PHYSICAL REVIEW B 84, 134304 (2011)
Coupled magnetoelastic waves in ferromagnetic shape-memory alloys
V. G. Bar’yakhtar,
1
A. G. Danilevich,
1,*
and V. A. L’vov
1,2
1
Institute of Magnetism, 36-b, Vernadsky Str. 36-b, Kyiv 03142, Ukraine
2
Department of Radiophysics, Taras Shevchenko University, 2, Glushkov Ave., Kyiv 01601, Ukraine
(Received 13 June 2011; revised manuscript received 23 September 2011; published 24 October 2011)
The theory of the spectra of coupled magnetoelastic waves in ferromagnetic shape-memory alloys (FSMA)
is developed. The possibility of an abnormally strong coupling of spin waves with the soft elastic mode at
approaching the martensitic transformation (MT) temperature is disclosed. In particular the magnetoelastic
waves in Ni–Mn–Ga single crystals are considered. A considerable (by an order of magnitude) reduction of the
shear elastic modulus and an appropriate lowering of the transversal velocity of sound in the applied magnetic
field are predicted. Optimum conditions for the experimental observation of the predicted effects are specified.
DOI: 10.1103/PhysRevB.84.134304 PACS number(s): 75.80.+q, 64.60.−i, 62.20.de, 75.47.Np
I. INTRODUCTION
The magnetoelastic interaction results in the coupling of
spin waves with sound/ultrasound waves propagating in mag-
netically ordered crystals.
1−3
The effect of coupling strongly
increases when the frequency of a spin wave approaches
the sound frequency. In this case the “repulsion” of the
quasispin and quasisound branches of the wave spectrum is
observed. The coupled magnetoelastic waves are studied for a
long time (see, e.g., Refs. 4–7). Waves observed in crystals
undergoing the spin-reorientation transitions are of special
interest, because the manifestations of the magnetoelastic
coupling become more pronounced in a vicinity of the phase
transition points.
8,9
This happens because the energy gap in
the spin-wave spectrum decreases at approaching the phase
transition point. The magnetoelastic coupling leads to a sub-
stantial reduction of the velocity of quasisound magnetoelastic
waves when the energy gap becomes comparable in magnitude
with the value of repulsion of the quasispin and quasisound
branches of the spectrum. In this case the theory predicts
the possibility of a drastic decrease in the sound velocity
through zero value. This means that the linear dispersion law
for a quasisound converts into the quadratic one at the phase
transition temperature.
5,8,9
In practice a substantial reduction
of the quasisound wave velocity is observed in a vicinity of
the spin-reorientation transitions.
Ferromagnetic shape-memory alloys (FSMAs) are consid-
ered now as a new class of materials that is of importance in
both applied and academic aspects.
10–12
Specific properties of
these alloys are caused by their martensitic transformations
(MTs). The MT of FSMA is accompanied by a spontaneous
deformation of the crystal lattice and a pronounced softening
of the shear modulus. In particular the Ni–Mn–Ga alloys
exhibiting the cubic-tetragonal MT are intensively studied.
The most intriguing feature of these materials is a giant
(>5%) magnetically induced deformation, which is caused
by a transformation of the twin structure of a single crystalline
specimen in the external magnetic field (for more details, see,
e.g., Refs. 10–12). According to the magnetoelastic model of
FSMA, the transformation of the twin structure is caused, in
turn, by the ordinary magnetostriction.
12
A drastic increase
of magnetostriction was observed experimentally
13,14
in a
Ni–Mn–Ga alloy at the MT temperature and described within
the framework of magnetoelastic model.
15
As far as we know, the coupled magnetoelastic waves in
FSMAs were not considered yet. The experimentally observed
increase in the magnetostriction
13
and a softening of the shear
elastic modulus
16–19
of a Ni–Mn–Ga single crystal in a vicinity
of the MT temperature suggest the idea of a strong effect of
the magnetoelastic coupling on the spin-wave and sound-wave
spectra.
In the present article a theory of magnetoelastic coupling
is applied to the description of magnetoelastic waves in
the cubic phase of FSMA. Special attention is paid to the
case when the temperature of the single crystalline specimen
of the alloy approaches the temperature of cubic-tetragonal
phase transition. The general theoretical relations are used
for the quantitative description of the spectrum of coupled
(magnetoelastic) waves in a Ni–Mn–Ga single crystal.
II. FORMALISM
To describe the coupled magnetoelastic waves, we present
the free energy of a Ni–Mn–Ga single crystal in the form
F = F
m
+ F
e
+ F
me
. (1)
The first term is the energy of the magnetic subsystem of a
cubic crystal,
F
m
=
α
2
∂ M
∂x
i
∂ M
∂x
k
+
K
1
M
4
(
M
2
x
M
2
y
+ M
2
x
M
2
z
+ M
2
y
M
2
z
)
+
K
2
M
6
M
2
x
M
2
y
M
2
z
− MH, (2)
where M and H are the magnetization and magnetic field
vectors, α is the exchange energy constant, and K
1
and K
2
are
the magnetocrystalline anisotropy constants.
4
The second term
F
e
=
3
2
(C
11
+ 2C
12
)u
2
1
+
1
6
C
′
(
u
2
2
+ u
2
3
)
+ 2C
44
(
u
2
4
+ u
2
5
+ u
2
6
)
, (3)
is the elastic energy expressed in terms of the linear combi-
nations of components of the strain tensor: u
1
= (ε
xx
+ ε
yy
+
ε
zz
)/3, u
2
=
√
3(ε
xx
− ε
yy
), u
3
= 2ε
zz
− ε
xx
− ε
yy
, u
4
= ε
yz
,
u
5
= ε
xz
, and u
6
= ε
xy
. The coefficients C
11
, C
12
, C
44
, and
C
′
= (C
11
− C
12
)/2 are the elastic modules of a cubic crystal.
134304-1 1098-0121/2011/84(13)/134304(5) ©2011 American Physical Society