Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions
We report on results of sound-velocity and sound-attenuation measurements in the Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in external magnetic fields up to 5 T, applied along several directions with respect to crystallographic axes, and at temperatures down to 1.7 K. The experimental data are analyzed with a microscop...
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nasplib_isofts_kiev_ua-123456789-1189142025-02-23T17:48:37Z Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions Zvyagina, G.A. Zhekov, K.R. Bilych, I.V. Zvyagin, A.A. Gudim, I.A. Temerov, V.L. Eremin, E.V. Низкотемпературный магнетизм We report on results of sound-velocity and sound-attenuation measurements in the Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in external magnetic fields up to 5 T, applied along several directions with respect to crystallographic axes, and at temperatures down to 1.7 K. The experimental data are analyzed with a microscopic theory based on exchange-striction coupling and phenomenological theory resulting in a qualitative agreement between theoretical results and experimental data. The study was supported by Grant of the Russian Federation president on support of sciences schools no. 4828.2012.2 and The Ministry of education and science of Russian Federation, project no. 8365. 2013 Article Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions / G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, E.V. Eremin // Физика низких температур. — 2013. — Т. 39, № 11. — С. 1202–1214. — Бібліогр.: 14 назв. — англ. 0132-6414 PACS: 72.55.+s, 74.25.Ld https://nasplib.isofts.kiev.ua/handle/123456789/118914 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемпературный магнетизм Низкотемпературный магнетизм |
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Низкотемпературный магнетизм Низкотемпературный магнетизм Zvyagina, G.A. Zhekov, K.R. Bilych, I.V. Zvyagin, A.A. Gudim, I.A. Temerov, V.L. Eremin, E.V. Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions Физика низких температур |
| description |
We report on results of sound-velocity and sound-attenuation measurements in the Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in external magnetic fields up to 5 T, applied along several directions with respect to crystallographic axes, and at temperatures down to 1.7 K. The experimental data are analyzed with a microscopic theory based on exchange-striction coupling and phenomenological theory resulting in a qualitative agreement between theoretical results and experimental data. |
| format |
Article |
| author |
Zvyagina, G.A. Zhekov, K.R. Bilych, I.V. Zvyagin, A.A. Gudim, I.A. Temerov, V.L. Eremin, E.V. |
| author_facet |
Zvyagina, G.A. Zhekov, K.R. Bilych, I.V. Zvyagin, A.A. Gudim, I.A. Temerov, V.L. Eremin, E.V. |
| author_sort |
Zvyagina, G.A. |
| title |
Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions |
| title_short |
Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions |
| title_full |
Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions |
| title_fullStr |
Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions |
| title_full_unstemmed |
Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions |
| title_sort |
magnetoelastic studies of nd₀,₇₅dy₀,₂₅fe₃(bo₃)₄ in the external magnetic field: magnetic phase transitions |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2013 |
| topic_facet |
Низкотемпературный магнетизм |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118914 |
| citation_txt |
Magnetoelastic studies of Nd₀,₇₅Dy₀,₂₅Fe₃(BO₃)₄ in the external magnetic field: Magnetic phase transitions / G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, E.V. Eremin // Физика низких температур. — 2013. — Т. 39, № 11. — С. 1202–1214. — Бібліогр.: 14 назв. — англ. |
| series |
Физика низких температур |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11, pp. 1202–1214
Magnetoelastic studies of Nd0.75Dy0.25Fe3(BO3)4
in the external magnetic field: Magnetic phase transitions
G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, and A.A. Zvyagin
B.I. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: zvyagina@ilt.kharkov.ua
I.A. Gudim, V.L. Temerov, and E.V. Eremin
L.V. Kirensky Institute of Physics, Syberian Branch of the Russian Academy of Sciences
Krasnoyarsk 660036, Russia
Received May 31, 2013
We report on results of sound-velocity and sound-attenuation measurements in the Nd0.75Dy0.25Fe3(BO3)4 in
external magnetic fields up to 5 T, applied along several directions with respect to crystallographic axes, and at
temperatures down to 1.7 K. The experimental data are analyzed with a microscopic theory based on exchange-
striction coupling and phenomenological theory resulting in a qualitative agreement between theoretical results
and experimental data.
PACS: 72.55.+s Magnetoacoustic effects;
74.25.Ld Mechanical and acoustical properties, elasticity, and ultrasonic attenuation.
Keywords: rare earth ferroborates, magnetoelastic interaction, magnetic phase transitions.
1. Introduction
During the last decades multiferroic systems came into
the focus of solid-state physics. Crystals belonging to the
family ReFe3(BO3)4 (Re3+ = Y, La–Nd, Sm–Er) borates
with the trigonal structure (the spatial group R32) have
interesting optical, magnetic, and magnetoelectric proper-
ties. Furthermore, multiferroic effects have been discov-
ered in some of them [1]. That is why this family of crys-
tals is a subject of intensive study nowadays. Their specific
magnetic properties are caused by the presence of two
types of magnetic ions: iron and rare earth ones. Antifer-
romagnetic ordering in iron subsystem develops in most of
compounds (at the Néel temperature TN = 30–40 K). A
spontaneous and magneto-induced electrical polarization
also develops in some of them in the magnetically ordered
state. Re3+ ions produce the main contribution to the mag-
netic anisotropy of the ferroborates, while iron ions are in
an orbital singlet state and a magnetic anisotropy induced
in them can be mostly due to the weak magnetic dipole–
dipole interaction. Hence magnetic structures, realized in
these crystals, depend on the type of Re3+ ion and are very
diverse. These compounds can be easy-axis (EA) antiferro-
magnets (Tb, Dy-based ferroborates), and easy-plane (EP)
antiferromagnets (Nd, Sm-based ferroborates), or they can
spontaneously transform from the EP to an EA state (Gd,
Ho-based ferroborates). In binary compounds of the type
Nd1–xDyxFe3(BO3)4 contributions of the Re3+ ions to the
magnetic anisotropy can have the competitive character
with one another and spontaneous reorientation from the
EP to the EA state is possible.
Indeed, in [2] it was reported the discovery of the spon-
taneous spin reorientation in the compound
Nd0.75Dy0.25Fe3(BO3)4. According to [2] the antiferro-
magnetic structure with the magnetic moments oriented in
the basic plane (EP anisotropy), which is formed in the
crystal below TN = 32 K, transforms spontaneously at TR =
= 25 K to an EA magnetic configuration. Studies of the
specific heat, magnetization [3] and our magnetoacoustic
investigations [4,5] of Nd0.75Dy0.25Fe3(BO3)4 crystal have
shown that the restructuring of its magnetic structure does
not reduce to a simple superposition of the features charac-
teristic of the NdFe3(BO3)4 and DyFe3(BO3)4. In this
compound we have detected new phase transitions (PTs):
the spontaneous PT (with the temperature at Tcr1 = 16 K),
and the one, induced by the external magnetic field, ap-
plied along the trigonal crystal axis C3, and applied in the
basic plane. We have constructed the H–T phase diagrams
for the cases 3|| ,H C 2||H C and 2 ,⊥H C and have de-
© G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, and E.V. Eremin, 2013
Magnetoelastic studies of Nd0.75Dy0.25Fe3(BO3)4 in the external magnetic field: Magnetic phase transitions
tected that this compound exhibited several PT lines and,
correspondingly, several magnetic phases. Magnetic
structures that are implemented in these phases are not
defined yet. Although, based on data from the experi-
ments [2–5], it can be assumed that in the absence of the
magnetic field the most of the low-temperature phase
(below Tcr1 = 16 K) corresponds to the EA configuration.
Phase, which is realized in the range 25 K < T < TN, was
supposed to be the EP one [2]. Magnetic configuration in the
range 16 K < T < 25 K is now the subject of debate. In [4,5],
we have suggested that the crystal Nd0.75Dy0.25Fe3(BO3)4
should be considered as the multisublattice antiferromag-
net. Then, detected magnetic PTs can be associated with
spin reorientation of the several magnetic sublattices of
this magnetic material.
It should be noted that the investigation of the behavior
of elastic characteristics of magnetic materials in external
magnetic field is a sensitive way of studying magnetically
ordered systems [6]. Study of the behavior of the sound
velocity and absorption as function of the temperature and
magnetic field allows an accurate determination of critical
temperatures and fields, as well as the order of magnetic
phase transitions in magnets. In the present work we have
performed the study of the magnetoelastic properties of the
Nd0.75Dy0.25Fe3(BO3)4 single crystal in a tilted external
magnetic field. We have determined the range of angles
where detected PTs [4,5] existed and corresponding mag-
netic phases were stable.
2. Experimental
Isometric Nd0.75Dy0.25Fe3(BO3)4 single crystals were
grown from a fluxed solution based on bismuth trimolyb-
date by the procedure described in detail in [3]; crystal
sizes up to 10–12 mm were obtained. We worked with a
crystal consisting of a transparent hexahedral prism, green
in color and of the order of 5 mm high, in a direction close
to an axis of symmetry of the third order (C3). Experimen-
tal sample with characteristic dimensions 1.5×1×1 mm was
prepared from it. The backward x-ray reflection method
(the Laue method) was used to orient the samples. The
measurements of the relative changes of the velocity and
attenuation of acoustic modes were performed using the
automatized setup described in [7]. The working frequency
was 54.3 MHz. The temperature behavior of the velocity
and absorption of acoustic modes (in the absence of an
external magnetic field or at fixed value of the field) and
the magnetic-field behavior of the same characteristics at a
fixed temperature were studied. The accuracy of the rela-
tive measurements of samples with the thickness ~0.5 mm
was about 10–4 in the velocity and 0.05 dB in the attenua-
tion. The range of the temperature was 1.7–50 K, and the
magnetic field up to 50 kOe was used.
3. Results
3.1. Zero magnetic field
At temperatures below 50 K we have observed three fea-
tures in the behavior of the velocity of transverse and longi-
tudinal acoustic modes: at the temperatures TN = 32 K,
Tcr2 = 25 K and Tcr1 = 16 K. They were always accompa-
nied by anomalies in the absorption at corresponding tem-
peratures. Anomaly at TN corresponds to the transition
of the crystal to the magnetically ordered state, and the
features at Tcr2 and Tcr1 are related to spin-reorientation
PTs [4,5].
Figure 1 illustrates the typical temperature behavior of
the velocity of the acoustic modes, for example of the C44
mode. The following notation is used in the figures: /s s∆
is the relative changes in the velocity of acoustic waves (q
is the wave vector and u is the polarization) propagating
along the x, y, and z axes of the standard Cartesian coordi-
nate system for trigonal crystals 2(y C and 3).z C Ab-
sorption behavior was illustrated in [4,5], and will not be
analyzed below. Figure 1 also shows the behavior of the
specific heat and magnetization of the crystal in the same
temperature range [3] for comparison. Note that the only
one anomaly at TN is clearly observed in the behavior of
the specific heat. The Néel temperature cannot be seen
from the behavior of the magnetization, however, peaks at
Tcr2 and Tcr1 are clearly visible. At the same time, the be-
havior of the acoustic characteristics exhibits all three crit-
ical temperatures associated with PTs in the magnetic sub-
system of the crystal. This means that the coupling
between magnetic and elastic subsystems is significant in
this compound, which is characteristic to multiferroics.
Therefore, the observation of the behavior of the elastic
properties of the crystal, and their response to an external
magnetic field, allows us to draw conclusions about the
state of its magnetic subsystem.
Fig. 1. (Color online) Temperature dependence of the sound veloci-
ty of C44 acoustic mode at H = 0 (red line), and at H = 10 KOe
[H z (blue line); H y (green line)] in Nd0.75Dy0.25Fe3(BO3)4.
For comparison we present the temperature behavior of the spe-
cific heat () and magnetization () (cf. [3]).
0 10 20 30 40
–1
0
1
0
0.7
0 10 20 30 40
0
200
TN
Tcr2Tcr1
C44
Mdc
Hy =10 kOe
Hz = 10 kOe
H = 0
C44( || , || )q z u y
T, K
Cp
∆s
s/
, 1
0–2
M
dc
, e
m
u/
g
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11 1203
G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, and E.V. Eremin
3.2. The external field 3H C
The application of an external magnetic field H z
shifts the features at cr2T and cr1T to lower temperatures,
however, the position of the feature at NT remains virtual-
ly unchanged (cf. Fig. 1). Two closely spaced features ex-
hibiting hysteresis were found in the magnetic field depen-
dences of the velocity of the acoustic modes in the fields
cr1
zH and cr2
zH (Fig. 2).
An increase of the temperature shifts them to the direc-
tion of weaker fields. Both features in the fields cr1H and
cr2H were registered in the temperature interval from 1.7
to 16 K cr1( ),T while only one feature in the field cr2H
was detected in the range from 16 K cr1( )T to 25 K cr2( ).T
The critical fields of the features, which we observed in the
behavior of the acoustic modes, and the temperature, cor-
responding to them, are correlated with the values of the
fields and temperatures, at which magnetization anomalies
were found according to the measurements performed in
[3], see Fig. 2. That is why, the observed anomalies have
been interpreted by us [4] as the manifestation of magnetic
reorientation PTs.
We have constructed [4] the low-temperature fragment
of the H–T phase diagram of the Nd0.75Dy0.25Fe3(BO3)4
crystal for 3H C (z) (see Fig. 3(a)). The diagram was
found to be more complex than the one presented in [2],
and differs from the latter by the presence of the lines 1
and 4. Our investigations show that a few (at least three)
low-temperature magnetically ordered phases exist in the
compound studied in the field .H z
3.3. The external field H applied along x and y axes
When an external magnetic field was applied in the
basal plane of the crystal (both for H y and ),H x the
features at cr2T and cr1T were shifted slightly toward low-
er temperatures. The position of the feature at NT was
essentially unchanged when the external field was applied.
The example of the typical temperature behavior of sound
Fig. 2. (Color online) Magnetic field ( )H z dependence of the
sound velocity (red line, the experiment; black line, calculations)
of C44 acoustic mode and magnetization (blue symbols) in
Nd0.75Dy0.25Fe3(BO3)4 at T = 2 (a) and T = 4 (b) K.
0 10 20 30 40 50–1.5
–1.0
–0.5
0
0.5
–5
0
5
10
15
20
25
30
35
(theory)∆s s/
H, kOe
/ (experimental)∆s s
Hz
cr2
Hz
cr1
T = 2 K
Magnetization
H C || 3
10 12 14 16 18 20 22
–0,6
–0.4
–0.2
0
0.2
0
10
20
30
Magnetization
∆
s
s/
, 1
0–2
M
dc
, e
m
u/
g
(theory)∆s s/
/ (experimental)∆s s
H, kOe
∆s
s/
, 1
0–2
M
dc
, e
m
u/
g T = 4 K
H C || 3
Hz
cr2Hz
cr1
Fig. 3. (Color online) External magnetic field — temperature phase
diagrams for Nd0.75Dy0.25Fe3(BO3)4: ( )H z (a); H y (b);
H x (c). The symbols are related to the features observed in our
various magnetoacoustic experiments at low temperatures and
small values of the field.
0 10 20 30
10
20
30
40
50
1
2
T, K
H z ||
TN
H
, k
O
e
0
10
20
30
40
50
H
, k
O
e
TN
5 10 15 20 25 30 35
2
1
3
H y ||
T, K
(a)
(b)
5 10 15 20 25 30 35
T, K
0
10
20
30
40
50
H
, k
O
e
TN
2
1
H x || (с)
4
1204 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11
Magnetoelastic studies of Nd0.75Dy0.25Fe3(BO3)4 in the external magnetic field: Magnetic phase transitions
velocity in external magnetic field H y is shown in
Fig. 1. From the analysis of the magnetic field behavior of
the acoustic modes in the external magnetic field applied in
basal plane we have concluded [5] that in the temperature
range 1.7 K < < 15T K the acoustic characteristics are
essentially field-independent (at least for ).H y But in
the temperatures range 15–32 K the behavior of the acous-
tic characteristics depends on the direction of the magnetic
field in the basal plane. We believe that this anisotropic
behavior of the acoustic characteristics observed in the
Nd0.75Dy0.25Fe3(BO3)4 is due to the existence of another
easy axis, which is parallel to C2 symmetry axis in the bas-
al plane. We have observed analogous magnetic behavior
in the NdFe3(BO3)4 at the temperature range, where an
easy plane, antiferromagnetic commensurate structure is
realized [8]. We have plotted the low-temperature part of
the H–T phase diagrams of the crystal for fields directed
along and perpendicular to the second order symmetry axis
in the basal plane of the crystal (Figs. 3(b) and 3(c), re-
spectively). To our opinion the line 3 in the H–T phase
diagrams ( )H y is the line of a spin-flop transition.
3.4. Behavior in the magnetic field tilted from the
crystallographic axes
Constructed phase diagrams (Fig. 3) imply the exis-
tence of several lines of PTs in the crystal in the ordered
magnetic states, and, consequently, of several magnetic
phases. The temperature borders (at H = 0) of those phases
are related to three intervals:
1) 1.7 K < < 16T K cr1( );T
2) 16 K cr1( )T < < 25T K cr2( );T
3) 25 K cr2( )T < < 32T K ( ).NT
Let us analize the behavior of the acoustic characteris-
tics of the crystal in each of those intervals when the mag-
netic field is tilted from z axis (C3) in the planes zy and zx,
and when the field is tilted from y (C2 axis) in the plane xy.
3.4.1. The first temperature range. Here the evolution
of the magnetic field behavior of the sound velocities re-
lated to various acoustic modes with the tilt angle of the
magnetic field from z axis from 0° to 90° in the zy plane
at the lowest temperature of the experiment, 1.7 K, is
shown in Fig. 4.
It is seen that the growth of the tilt angle up to 30° basi-
cally does not change the values of critical PT fields cr1
zH
and cr2
zH (Fig. 4(a)). The form of anomalies (jumps) also
remains the same. Further increase of the angle α ≥ 30°
produces the shift of cr1
zH and cr2
zH to the higher values
(Fig. 4(b)). The form of anomalies at cr1H and cr2H and
the character of the magnetic field dependences above PT
points at 18α ≥ ° is also partly changed. Perhaps, those
transitions are realized at higher values of the angle, how-
ever, their critical values of the magnetic field exceed max-
imal possible in our experiments value of the field 50 kOe.
Hysteretic character of anomalies at cr1
zH and cr2
zH re-
mains the same in the total angle range, where we regis-
tered PTs (0 65 ).≤ α ≤ ° When = 90 ,α ° i.e., at ,H y the
sound velocity basically does not depend on the value of the
field (Fig. 4(b)). The dependence of cr1
zH and cr2
zH on the
tilt angle α from the axis z (C3) in the plane zy at 1.7 K in
the polar coordinates is given in Fig. 4(c). The values cr1
zH
and cr2
zH (in kOe) for each of the value of the tilt angle are
plotted at respective radia-vectors.
In Fig. 5(b) angle dependences of cr1
zH and cr2
zH for
two more temperature are presented: For 6 K for the tilting
of the magnetic field from z axis in the zx plane (a), and for
10 K for the tilting in the zy plane. From the comparison
of the angle-dependent phase diagrams (Fig. 4(c), and
Fig. 4. (Color online) Magnetic field dependencies of the sound
velocity in Nd0.75Dy0.25Fe3(BO3)4 at T = 1.7 K for negative (a)
and positive (b) values of the tilt angle from C3 axis. Magnetic
field phase diagram for Nd0.75Dy0.25Fe3(BO3)4 for the magnetic
field tilted from z axis in the zy plane at T = 1.7 K (c).
0 10 20 30 40 50 60
–7
–6
–5
–4
–3
–2
–1
0
1
0° H z ||
9°
13°
18°
22°
27°
45°
63°
90° H y ||
15 20 25 30 35
–3
–2
–1
0
1
2
α = H z
0°
–13°
13°
–9°
9°
H, kOe
Hz
cr2Hz
cr1
∆s
s/
, 1
0–2
H, kOe
Hz
cr2
Hz
cr1
∆s
s/
, 1
0–2
(a)
(b)
α = H z
0
10
20
30
40
+α
30°
60°
30°
60°
3030 20 10 10 20
–α
Hz
cr2
Hz
cr1
Hy, kOe Hy, kOe
H
z,
kO
e (с)
α = H z
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11 1205
G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, and E.V. Eremin
Fig. 5(b)) we can see that the features at cr1
zH and cr2
zH
are registered at the tilt angles 0 65≤ α ≤ ° in the zy plane,
as well as in the zx plane in the first temperature interval.
However, when the tilt angle in the zx plane exceeds
30°, for all investigated temperatures in the first interval
the smooth increase of the sound velocity is observed al-
ready in the region above the transition cr1,2< zH H . The
anomalies themselves at cr1
zH and cr2
zH look like cusps,
and we cannot see hysteresis. Figure 5(a) illustrates the
behavior of the sound velocity related to the acoustic
mode q x and u z at large tilt angles (α ≥ 54°) in the
zx plane at 6 K. With the growth of the tilt angle the fea-
tures at cr1
zH and cr2
zH are shifted to higher values of the
field. At larger tilt angles α ≥ 72° the anomalies at cr1
zH
and cr2
zH in the applicable field range cannot be ob-
served, however, the specifics of the monotonous in-
crease of the sound velocity in the range below PTs keeps
unchanged up to α ≥ 90°, i.e., for .H x
We have also observed similar characteristic mono-
tonous increase of the sound velocity in the case H x
for the sound mode (q z and )u x when the field is
tilted in the xy plane in the angle range 90° ≥ β ≥ 20°
[here β is the tilt angle from the y axis (C2) in the xy
plane], see Fig. 5(a). For small values of the angle 0 ≤ β ≤
20° the sound velocity practically does not depend on the
value of the magnetic field.
3.4.2. The second temperature range. The behavior of
acoustic characteristics has been studied at the temperature
17 K, when the field had been tilted from the z axis in the
zy and zx planes, and also in the xy plane when the field
had been tilted from the y axis. At that temperature, ac-
cording to the phase diagrams (Fig. 3), PTs were observed
at field values cr2 ,zH cr3,yH cr3
yH ′ and cr3 .xH ′ Let us de-
Fig. 5. (Color online) Magnetic field dependencies of the sound
velocity in Nd0.75Dy0.25Fe3(BO3)4 at T = 6 K for several values of
tilt angles from C3 axis (the angle α) and from C2 axis (the angle
β) in the zy and xy planes, respectively (a). Magnetic field phase
diagram for Nd0.75Dy0.25Fe3(BO3)4 for the magnetic field tilted
from z axis in the zy and zx planes at T = 6 and 10 K (b).
0
10
20
30
40
30°
60°
30°
60°
0 10 20 30 40 50 60
54°
H x ||
63°
68°
72°
90°
90°
81°
63°
6 K
3030 20 10 10 20
–α
H, kOe
Hz
cr2
Hz
cr1
Hz
cr2
Hz
cr1
Hy, kOe Hy, kOe
H
z,
kO
e
∆s
s/
, 1
0–2
–2
0.4
0.6
0.8
0.2
0
q z u x || , ||
q x u z || , ||27°
0°
54°
H x ||
H y ||
+α
6 K 10 K
(a)
(b)
α = H z
β = H y
α = H z
Fig. 6. (Color online) Magnetic field dependencies of the sound
velocity in Nd0.75Dy0.25Fe3(BO3)4 at T = 17 K for several values
of tilt angles from C3 axis in the zx plane (a) and in the zy plane
(b). Magnetic field phase diagram for Nd0.75Dy0.25Fe3(BO3)4
for the magnetic field tilted from z axis in the zy and zx planes at
T = 17 K (c).
0
10
20
30
α
30°
60°
30°
60°
0 10 20 30 40 50 60
–4
–3
–2
–1
0
1
0°
H z ||
18°
36°
45°
54°
27°
63°
72°
90° H y ||
20 30 40 50 60
–3
–2
–1
0
1
54°
45°
36°
27°
9°
0°
20 10 10 20
α
H, kOe
Hz
cr2
∆s
s/
, 1
0–2
H, kOe
Hz
cr2
Hy
cr3
Hz
cr2
Hx, kOe Hy, kOe
H
z,
kO
e
∆s
s/
, 1
0–2
(a)
(b)
(с)
100
H x ||
63°
72°
90°–4
H z ||
81°
Hy
cr3
α = H z
α = H z
α = H z
1206 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11
Magnetoelastic studies of Nd0.75Dy0.25Fe3(BO3)4 in the external magnetic field: Magnetic phase transitions
termine the range of values of angles, where we have ma-
naged to register those phase transformations (see Figs. 6
and 7). The anomalies at the field value cr2 ,zH when the
field is tilted in the zy and zx planes from z axis are ob-
served in the angle ranges 0 ≤ α ≤ 65° and 0 ≤ α ≤ 54°,
respectively. The increase of the tilt angle up to 30° in the
zy and zx planes does not basically change the critical val-
ue of the field cr2.zH Further increase of the tilt angle in
the zy plane yields the shift of cr2
zH to higher values.
The deviation in the plane zx for angles α ≥ 30° also
shifts the anomaly in cr2
zH towards higher fields and
yields its smearing. Monotonous growth of the sound ve-
locity is observed in the region before transition, Fig. 6.
Hence, the angle dependence of the critical field cr2
zH in
the second range is similar to the behavior cr1
zH and cr2
zH
in the first temperature range.
The features at cr3,yH cr3
yH ′ and cr3
xH ′ are observed
when the field is tilted from the y and x axes in the xy and
zx planes or angles larger than 40°, see Fig. 6. The increase
of the tilt angle in the zy plane yields small shifts of cr3
yH
and cr3
yH ′ towards larger values. The value of cr3
xH ′ is
weakly dependent on the growth of the tilt angle in the zx
plane. The features at cr3,yH cr3
yH ′ and cr3
xH ′ are ob-
served for the deviations in the xy plane in the range of tilt
angles from the y and x axes smaller than 30°.
3.4.3. The third temperature range. In that interval, ac-
cording to the phase diagrams, Fig. 3, only phase transfor-
mations at cr3,yH cr3
yH ′ and cr3
xH ′ are realized. We have
performed the investigations of angle dependences of
cr3,yH cr3
yH ′ and cr3
xH ′ at the temperature 28 K. It turns
out that at such high enough temperature (the value is close
to = 32NT K) the scale of anomalies, which we attribute
to phase transformations, is not large. The peculiarities
themselves are smeared, and, thus, the investigation of the
angle dependencies of cr3,yH cr3
yH ′ and cr3
xH ′ has the
qualitative character.
Nevertheless, the anomalies at cr3,yH cr3
yH ′ and cr3
xH ′
can be distinguished at tilt angles less than 45° from the y
and x axes in the zy and zx planes, see Fig. 8. The values
cr3,yH cr3
yH ′ and cr3
xH ′ become larger with the growth of
tilt angles.
Magnetic field behavior of some acoustic modes, when
the field is tilted in the xy plane appeared somehow unex-
pected. In such a geometry of the experiment we have the
opportunity to study the behavior of only those acoustic
modes, that have q y or .q x
The magnetic field dependences of the sound velocity
of the transverse mode q y and u z for various values
of tilt angle in the xy plane are presented in Fig. 9. For that
mode the anomalies at cr3
yH and cr3
yH ′ in the field, di-
rected exactly along y ( )H y are weakly manifested. That
is why, we define in the figure by arrows the values of the
fields cr3
yH and cr3 ,yH ′ which are determined from the
behavior of the mode ,q z ,u x see Fig. 8. The feature
at cr3
xH ′ at H x can be clearly seen. As one can see from
the figure, the anomalies at cr3,yH cr3
yH ′ and cr3
xH ′ can be
detected only until relatively small (less than 9°) tilt angles
from the y and x axes, respectively. However, with increas-
ing the angle (larger than 18°) in the fields cr3
yH≈ for
H y the new anomaly appears, which becomes more
pronounced with the growth of the angle and reaches its
maximum value at β ≈ 45°.
Hence, in the third temperature range the anomalies in
the magnetic field behavior of the sound velocity (attenua-
tion) of at least two acoustic modes are observed at any tilt
angles of the magnetic field in the xy plane. Let us note that
in the second range the behavior of the same modes when
the field is tilted in the xy plane was different, see Fig. 7.
Fig. 7. (Color online) Magnetic field dependencies of the sound
velocity in Nd0.75Dy0.25Fe3(BO3)4 at T = 17 K for several values
of tilt angles from C2 axis (a). Magnetic field phase diagram for
Nd0.75Dy0.25Fe3(BO3)4 for the magnetic field tilted from y axis in
the yx plane at T = 17 K (b).
0 10 20 30 40 50–3
–2
–1
0
0°
18°
36°
45°
54°
27°
63°
72°
90° H || x
H, kOe
Hy
cr3
∆s
s/
, 1
0–2
(a)
H || y
81°
0
5
10
15
30°
60°
105 Hx, kOe
H
y,
kO
e
( )b
Hy
cr3
9°
β
90°
β = H y
0°
β = H y
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11 1207
G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, and E.V. Eremin
4. Theoretical analysis
4.1. Microscopic consideration
The following theoretical calculation was used to ex-
plain the observed experimental data. In magnetic mate-
rials the dominant contribution to the spin-lattice interac-
tions arises mostly from the exchange-striction coupling.
In our calculations we assume that in the multiferroic un-
der study the spatial dependence of the magnetic anisotro-
py (i.e., of the magnetic relativistic interaction) is weaker
than the spatial dependence of the exchange integrals. In
this case, one can expect that mostly longitudinal sound
waves interact with the spin subsystem. The magneto-
acoustic interaction is considered then in the standard way
in the framework of the perturbation approach [9]. Accord-
ing to Refs. 9, 10 the renormalization of the longitudinal
sound velocity of such a model can be written as
1 2
2= ,
( )
A As
s N
+∆
−
ωq
(1)
where
2 2 2 2
1 0 0 0
= , ,
= 2 | ( ) | | ( ) | ( ) ,z z z
x y z
A G S T Gα α
α
〈 〉 χ + χ∑ ∑ k k
k
q q
2
2 0 0
= , ,
= ( ) ( ) .
2
z z
x y z
TA H S H α α
α
〈 〉 + χ∑ ∑ k k
k
q q (2)
Here, N is the number of spins in the system, k is the
wave vector of magnetic excitations, = sqωq is the low-q
dispersion relation with sound velocity s in the absence of
spin-phonon interactions, 0
zS〈 〉 is the average magnetiza-
tion along the direction of the magnetic field, , ,x y zχk are
non-uniform magnetic susceptibilities, and the subscript 0
corresponds to k = 0. The renormalization parameter
Fig. 8. (Color online) Magnetic field dependencies of the sound
velocity in Nd0.75Dy0.25Fe3(BO3)4 at T = 28 K for several values
of tilt angles from C3 axis: in the zx plane (a) and in the zy plane
(b). Magnetic field phase diagram for Nd0.75Dy0.25Fe3(BO3)4
for the magnetic field tilted from z axis in the zy and zx planes at
T = 28 K (c).
0
10
20
α
30°
60°
30°
60°
0 10 20 30 40 50 60
–1
0
0°
H z ||
18°
36°
45°
27°
63°
90° H y ||
20 30 40 50 60
0.5
0
1.0
54°
45°
13°
27°
9°
0°
20 10 10 20
α
H, kOe
∆s
s/
, 1
0–2
H, kOe
Hy
cr3
Hx, kOe Hy, kOe
H
z,
kO
e
∆s
s/
, 1
0–2
(a)
(b)
(с)
100
H x ||
63°
90°
H z ||
Hy
cr3
α = H z
α = H z
α = H z
Fig. 9. (Color online) Magnetic field dependencies of the sound
velocity in Nd0.75Dy0.25Fe3(BO3)4 at T = 28 K for several values
of tilt angles from C2 axis (a). Magnetic field phase diagram for
Nd0.75Dy0.25Fe3(BO3)4 for the magnetic field tilted from y axis in
the yx plane at T = 28 K (b).
0 10 20 30 40 50
–1
0
0°
18°
36°
45°
54°
27°
63°
72°
90°H || x
H, kOe
Hy
cr3
∆s
s/
, 1
0–2
(a)
H || y
81°
( )b
9°
β
60
0
5
10
15
30°
60°
105 Hx, kOe
H
y,
kO
e
Hy
cr3
90°
β = H y
β = H y
1208 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11
Magnetoelastic studies of Nd0.75Dy0.25Fe3(BO3)4 in the external magnetic field: Magnetic phase transitions
1 2A A+ is proportional to the spin-phonon coupling con-
stants (which have to be determined independently)
1= e (e 1) ,i i mnnm nm
mn
J
G
m
α
α ∂
−
∂∑ kR qR
k qu
R
1= e (e 1)(e 1)i i inm nm nm
n
H
m
− −α − − ×∑ kR qR qR
k
2
.mn
n m
J α
−
∂
×
∂ ∂q qu u
R R
(3)
Here, m is the mass of the magnetic ion, mnJ α denote ex-
change integrals, qu is the polarization of the phonon with
wave vector q, and nR is the position vector of the nth site
[9,10]. In our calculations we used these quantities as fitting
parameters. Our simplified theory reproduces the main fea-
tures of the experimentally observed behavior, see Fig. 2.
It turns out that the theory reproduces the low-temperature
behavior of the sound velocity rather well. On the other
hand, for the higher-field region of the phase diagram the
agreement is not so good. We suppose that in that phase
inhomogeneous spin distribution can take place like in Nd
ferroborate [8], resulting in nonzero inhomogeneous mag-
netic susceptibility, contribution of which was neglected in
calculations shown in Fig. 2.
4.2. Phenomenological approach
To understand the features of the magnetic structure of
the studied magnetic system of Nd0.75Dy0.25Fe3(BO3)4, we
have also constructed the phenomenological theory, based
on the consideration of a six-sublattice antiferromagnet, cf.
[4,5,11]. We have chosen six magnetic sublattices due to
the following reasons. We have assumed that each magnet-
ic ion (Fe, Nd, and Dy) is in the magnetically ordered state
below NT (at least in the ground state), and that each of
those magnetic ions form two magnetic sublattices. Then,
we have assumed that the main interaction is the exchange
between iron magnetic sublattices. We also suggested that
the single-ion magnetic anisotropy affects only rare earth
ions, the EA anisotropy for Dy magnetic sublattices, and the
EP one for the Nd sublattices, because iron ions are in the
orbital singlet state. Finally, we have supposed that there
exists weak interaction between iron and rare earth magnetic
sublattices, and we have neglected direct interactions be-
tween rare earth sublattices. In the lowest approximation we
do not take into account the weak magnetic anisotropy in the
basal plane (the plane, perpendicular to C3 axis).
The ground-state energy of such a model system in the
external magnetic field H has the form
____________________________________________________
0 1 0 2 1 1 1 2 2 1= [ cos( ) cos( ) cos( ) cos( ) cos( )H H H H HE H M M m m m− −θ + θ + −θ + θ + θ − ϕ + θ − ϕ + −θ + ψ +
2 2 22 2 2 2
2 2 0 1 2 1 1 1 2 2 2 1 2
1 1cos( )] cos( ) [ ( ) ( )] [ ( ) ( )]sin sin sin sin
2 2Hm JM m K m K+ −θ + ψ + θ − θ + ϕ + ϕ − ψ + ψ +
1 0 1 1 1 1 2 2 1 2 2 2 0 2 1 1 1 2[cos( ) cos( ) cos( ) cos( )] [cos( ) cos( )J M m J M m+ θ − ϕ + θ − ϕ + θ − ϕ + θ − ϕ + −θ + ψ + −θ + ψ +
2 1 2 2cos( ) cos( )] .+ −θ + ψ + −θ + ψ (4)
_______________________________________________
Here we denote by 0M the magnitude of the iron mag-
netic sublattice with 1,2θ being the angles between two
iron sublattices and C3 axis, by 1,2m the magnitudes of
rare earth magnetic sublattices of Nd and Dy ions, respec-
tively (with 1,2ϕ and 1,2ψ being angles between Nd and
Dy magnetic sublattices, respectively, and the C3 axis),
Hθ denotes the angle between the direction of the external
magnetic field and C3 axis, > 0J defines the iron–iron
antiferromagnetic exchange interaction, 1,2 > 0K are the
EP and EA magnetic anisotropies for Nd and Dy magnetic
sublattices, respectively, and, finally, 1,2J are the coupl-
ings between iron and rare earth magnetic sublattices.
To determine the steady-state magnetic configurations
of the considered model we minimize the expression for
the ground-state energy with respect to the angles 1,2 ,θ
1,2ϕ and 1,2.ψ The analysis is very complicated. For ex-
ample, even for = 0,Hθ i.e., for the external magnetic
field directed along C3 axis, we have found 27 solutions of
the minimization conditions. They correspond to the anti-
parallel, parallel and tilted configurations of each pair of
magnetic sublattices.
Let us consider those solutions. The phase with each of
three pairs of sublattices being antiparallel (the antiferromag-
netic solution) is related to the case with 2 2 2= = =θ ϕ ψ π
and 1 1 1= = = 0θ ϕ ψ (or vice versa). Such an antiferromag-
netic state has the energy 2
0= .AFE JM− Obviously, that
phase has the minimal energy in the absence of the exter-
nal magnetic field.
There are three solutions with two pairs of magnetic
sublattices being antiparallel, and one pair of sublattices in
the tilted (spin-flop-like) state. The first such phase has
2 2= = ,ϕ ψ π 1 1= = 0ϕ ψ and 2 1=θ −θ with
1
0
cos = .
2
H
JM
θ (5)
It has the energy 2
2 2 0= ( /2) ( / ).AFE M H H J− The second
solution has the values of angles 2 2= = ,θ ψ π
1 1= = 0,θ ψ and 2 1=ϕ −ϕ with
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11 1209
G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, and E.V. Eremin
1
1 1
cos = .H
m K
ϕ − (6)
It has the energy 2 2 2
2 4 1 0 1 1= ( / ) .AFE H K JM m K− + Final-
ly, the third such solution is related to the values of angles
2 2= = ,θ ϕ π 1 1= = 0θ ϕ and 2 1=ψ −ψ with
1
2 2
cos = .H
m K
ψ (7)
This phase has the energy 2
2 6 2= ( / )AFE H K− −
2 2
0 2 2.JM m K− −
There are three solutions with two pairs of magnetic
sublattices being antiparallel, and one pair of sublattices
being parallel (and directed along the field). The first such
solution has 2 2= = ,ϕ ψ π 1 1= = 0ϕ ψ and 2 1=θ −θ
with 1 = 0.θ It has the energy 2
2 1 0 0= 2 .AFE HM JM− +
The second one has the values of angles 2 2= = ,θ ψ π
1 1= = 0,θ ψ and 2 1=ϕ −ϕ with 1 = 0.ϕ It has the energy
2
2 3 1 0= 2 .AFE Hm JM− − The third such solution is related
to the values of angles 2 2= = ,θ ϕ π 1 1= = 0θ ϕ and
2 1=ψ −ψ with 1 = 0.ψ This solution has the energy
2
2 5 2 0= 2 .AFE Hm JM− − Possible field-governed PTs bet-
ween these three phases and three previous phases are of
the second order.
There are 12 phases in which only one pair of sublattic-
es is antiparallel.
Among them there are three solutions with one pair of
sublattices being antiparallel, and two others being in the
spin-flop-like (tilted) state. The first one of those phases
has 2 = ,ψ π 1 = 0ψ with 2 1= ,θ −θ 2 1=ϕ −ϕ and with
1 1
1 2
0 1 1
( 2 )
cos = ,
2 ( 2 )
H K J
M K J J
+
θ
+
1
1 2
1 1 1
( )
cos = .
( 2 )
H J J
m K J J
−
ϕ
+
(8)
This state has the energy
2 3 2 2 2 2
4 1 1 0 1 1 0 1= (2 2AFE m K J J M HK J K M HJ+ + +
2 2 2 2 2 2 2 2
1 1 1 1 1 1 0 12 2 8 2H K J H K J m K JJ JK M HJ+ − + + −
2 3 2 4
1 1 0 1 1 1 18 4 8H J K J JM HJ m K J− + + −
2 2 2 3 2 2
1 1 1 1 12 8 )/2( 2 )H K J H J K J J− − + .
The second one is related to the following set of angles
2 = ,ϕ π 1 = 0ϕ with 2 1= ,θ −θ 2 1=ψ −ψ and
2 2
1 2
0 2 2
(2 )
cos = ,
2 ( 2 )
H J K
M K J J
−
θ
− +
2
1 2
2 2 2
( )
cos = .
( 2 )
H J J
m K J J
−
ψ
− +
(9)
It has the energy
2 3 2 2 2 2
8 2 2 0 2 2 0 2= (2 2AFE m K J J M HK J K M HJ− − + +
2 2 2 2 2 2 2 2
2 2 2 2 2 2 22 2 8 8H K J H K J m K JJ H J K J+ + − − +
2 3 2 4 2 2
2 0 2 0 2 2 2 2 2 22 4 8 2JK M HJ JM HJ m K J H K J+ − + − +
2 3 2 2
2 2 28 )/2( 2 )H J K J J+ − .
Finally, the third one corresponds to the set of angles
2 = ,θ π 1 = 0,θ 2 1= ,ϕ −ϕ 2 1=ψ −ψ with
1
1 1
cos = ,H
m K
ϕ − 1
2 2
cos = .H
m K
ψ (10)
The energy of such a state is equal to
2 2 2
12 2 1 0 1 2= (AFE H K H K JM K K− − +
2 2 2 2
1 1 2 2 2 1 1 2)/ .m K K m K K K K+ −
Six such solutions have one pair of sublattices antipa-
rallel, one pair parallel (and directed along the field), and
one pair in the tilted state. The angles, corresponding to
antiparallel state are either zero or ,π the angles related to
the parallel state are zero, and the angles for sublattices in
the tilted states differ from each other by their signs. The
solutions are
1 1
1
0
2
cos =
2
H J m
JM
−
θ (11)
with the energy
2
2 1 1 1 0= (2 8 4AFE H HJ m Hm J JM H− − + − +
2 2
0 1 1 1 12 8 )/2 ;JM J m J m J+ +
1
0
cos =
2
H
JM
θ (12)
with 6 2 0 2 2= (2 4 4 )/2 ;AFE H H m J JM J m J− + − −
1 0
1
1 1
2
cos =
J M H
m K
−
ϕ (13)
with the energy
2
3 0 1 1 0= ( 2 4AFE HM K HJ M H− − + +
2 2 2 2 2
0 1 1 0 1 1 14 )/ ;JM K J M m K K+ + +
1
1 1
cos = H
m K
−
ϕ (14)
with the energy
2 2 2 2
11 2 1 0 1 1 1 1= ( 2 )/ ;AFE H Hm K JM K m K K− − +
1
2 2
cos = H
m K
ψ (15)
1210 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11
Magnetoelastic studies of Nd0.75Dy0.25Fe3(BO3)4 in the external magnetic field: Magnetic phase transitions
with the energy
2 2
7 0 2 0 2= (2AFE HM K H JM K− + − −
2
2 0 2 2 24 )/J M H m K K− + ;
and
1
2 2
cos = H
m K
ψ (16)
with the energy
2 2 2 2
10 1 2 0 2 2 2 2= (2 )/AFE Hm K H JM K m K K− + + + .
There are three solutions with one pair of sublattices be-
ing antiparallel, and two other pairs being parallel and di-
rected along the magnetic field. The angles for antiparallel
states are equal to zero and ,π and the angles in parallel
states are equal to zero. The energies of those phases are
2
1 0 1 0 1 0 1= (2 2 ) 4 ,AFE H M m JM J M m− + + +
2
5 0 2 0 2 0 2= (2 2 ) 4 ,AFE H M m JM J M m− + + +
2
9 1 2 0= (2 2 )AFE H m m JM− + − .
Then there exists a solution with all three pairs of sublat-
tices being in the tilted phases. For that phase we get differ-
ent signs for angles belonging to opposite sublattices with
2 1 1 2 1 2
1 2 2
0 2 1 1 2 1 2
(2 2 )
cos = ,
2 (2 2 )
H J K J K K K
M J K J K JK K
+ −
θ
− −
2
2 1 2 1 2 2
1 2 2
1 2 1 1 2 1 2
( 2 2 )
cos = ,
(2 2 )
H JK J K J J J
m J K J K JK K
− + −
ϕ
− −
2
1 2 1 1 2 1
1 2 2
2 2 1 1 2 1 2
( 2 2 )
cos =
(2 2 )
H JK J K J J J
m J K J K JK K
− + + −
ψ
− −
(17)
with the energy
____________________________________________________
2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 3 4 2 2 2
3 2 2 2 1 2 2 1 1 1 1 1 2 2 2 2 1 1 1 1 1 2 1 2= (8 8 8 16 8 8FE m K J K J m K J JK m K J K J m K J K J m K J H J J K− + − − − + +
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3
1 2 1 2 1 2 1 2 2 1 2 1 2 1 1 2 12 8 2 2 16 2 16H J K K H J J K H K K J H K K J H JK J H J K K H J J K −+ − − + − − +
2 3 2 2 2 2 2 3 2 2 3 2 2 4 2 4 3 2 2
1 2 2 1 1 2 1 2 1 1 2 2 2 1 2 1 1 2 1 216 16 8 8 8 8 8H J J K m K J J K m K J JK H J K H J K H J K J H K− + + + − + + +
2 4 2 2 3 2 2 2 4 2 2 3 2 2 2 2 2 2
2 2 2 1 2 2 1 1 1 1 2 1 1 2 2 1 2 2 2 18 2 8 2 8 8m K J K m K J K m K J K m K J K H J K JK H JK J K+ + − − − + +
2 2 2 2 2 2 2 2 2 3 2 3 2
1 1 2 1 2 1 2 2 1 1 2 1 2 0 1 2 0 2 1 08 8 8 4 4H JK J K J H K K J H J K J K HJ K K M HJJ K M HJJ K M+ − − − + − +
2 2 2 2 2 2 2 2
1 1 2 0 2 1 2 0 1 1 2 0 2 1 2 02 2 2 2HJ J K K M HJ J K K M HJJ K K M HJJ K K M+ + − + +
2 2 2 2 2
1 2 1 2 0 1 2 1 2 0 1 2 2 1 1 24 4 )/2( 2 2 ) .HJ J JK K M HJJ J K K M JK K J K J K+ − − +
There are three solutions with two pairs of sublattices being in the tilted phases and one pair being parallel and directed
along the field. The angles for parallel states are zero, and the ones for tilted states have different signs. The first such a
state has the angles
2 2 1 1 2
1 2
0 1 2 2
( 2 ) 2
cos = ,
2 ( )
H K J J m K
M J K J
− −
θ
−
1 2 1 2 1
1 2
2 1 2 2
(2 2 ) 4
cos =
2 ( )
H J J J J m
m J K j
− +
ψ
−
(18)
with the energy
2 2 2 2 2 2 4 2 3
1 1 2 0 2 1 1 2 0 2 1 2 0 1 2 1 2 0 1 2 0 2 0 2= (8 2 2 16 8 8FE J K HM J m J K JM HJ J K JM m J J K HM m J M HJ M H J+ − + − + +
2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 1 2 0 2 1 2 0 1 1 2 0 1 1 2 0 1 2 1 2 0 2 1 2 2 0 14 8 2 16 8 16H J m K JM HJ J K HM m J K JM m J K M m J J K M HJ J K J M m+ + − + − − − −
2 2 2 2 2 2 2 2 2 2 2 3
1 2 2 0 2 1 2 0 1 2 2 0 1 2 0 2 1 2 2 1 1 2 1 2 1 0 2 14 8 4 4 4 16J K m M J J K JM H J K J M H J K M H J J K H J m J K Hm J J HM J m− − + − − + − −
2 3 2 2 2 4 2 2 3 2 2 3 2 2 2 2 2
0 2 2 1 2 2 2 0 2 0 1 2 1 1 2 0 2 1 2 1 1 0 22 2 2 8 2 4 2JM HJ K H m J K m M J M m K J J m M K J K m H J M H K− − + + + − + +
2 2 2 2 4 2 3 2 2 2 2 2 2 2
1 0 2 1 2 1 1 2 0 1 1 2 0 1 0 2 0 2 2 0 1 2 22 2 16 8 4 4 )/2 ( ) .J M H K J K m H J J M m J J M H J M H J M H K J M J K J+ + + + − − −
The second one has angles
1 1 2 2 1
1 2
0 2 1 1
( 2 ) 2
cos = ,
2 ( )
H K J J m K
M J K J
+ −
θ
+
1 1 2 2 1
1 2
0 2 1 1
( 2 ) 2
cos =
2 ( )
H K J J m K
M J K J
+ −
ϕ −
+
(19)
with the energy
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11 1211
G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, and E.V. Eremin
2 4 4 3 2 2 2 2 2 2 2
2 1 1 1 2 1 0 1 1 0 2 2 1 2 1 1 1 2 1 1 0 1= ( 2 4 2 2 4 4FE m K J Hm J JM HJ J JM J m K H J J m K J J J JM HK− − + − + + − − +
2 2 2 2 2 3 2 2 2 2
2 1 2 1 1 2 1 1 2 2 1 1 0 2 1 1 2 2 2 1 0 2 1 28 8 16 2 2 8 2Hm J J K H J J K HJ J m K J JM J K H H K J J m K JM J K m+ + − − + + + −
2 2 3 2 2 2 2 2 2 2
0 2 1 1 1 2 2 1 2 2 1 2 1 12 2 4 )/2( ) .JM J K H m K J H J K J m HK J K J− − − − +
_______________________________________________
The third such state has the values of the angles
1 0
1
1 1
2
cos = ,
J M H
m K
−
ϕ
2 0
1
2 2
2
cos =
H J M
m K
−
ψ (20)
with the energy
2
3 0 1 2 2 1 0 2 1 2 0= (2 4 4FE HM K K HK J M K H HK J M− + − − +
2 2 2 2 2 2
1 0 1 2 1 1 2 2 1 04K H JM K K m K K K J M+ − − − +
2 2 2 2
2 2 1 1 2 0 1 24 )/m K K K J M K K+ + .
There are three solutions with one pair of sublattices be-
ing in the tilted phase, and two other pairs being parallel
(and directed along the field). The states with parallel sub-
lattices have angles equal to zero, and those in the tilted
states have opposite signs of angles for the pair of sublat-
tices with
1 1 2 2
1
0
2 2
cos =
2
H J m J m
JM
− −
θ − (21)
with the energy
2
2 3 1 1 2 2 1= (2 8 8 4FE H HJ m HJ m Hm J− − + +
2 0 0 1 1 0 2 24 2 2Hm J JM H JM J m JM J m+ − + + +
2 2 2 2
1 1 1 1 2 2 2 28 16 8 )/2 ;J m J m J m J m J+ + +
with
1 0
1
1 1
2
cos =
H J M
m K
− +
ϕ (22)
and the energy
2 2
2 2 0 1 1 0 2 1 0 1= ( 2 4 2FE HM K H HJ M Hm K JM K− + − − + +
2 2 2 2
1 0 2 0 2 1 1 1 14 4 )/ ;J M J M m K m K K+ + +
and with
2 0
1
2 2
2
cos =
H J M
m K
−
ψ (23)
with the energy
2
2 1 0 2 1 2 2 0= (2 2 4FE HM K Hm K H HJ M− + + − −
2 2 2 2 2
0 2 1 0 1 2 2 0 2 2 24 4 )/ .JM K J M m K J M m K K− − + +
Finally, there exists a ferromagnetic solution, where all
sublattices are parallel and directed along the field, i.e.,
1 2 1 2 1 2= = = = = = 0θ θ ϕ ϕ ψ ψ with the energy
2
0 1 2 0 1 0 1 2 0 2= 2 ( ) 4 4 .FE H M m m JM J M m J M m− + + + + +
Obviously, this ferromagnetic state is realized at large val-
ues of the external magnetic field.
When the value of the external magnetic field grows, one
should, generally speaking, observe all possible field-
induced transitions between the above presented solutions.
As a rule, the transitions between antiparallel and tilted state
of the same pair of sublattices are first order transitions
(spin-flop-like), while the transitions between the tilted and
parallel directions of sublattices are of the second order.
We have also obtained analytical solutions for
= /2,Hθ π i.e., the external magnetic field is directed in
the basal xy plane, perpendicular to C3. Notice that this
case cannot be applied to the considered system directly,
because of the in-plane magnetic anisotropy in
Nd0.75Dy0.25Fe3(BO3)4. In that case the symmetry implies
2 1= ;θ π − θ 2 1=ϕ π − ϕ and 2 1= .ψ π − ψ There are sev-
eral solutions, which can be realized.
In the case = 0H we have 1 1 1= = = 0.θ ϕ ψ
For weak nonzero H the tilt angles are
1 2 2 1 1 2
1 2 2
0 1 2 2 1 1 2
( 2 2 )
sin = ,
( 4 4 )
H K K J K J K
M JK K J K J K
− + −
θ
+ −
2
2 1 2 2 1 2
1 2 2
1 1 2 2 1 1 2
( 4 4 2 )
sin = ,
( 4 4 )
H JK J J J J K
m JK K J K J K
− + − −
ϕ
+ −
2
1 1 2 1 2 1
1 2 2
2 1 2 2 1 1 2
( 4 4 2 )
sin = .
( 4 4 )
H JK J J J J K
m JK K J K J K
+ − +
ψ
+ −
(24)
When the value of the external field becomes larger
pairs of magnetic sublattices become parallel to the direc-
tion of the field, step by step. First, the one pair of rare
earth sublattices become field-directed, e.g., Nd one with
1cos = 0,ϕ and with
2 1 1 2 2
1 2
0 2 2
2 2
sin = ,
( 4 )
HK J m K HJ
M JK J
− + +
θ
+
2 1 2 1
1 2
2 2 2
2 4
sin =
( 4 )
JH J H J J m
m JK J
+ −
ψ
+
(25)
at
2 2
1 1 2 2 1 1 2
2
2 1 2 2 1 2
( 4 4 )
=
4 4 2
m JK K J K J KH
JK J J J J K
+ −
− + − −
1212 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11
Magnetoelastic studies of Nd0.75Dy0.25Fe3(BO3)4 in the external magnetic field: Magnetic phase transitions
(or, vice versa, the angle 1ψ for Dy sublattices becomes
/2π with the angle 1φ for Nd sublattices being tilted, de-
pending on the relative values of 1,2J and 1,2 ).K
Then two pairs of rare earth sublattices become parallel
to the direction of the field, 1 1cos = cos = 0,ϕ ψ in the
considered case at
2
2 2 2 1 2 1 2= [ ( 4 ) 4 ]/( 2 )H m JK J J J m J J+ + + .
Here the tilt angle for iron sublattices is
1 1 2 2
1
0
2 2
sin = .
H J m J m
JM
− + +
θ (26)
Finally, at the largest value of the field, in the consi-
dered case at 0 1 1 2 2= 2 2 ,H JM J m J m− − all magnetic
sublattices become parallel to the direction of the external
field.
The PTs for 3⊥H C are of the second order for the
considered model.
The analysis can be significantly simplified when we
take into account that 1,2 1,2.J K J In this case the
tilted states for the field directed along C3 for Nd sublattic-
es have angles
1
1 1
cos ,H
m K
ϕ ≈ − (27)
for Dy sublattices the angles are
1
2 2
cos ,H
m K
ψ ≈ (28)
and for the iron sublattices the tilt angle is
1
0
cos .
2
H
JM
θ ≈ (29)
Then the main PTs occur between the states with antiparal-
lel directions of rare earth sublattices to the tilted ones
(with the iron sublattices being in the antiparallel state),
following by transition in the iron sublattice subsystem
from the antiparallel to the tilted state. These PTs between
the antiparallel states of sublattices to the tilted ones are of
the first order (of the spin-flop type). Then at much higher
values of the field, directed along z, first rare earth sublat-
tices flip to the ferromagnetic state along the field direction
(at the field values 1 1= | |H m K and 2 2= ),H m K and,
finally, iron sublattices flip to the ferromagnetic state at
0= 2 .H JM From the estimates of the critical values of the
field, connected to our experiments, we can assume that we
observe field-induced first order transitions, related to the
former case of the field, directed along C3, while we do not
have enough field strength to observe the PTs to the latter
case of the transitions to the parallel sublattices.
We suppose that in the real system the values of critical
fields are very close to each other, and, hence, it is difficult
to distinguish all such PTs, especially at the conditions of
the experiment 0T ≠ (at least in the second and third tem-
perature ranges).
On the other hand, in this approximation for = /2Hθ π
(field, directed in the basal xy plane), the tilt angles become
1
0
sin ,H
M J
θ ≈ −
1
1 1
sin ,H
m K
ϕ ≈ −
1
2 2
sin .H
m K
ψ ≈ (30)
When the field value grows the first rare earth magnetic
sublattices flip to the state, parallel to the direction of the
field at 1 1= | |H m K and 2 2=H m K , and, finally, the iron
magnetic sublattices become parallel to the field direction
at 0= | | .H M J Those PTs are of the second order. In our
experiments we observe first order transitions related to the
value of the in-plane magnetic anisotropy. This case cannot
be studied analytically yet.
The theoretical analysis of the direction of the external
magnetic field tilted from C3 and tilted from C2 in the bas-
al plane is very complicated and have not yet permitted us
to obtain analytical results. Even for two magnetic sublat-
tices such an analysis can be realized analytically only for
small values of the tilt angles [12,13]. Our study for the
considered six-sublattice model of the antiferromagnet
implies that the critical values of tilt angles, at which the
first order phase transitions transform to the second order
ones [12] are larger than for the two-sublattice antiferro-
magnetic model. On the one hand, this is related to the
considered approximation (singlet orbital state of the iron
sublattices, and the absence of the direct interaction be-
tween rare earth sublattices themselves), and, on the other
hand, it qualitatively agrees with our experimental findings
in Nd0.75Dy0.25Fe3(BO3)4.
We can take into account nonzero temperatures, using
the method developed in Ref. 14. The results of [14] imply
that the values of the critical fields depend on T as
= ( = 0) ( ),c cH H T a T where ( )a T is the component of
the order parameter, characteristic for the considered mag-
netic phase, which decreases with the growth of T from
the maximal value at = 0T to zero at = .cT T Such a con-
sideration qualitatively describes the features of the ob-
tained phase H–T diagrams in Nd0.75Dy0.25Fe3(BO3)4.
5. Summary
In summary, we have performed low-temperature mag-
netoacoustic investigations of rare earth ferroborate
Nd0.75Dy0.25Fe3(BO3)4. We have observed how the fea-
tures of the sound velocity behavior in the magnetic field
were changed due to the tilting of the direction of the ex-
ternal magnetic field from the main crystallographic axes
C3 and C2 in all observed temperature regions, where
magnetic orderings exist. Our observations show that mag-
netic phase transitions exist for large enough values of the
tilting of the external magnetic field from C3 axis. In the
lowest temperature range the behavior for tilting in the zy
and zx planes is similar. In the intermediate temperature
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 11 1213
G.A. Zvyagina, K.R. Zhekov, I.V. Bilych, A.A. Zvyagin, I.A. Gudim, V.L. Temerov, and E.V. Eremin
phase, on the other hand, the behavior of the magnetic
field-induced phase transitions were different for devia-
tions of the field direction from C3 in the zy and zx planes.
Also in this temperature range we registered field-
governed phase transitions when the field is tilted from the
crystallographic axes in the xy (basal) plane. The latter can
be seen also, when we tilt the direction of the external
magnetic field from the basal plane towards C3 axis. In the
highest-temperature ordered phase we observed phase
transition for the field tilted from the basal plane towards z,
with the similar field behavior, comparing with the inter-
mediate phase. Contrary, the features for the tilting of the
field from C2 axis are manifested much weaker than those
in the intermediate temperature phase.
We have also performed comparison of our experimental
findings with the results of the developed microscopic and
phenomenological theories. Our theory agrees with our ear-
lier assumption [4,5] about multisublattice nature of the
magnetically ordered phases in the considered ferroborate.
According to the phenomenological consideration, there can
exist many field-induced magnetic phases in the system, the
transitions between some of them being very close to each
other. We can observe only few of them in the experimental-
ly available magnetic field and temperature intervals. Our
analysis permits us to assume that for higher field region
inhomogeneous magnetic state can take place for the mag-
netic field directed along C3. Our phenomenological analy-
sis qualitatively agrees with the mean field approach devel-
oped in Ref. 11 and with our experimental findings. We can
suppose that the large values of the field and tilt angle val-
ues, at which the first order phase transitions transform to
the second order ones, comparing to the standard two-
sublattice antiferromagnet, are related to the features of the
studied rare earth ferroborates. Namely, we think that it is
related, first, to the singlet state of iron ions (which, from
this viewpoint, have to be more magnetically isotropic,
while the magnetic anisotropy is caused mostly by the rare
earth magnetic ions). Second, it is connected with the rela-
tively weak coupling between rare earth magnetic ions with
iron ones, and with the almost absent interaction between
rare earth magnetic ions themselves.
Acknowledgment
The study was supported by Grant of the Russian Federa-
tion president on support of sciences schools no. 4828.2012.2
and The Ministry of education and science of Russian Feder-
ation, project no. 8365.
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