the small size of AFM domains and as-
sume that the surface properties of the sample do
strongly depend upon the average strain.
It should be mentioned that an equilibrium DS is
observed in the martensites in the course of a
thermoelastic phase transition. It arises from the
condition of strain compatibility of martensitic and
austenitic phases. The peculiar feature of this struc-
ture is that the domain (twin) size is very small
(∼ 1000 A° , see [8]) and compatibility conditions
relate only with the average-strain.
The present paper is aimed at the theoretical
investigation of the macroscopic magnetoelastic do-
main structure of the defectless easy-plane layered
AFM of CoCl2-type with the account of finite-size
(surface) effects. In the framework of the pheno-
menological model, we calculate the domain struc-
ture, sample magnetostriction and AFMR frequency
in the presence of an external magnetic field di-
rected perpendicular to the main symmetry axis of
the crystal.
1. Model
We consider a thin plate of a layered easy-plane
rhombohedral antiferromagnet of CoCl2-type. The
crystal symmetry group is D3d
5 . The plate is ori-
ented perpendicular to the 3-rd order crystal axis,
labeled as z, x-axis is chosen along the 2-nd order
in-plane symmetry axis. Below the Ne′el tempera-
ture, the magnetic structure of the crystal can be
described with two orthogonal dimensionless vec-
tors: ferromagnetic, m, and antiferromagnetic, l,
m2 + l2 = 1. In the absence of external fields m = 0
and l has 3 equivalent orientations* in the basis
plane (directed along three 2-nd order symmetry
axes). In the in-plane external magnetic field, the
magnetic structure is described by two parameters:
the modulus m of the ferromagnetic vector, and the
angle ϕ between the l and x-axes, neglecting small
deflections of the magnetic vectors from the basis
plane (see Fig. 1). The bulk free energy of the
crystal in this case can be written in the simplest
form (see, e. g., [9]):
Fvol = ∫ dv {2JM0
2m2 + 2 β(2)M0
2m
z
2 −
−
2
3 β⊥
(6)M0
2(1 − m2)3 cos 6ϕ + 4M0
2[λ
me
(l) (1 − m2) +
+ λ
me
(m)m2] [(u
xx
− u
yy
) cos 2ϕ + 2uxy sin 2ϕ] +
+
1
2c66[(u
xx
− uyy)2 + 4u
xy
2 ] − 2M0H0m sin (ϕ − ψ)},
(1)
where the constant J describes the interplanar AFM
exchange; β(2), β⊥
(6) are the effective anisotropy
constants; 2M0 is the saturation magnetization; λme
are the magnetostrictive constants; uik (i, k = x, y)
are the strain tensor components; c66 is the elastic
modulus, principal for the case under consideration,
and the external magnetic field is defined as
H0 = |H0|, tan ψ = H0y/H0x . In the expression (1)
we have omitted the isotropic part of strain tensor,
uxx + uyy , qualitatively insignificant for the pre-
Fig. 1. Orientation of l vectors inside the domains, H0 is the
external magnetic field.
* We don’t distinguish between the states with l and − l. Besides, these three directions are usually provided by hexagonal
anisotropy, as it was suggested in [4]; but in rhombohedral AFM this anisotropy can be also caused by the difference in strain
components u
xz
, u
yz
that results in 60° in-plane anisotropy for l vector. In what follows, however, we shall suppose that there is
an in-plane magnetic anisotropy which effectively includes both these factors. So, corresponding strain components will be omitted
for simplicity.
E. V. Gomonaj and V. M. Loktev
700 Fizika Nizkikh Temperatur, 1999, v. 25, No 7
sent problem. The effective magnetostrictive con-
stants λme
(l) and λme
(m) originate from the relativistic
(dipole-dipole or spin-orbit) interactions and, as
was shown in [10], can essentially depend upon the
concrete electronic and crystal structure of a com-
pound.
Herein we consider the model in which the
surface effects are accounted through the surface
tension. The shape-dependent part of the surface
energy for the simplest case of disk with radius R
and thickness h is given by the expression
Fsurf =
πRh
2 σsurf [〈u
xx
− u
yy
〉2 + 4〈u
xy
〉2], (2)
where σsurf is the surface tension coefficient for the
(100) and (010) faces, notion 〈...〉 means the aver-
aging over the sample volume:
〈u
ij
〉 =
1
πR2h
∫u
ij
dv (3)
and we have neglected the contribution from the
disc sides (001) (which is appropriate if h << R).
In the expressions (1), (2) we have neglected the
magnetostatic energy which contribution, as will be
shown below, is much smaller than that from the
surface energy.
For small external fields, H0 << JM0 , the fer-
romagnetic moment m << 1 can easily be excluded
from (1). So, neglecting β(2) << J,
Fvol = ∫dv
−
2
3β⊥
(6)M0
2 cos 6ϕ −
H0
2
2J sin2(ϕ − ψ) +
+ 4M0
2λ
me
(l) [(u
xx
− u
yy
) cos 2ϕ + 2u
xy
sin 2ϕ] +
+
1
2c66[(uxx
− u
yy
)2 + 4u
xy
2 ]
. (4)
The local orientation of the vector l can then be
found by minimization of the functional
F = Fvol + Fsurf (5)
with respect to ϕ(r), u(r) functions. The correspond-
ing integral equations have the form:
u
xx
− u
yy
= −
4M0
2λ
me
(l)
c66
cos 2ϕ +
4M0
2λ
me
(l) σsurf
c66(c66R + σsurf)
〈cos 2ϕ〉
2u
xy
= −
4M0
2λme
(l)
c66
sin 2ϕ +
4M0
2λme
(l) σsurf
c66(c66R + σsurf)
〈sin 2ϕ〉
,
(6)
1
3Hs f
2 sin 6ϕ = sin 2ϕ [H0
2 cos 2ψ + 2H
MD
2 〈cos 2ϕ〉] − cos 2ϕ [H0
2 sin 2ψ + 2H
MD
2 〈sin 2ϕ〉] . (7)
Here we have introduced the characteristic fields convenient for further calculations: Hsf = = 2M0 √6β⊥
(6) J
— spin-flop field, and HMD = 4M0
2 λme
(l) [(2Jσsurf)/c66(c66R + σsurf)]
1/2 — the field of monodomenization.
Equation (7) evidently shows that the surface produces the same effect as an external magnetic field, the
effective internal field being defined as
Heff
2 = √H0
4 + 4H
MD
4 (〈cos 2ϕ〉2 + 〈sin 2ϕ〉2) + 2H0
2H
MD
2 〈cos 2(ϕ − ψ)〉2 (8)
tan 2ψeff =
H0
2 sin 2ψ + 2H
MD
2 〈sin 2ϕ〉
H0
2 cos 2ψ + 2H
MD
2 〈cos 2ϕ〉
.
Thus, it can be stressed that in the case of AFM
it is the elastic strain that plays the role similar to
the magneto-dipole interaction in ferromagnets.
Note, that for infinite sample (R → ∞) HMD → 0
and effective field identically coincides with the
external field H0 ; for such a situation ψeff → ψ. o.
Equation (7) has different solutions depending
on the physical situation considered below.
1.1. Mobile domain walls
In this case the average strain can follow the
changes caused by an external magnetic field. Up to
a certain field value, H0 = H1c , specified below,
the effect of the magnetic field is compensated by
the average strains, so that the effective field inside
the sample Heff = 0*. Equation (7) has three non-
On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet
Fizika Nizkikh Temperatur, 1999, v. 25, No 7 701
trivial solutions ϕ1 = 0, ϕ2,3 = ± 2π/3, correspond-
ing to 3 equivalent equilibrium orientations of the
l vector, i.e., to three magnetoelastic domains (as
was already pointed out before). Evidently, each of
them is distorted orthorhombically [in correspon-
dence with [11,12], see formulas (6)]. Such a dis-
tortion for easy-plane AFM was observed in [13].
Moreover, magnetoelasticity proves to be a crucial
factor for the existence of an equilibrium DS.
In neglecting the domain wall energy, one can
find the relative volume ξj(j = 1, 2, 3) of each
domain from the following equations, obtained
from (8):
H0
2 cos 2ψ + 2H
MD
2 〈cos 2ϕ〉 = 0 (9)
H0
2 sin 2ψ + 2H
MD
2 〈sin 2ϕ〉 = 0, (10)
we take it into account that:
〈cos 2ϕ〉 = ∑ξ
j
j
cos 2ϕj , ∑
j
ξ
j
= 1. (11)
The ultimate expression for ξj is
ξ
j
=
1
3
1 −
H0
2
H
MD
2 cos 2(ϕ
j
− ψ)
, j = 1, 2, 3. (12)
Note, that in this case ξj are the thermodynamic
variables as well as strain components and ϕ; the
equality of the chemical potentials (free energy
densities) of different «phases» (domains) is satis-
fied automatically.
According to (12), the volume fraction of the
domains depends upon the value of the external
magnetic field H0 . At zero field, H0 = 0, all three
types are equally distributed, so that the symmetry
of the sample does not change after the transition
into the polydomain antiferromagnetic state. In
nonzero field the fraction of the most energetically
«unfavorable» domain (l vector lies closely to the
direction of the magnetic field, say, domain 1 for
0 < ψ < π/6, see Fig. 1) diminishes. At H0 =
= H1c = HMD/(cos 2ψ)1/2 the domains of the 1-st
type disappear. Further behavior of the system at
H0 > H1c can be found out from the equations (7),
(10) and (11) with j = 2, 3. In this case the internal
effective field is no longer zero, but it is directed
along x-axis (ψeff = 0)**, so, ϕ2 = −ϕ3 = ϕ, and
cos 2ϕ =
3HMD
2
4H
sf
2 −
−
1
2
1 +
3HMD
2
2H
sf
2
2
+ 3
H0
2 − H1c
2
H
sf
2 cos 2ψ
1/2
, (13)
ξ2,3 =
1
2
1 +−
H0
2 sin 2ψ
H
MD
2 sin 2ϕ
.
(14)
In other words, in the two-domain structure the l
vectors inside the domains start to rotate and simul-
taneously the fractions of the different domains
change. The process of monodomenization is com-
pleted at some critical field H0 = Hc which can be
found from equation (13) along with the condition
ξ2 = 0 or in other words,
H
c
2 sin 2ψ = H
MD
2 sin 2ϕ . (15)
For the case ψ = 0 both domains 2 and 3 disappear
simultaneously at
H0 = H2c ≡ √H
sf
2 + 2H
MD
2 , (16)
when all the l vectors achieve the direction perpen-
dicular to the external field. Effective field (16) of
monodomenization is defined both by the magnetic
anisotropy (due to Hsf) and by the surface effect
(due to HMD). For the symmetric case ψ = π/6
monodomenization is completed at H0 = H1c when
domains 1 and 2 disappear and l vector in the third
domain is aligned perpendicular to the external
field. For the general case, 0 < ψ < π/6, critical
field H1c < Hc < H2c ; after the process of mono-
domenization is finished, further change of ϕ angle
can be calculated from the equation
1
3
H
sf
2 sin 6ϕ = H0
2 sin 2(ϕ − ψ). (17)
The considered model gives rise to thermody-
namically equilibrium domain structure at any mag-
netic field value. Really, the difference in free
energy of the polydomain and monodomain state
calculated from (1), (2) at the same external field
value,
* The similar equality holds true for DS corresponding to the so-called intermediate state that exists in the vicinity of the 1-st order
spin-reorientation phase transition in the easy-axis AFM [3].
** It is easy to check that another solution of (7), (8) with ψ
eff
= π/2 is energetically unfavorable.
E. V. Gomonaj and V. M. Loktev
702 Fizika Nizkikh Temperatur, 1999, v. 25, No 7
Fpoly − Fmono = −
V
2J
H
MD
2 +
1
2
H0
2 cos 2(ϕ − ψ) +
+
1
18
H
sf
2 (1 − cos 6ϕ)
≤ 0 (18)
is nonpositive, which makes the polydomain state
thermodynamically preferable. So, in the model
proposed the behavior of the DS in the external
magnetic field is absolutely reversible.
1.2. Immobile domain walls
The domain walls cannot move freely, so, the
ratio of the domains is fixed and only the rotational
processes take place inside the domains. Orientation
of l vectors can be calculated from equations (7),
(11) with the given ξj values which are defined by
technological factors. For small external field the
equation (7) has 3 solutions corresponding to differ-
ent domains. Monodomenization of the sample is
completed when all the l vectors are aligned perpen-
dicular to the external field direction.
For illustration let us consider the symmetrical
case ψ = 0. If initially the domains have been pro-
duced by stray field at random, then ξ1 = ξ2 =
= ξ3 = 1/3 and
ϕ1 = 0, ϕ2 = − ϕ3 = ϕ,
cos 2ϕ =
H
MD
2
2H
sf
2 −
1
2
1 +
H
MD
2
Hsf
2
2
+ 3
H0
2
H
sf
2
1/2
. (19)
The field of monodomenization Hc
immob =
= (Hsf
2 + 2⁄3HMD
2 )1/2 in this case is smaller than the
corresponding value for the case of mobile domain
walls [Hc
immob < H2c , compare with formula (16)].
For arbitrary ψ the value of monodomenization field
can be much greater.
After the magnetic field is removed, such a
domain structure will not restored, at least, in
principal, because l vectors will tend to lie along
the nearest easy axis, which for general field orien-
tation is the only one. So, once cycled in the
magnetic field, the sample becomes monodomain
and the behavior of the DS in this case is absolutely
irreversible.
In the real experiments, the behavior of the DS
in antiferromagnets of CoCl2-type are partly irre-
versible, so, we can assume some intermediate case
when most of the domain walls are mobile but some
of them are pinned by the defects or different
imperfections of a crystal and contribute to the
certain irreversibility mentioned and observed.
2. Magnetostriction
In the experiments [5,7] the magnetic field was
arbitrarily oriented in the easy plane of the crystal,
and magnetostriction was measured along and per-
pendicular to the field direction. Corresponding
macroscopic elongations, (∆l/l )|| and (∆l/l )⊥ can
be calculated according to the general formula
(∆l/l )n = ∑ ni < u
ik
> n
k
, (20)
where n is a unit vector in the direction of measure-
ment, < uik> is the averaged strain tensor. Substi-
tuting (6) into (20) and neglecting of isomorphous
strain (uxx + uyy) one readily obtains:
∆l
l
||
= −
∆l
l
⊥
=
= −
4M0
2R
λ
me
(l) + (λ
me
(m)− λ
me
(l) )
H0
2H
E
2
c66R + σsurf
〈cos 2(ϕ − ψ)〉,
(21)
where HE = JM0, is the exchange field. In the
formula (21) we have taken into account the de-
pendence m(H0) ≈ H0/2HE , which is significant at
H0 ≤ 2HE .
Field dependence of elongation (∆l/l)|| calculated
from (21) for CoCl2 with ψ = 0, Hsf = 2 kOe;
HMD = 3.3 kOe, 2HE = 32 kOe, 4M0
2λme
(l) =
= − 4M0
2λme
(m) = 36 MPa, c66 = 34.7 GPa [7] is
shown in Fig. 2. We have considered two cases:
mobile (solid curve) and immobile (dash curve)
domain waIls. Figure 3 shows the same dependences
vs squared magnetic field, H0
2 ; the points corre-
spond to experimental data [7]. The difference be-
tween two theoretical curves is significant at low
field value. In the case of mobile domain walls the
theoretical dependence (solid curve) is in good
agreement with the experimental data. The HMD
value was taken to fit the experimental slope
(∆l/l )⊥vs H0
2 at H0 < 3 kOe.
The behavior of magnetostriction as seen from
formula (21) and Fig. 2 is governed by two proc-
esses. At small field, H0 ≤ Hc << HE , magne-
tostriction of the sample is changed due to the
process of monodomenization that influences the
average cos 2(ϕ − ψ) value (increasing section of the
curve in Fig. 2). After this process is finished,
variation of monodomain magnetostriction is de-
fined only by an increase of magnetization in the
external magnetic field (decreasing section of the
On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet
Fizika Nizkikh Temperatur, 1999, v. 25, No 7 703
curve in Fig. 2). It is seen from (21) that in the
monodomain state the slope (∆l/l )⊥vs H0
2 depends
upon the difference λme
(m) − λme
(l) only, that must be
considered as a phenomenological parameter.
3. AFMR spectra
Experimentally measured field dependence of the
low-frequency AFMR [7] shows that below 5 kOe
the resonance frequency is approximately two times
lower than the value extrapolated from measure-
ments at higher frequency. In the interval H0 ∼ 5–
7 kOe field dependence of AFMR is practically
vertical and starting from 7 kOe it turns out to be
in agreement with high-frequency measurements.
The characteristic field value coincides with the
value of monodomenization field Hc observed in
magnetostriction experiments.
The observed peculiarity in the AFM spectra can
be interpreted in the framework of the equilibrium
magnetoelastic domain model developed above. In-
deed, for the infinite samples the low AFMR fre-
quency is contributed by the magnetic anisotropy
field and magnetoelastic field as well, due to the
fact, that the crystal lattice is «frozen». On the
other hand, it was shown by Gann and Zhukov [14]
that for small samples the lattice relaxes together
with the antiferromagnetic vectors, and then, the
resonance frequency is defined mainly by the local
anisotropy field.
The similar effect of «unfreezing» of crystal
lattice can be achieved in the AFM with the magne-
toelastic DS, if the domain size is quite small. To
catch the effect, let us consider the in-plane oscilla-
tions of the magnetic moments together with acous-
tic waves in the polydomain sample with mobile
domain walls. Low-frequency AFMR can be found
on the basis of Lagrangian formalism with a La-
grangian taken in a standard form [15]:
L = ∫dv
ϕ
. 2
2Jg2 +
1
2
ρu
. 2
− Fvol , (22)
where g is gyromagnetic ratio, ρ is a crystal density,
u is a displacement vector, Fvol is given by formula
(4). Corresponding Euler-Lagrange equations have
the form:
ϕ
..
−
g2
2
H0
2 sin 2(ϕ − ψ) +
g2
3
H
s f
2 sin 6ϕ − 16g2λ
me
(l) JM0
2[(uxx − u
yy
) sin 2ϕ − 2uxy cos 2ϕ] = 0
u..
x
− s2
∂2
∂x2 + ∂2
∂y2
u
x
−
4λ
me
(l) M0
2
ρ
∂ cos 2ϕ
∂x
+
∂ sin 2ϕ
∂y
= 0
u..
y
− s2
∂2
∂x2 +
∂2
∂y2
u
y
−
4λ
me
(l) M0
2
ρ
∂ sin 2ϕ
∂x
−
∂ cos 2ϕ
∂y
= 0 ,
(23)
Fig. 2. Field dependence of magnetostriction of polydomain
crystal: solid line — theoretical, mobile domain walls; dash
line — theoretical, immobile domain walls; points — experi-
mental [7].
Fig. 3. Magnetostriction vs squared magnetic field in polydo-
main crystal: solid line — theoretical, mobile domain walls;
dash line — theoretical, immobile domain walls; points — ex-
perimental [7].
E. V. Gomonaj and V. M. Loktev
704 Fizika Nizkikh Temperatur, 1999, v. 25, No 7
where s = (c66/ρ)1/2 is the in-plane sound velocity
and equilibrium ϕ value depends upon the x and y
coordinates. Equations (23) describe the perturba-
tions over inhomogeneous (polydomain) state of the
sample.
The low-frequency branch of AFMR for
H0 < H1c can be then calculated from the following
equation
ω2 = g2
H
sf
2 + H
ME
2
1 − ∫
k2(|ak|
2 + |bk|
2)
k2 − (ω2/s2)
dk
,
(24)
where HME = 8M0
2λme
(l) √J/c66 is a magnetostriction
field; ak , bk are the Fourrier components of the
functions sin 2ϕ(r) and cos 2ϕ(r), correspondingly,
∫(|ak|
2 + |bk|
2) dk = 1.
The relation (24) shows that resonance fre-
quency depends upon the average domain size d.
For macroscopic domains with d >> λ ≡ s/(gHsf)
characteristic value of k ∼1/d << s/ω and the last
term in (24) can be neglected. In this case the
domain can be treated as infinite, corresponding
AFMR frequency is
Ω
AFMR
(∞) = g √H
sf
2 + H
ME
2 (25)
and we arrive to a standard situation with the
«frozen» lattice, AFMR gap is defined by anisot-
ropy and magnetoelasticity as well. In the opposite
case with d << λ, the Fourrier spectrum of func-
tions sin 2ϕ(r) and cos 2ϕ(r) has two significant
contributions with k = 0 and k = π/d. The corre-
sponding expression for AFMR frequency is
Ω
AFMR
= g
Hsf
2 +
H0
4H
ME
2
4H
MD
4
1/2
×
×
1 +
g2HME
2 d2
2π2s2
(|aπ/d
|2 + |bπ/d
|2)
< Ω
AFMR
(∞) ,
(26)
where we have taken into account that
|a0|
2 + |b0|
2 = 〈cos 2ϕ〉2 + 〈sin 2ϕ〉2 =
H0
4
4HMD
4
.
So, in the magnetically inhomogeneous sample the
crystal lattice does follow the oscillations of the
magnetic moments and thus diminishes the mag-
netoelastic contribution to the resonance fre-
quency [see formula (26)]. This effect should be
more pronounced in CoCl2 , where magnetoelastic
contribution into AFMR spectrum is of the same
order as an anisotropy one.
The field dependence of AFMR spectrum in
CoCl2 can be thus explained as follows. Suppose, at
zero field the sample has a well developed DS with
the average size d ≤ λ (for CoCl2 λ ∼ 10−7 m ). The
lattice is then proves to be partially «unfrozen» and
frequency is defined mainly by anisotropy field [see
expression (26)]. The external magnetic field af-
fects the AFMR frequency in two ways: through the
variation of average cosine and sine values and
through the increase of the average domain size*
d(H0) [last term in (26)]. As a result, the frequency
grows smoothly with the field H0 . After the do-
main size achieves macroscopic value d ∼ R >> λ,
the lattice becomes frozen, and resonance frequency
steeply jumps to the value corresponding to infinite
homogeneous sample [formula (25)]. Tentative be-
havior of AFMR vs magnetic field calculated from
(25), (26) with g = 6, Hsf = 2 kOe, HME = 1.5 kOe,
HMD = 3.3 kOe is shown in Fig. 4.
Additional decrease of AFMR frequency in the
polydomain sample can also result from the damp-
ing of the domain wall motion, stimulated by reas-
sembling of the DS in the external magnetic
field [16]. The ultimate value of AFMR frequency,
Ω~A FMR = √ΩAFMR
2 − δ2 , where δ is the damping
coefficient, can be significantly lower than ΩAFMR
(∞) .
* We suppose that the change of the DS proceeds by the growth of the domains of certain type at the expence of others.
Fig. 4. Theoretical field dependence of resonance AFMR fre-
quency (scheme): solid line — polydomain sample; dash line —
infinite homogeneous sample.
On the theory of equilibrium magnetoelastic domain structure in easy-plane antiferromagnet
Fizika Nizkikh Temperatur, 1999, v. 25, No 7 705
It can be shown that the model of immobile
domain walls gives another form of spectrum which
does not correlate with the experimental data.
4. Discussion
Above we propose the model which naturally
interprets the antiferromagnetic domains in the
easy-plane antiferromagnets as equilibrium. The
model is based on the magnetoelastic origin of the
domains, supposition of mobile domain walls and
additional condition imposed on the average strain
of the sample. For illustration we have considered
the simplest example when the condition in ques-
tion was related with the surface tension of the
sample which was the case if the surface energy
made a significant contribution into free energy of
the sample. As a consequence of the model, the
value of monodomenization field depends upon the
characteristic size R of the sample and thus vanishes
when R tends to infinity.
The results obtained could be generalized if R
implies the characteristic size of substructure unit of
the sample (i.e., dislocation-free region or crystal-
lite), defined by the technological factors. In the
case of fixed (glued) sample expression (2) should
be substituted with corresponding boundary condi-
tions for the sample faces. Anyway, finite-size ef-
fects can play a crucial role in the behavior of
antiferromagnets with significant magnetostriction
and should be taken into account in interpretation
of experimental results.
The predicted behavior of the DS of AFM is
analogous to that of ferromagnetic. Namely, up to
some critical external field value, the effective in-
ternal magnetic field is compensated due to the
reassembling of the domains. The value of critical
field is defined by the geometry and size of the
sample. It is interesting and important that in AFM
model considered the long range interactions in-
clude no magnetic component and are completely
provided by elasticity.
The size of the domains can be calculated with
the account of short-wavelength contribution to the
surface energy which compensates the increase of
volume energy resulting from the domain walls.
In our calculations we have not considered the
magnetostatic effects and domain walls themselves.
Demagnetization factor influences the DS in the
region of 1-st order (spin-flop) phase transition
which takes place in narrow interval of angle
ψ ≈ 0 (see [4]). For arbitrary orientation of the
external magnetic field in the easy plane of AFM
the magnetostatic contribution is (HMD/2HE) ∼
∼ 0.06 << 1 times smaller than the surface energy
and thus can be neglected.
The account of domain walls is necessary for
evaluation of the domain size that is out of scope of
the present paper. The structure of domain walls
can be calculated in an analogy with general ap-
proaches (see [3]) by taking into account magnetic
and elastic subsystems. It should be noted that in
the defect-free sample the interdomain boundary are
ideally conjugated and no stresses appear because of
compatibility conditions.
Conclusions
1. The domain structure of easy-plane AFM with
degenerated orientation of antiferromagnetic vector
can be treated as equilibrium in the finite-size
sample with the mobile domain walls. The effect
originates from the magnetoelastic nature of the
domains with the account of surface tension.
2. At zero magnetic field all types of the domains
are equally represented. The external magnetic field
effects the magnetic and elastic properties of the
sample in two ways. At small field value variation
of the domain structure gives rise to additional
average strain field which compensates the external
magnetic field, the orientation of the magnetic
moments inside domains being fixed. At some criti-
cal value, H0 = Hc , the sample becomes monodo-
main and magnetic field results in reorientation of
the magnetic moments.
3. Experimentally observed magnetostriction vs
magnetic field dependence for CoCl2 crystal is ade-
quately described in terms of the model under
consideration with mobile domain walls.
4. AFMR frequency of polydomain crystal can be
significantly lowered due to «unfreezing» of the
crystal lattice if the size of domain is quite small.
5. The above mentioned calculations can be con-
sidered as a basis for the following important and
quite general supposition: variation of the surface
energy (that is of Coulomb nature and so is not
small) can result in the formation of equilibrium
inhomogeneous state (or in other words, the equi-
librium domain structure of magnetoelastic or elas-
tic nature).
Acknowledgements
We are sincerely indebted to Prof. S. M. Ryab-
chenko, Dr. A. F. Lozenko and Dr. V. M. Kalita for
stimulating and constructive discussions of the
problem under consideration and for opportunity to
get acquainted with the results of the manu-
script [5] before publication. We are also grateful
to Prof. V. G. Bar’yakhtar for the interest to the
work and useful comments.
E. V. Gomonaj and V. M. Loktev
706 Fizika Nizkikh Temperatur, 1999, v. 25, No 7
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Fizika Nizkikh Temperatur, 1999, v. 25, No 7 707
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