gakel’–turov oscillations in iron borate

*Corresponding author. Tel.: #380-0652-230232; fax: #380-0652-232310.E-mail address: [email protected] (M.B. Strugatsky).

Journal of Magnetism and Magnetic Materials 219 (2000) 340}348

Gakel'}Turov oscillations in iron borate

Yu.N. Mitsay!, K.M. Skibinsky!, M.B. Strugatsky!,*, A.P. Korolyuk",V.V. Tarakanov", V.I. Khizhnyi"

!Solid-State Physics Chair, Department of Physics, Taurida National University, 95036 Simferopol, Yaltinskaya St. 4, Ukraine"Institute of Radio Physics and Electronics, National Academy of Sciences of the Ukraine, 61085 Kharkov, Academika Proskyri St. 12, Ukraine

Received 14 March 2000

Abstract

Dependence of the amplitude of transverse sound passed through a basal plate of easy-plane weak ferromagnet FeBO3

on a magnetic "eld H, parallel to the plate is investigated in experiment at ¹"77K. This dependence is of the oscillatingform. The phenomenon is caused by magnetic birefringence of transverse sound, being the analog of Cotton}Moutonoptical e!ect. In acoustics this e!ect is associated with magnetic additions appearing in elastic modules upon transitionfrom a paramagnetic phase to an antiferromagnetic one. However, the theory of magnetoacoustic birefringence forhomogeneously magnetized crystals, valid for MnCO

3, is unable to explain the experiment for FeBO

3. A model, taking

into account the experimental boundary conditions and inhomogeneous distribution of magnetization in the sample ofiron borate is constructed. Calculations based on this model describe the experiment satisfactorily. ( 2000 ElsevierScience B.V. All rights reserved.

PACS: 75.30.Gw; 75.50.!y; 75.80.#q

Keywords: Birefringence of sound; Magnetic anisotropy; Magnetoacoustic e!ect; Weak ferromagnet; Iron borate

1. Introduction

Gakel' observed [1] oscillations of the amplitude of transverse sound passing through a plate ofantiferromagnet (AF) MnCO

3during its magnetization. The geometry of the experiment was as follows:

kEC3oH. Here k is the wave vector of the sound, C

3is the 3-fold symmetry axis, and H is the external

magnetic "eld. A consistent theory of this phenomenon was constructed by Turov [2]. It is based ona "eld-dependent magnetic birefringence of transverse acoustic waves propagating along the acoustic axis ofan AF crystal, which is an analog of the Cotton}Mouton optical e!ect. Birefringence of sound is associatedwith magnetic additions appearing in elastic modules upon a transition from the paramagnetic phase to theantiferromagnetic one. According to the model [2], these additions are conditioned by addends in mag-netoelastic energy &l

iljukl, where l

iis a component of the antiferromagnetic vector and u

klis a component

0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 4 1 3 - 3

of the elastic deformation tensor. In this approach, one of the linearly polarized modes of transverse sound isnot in coupling with the magnetic subsystem (nonmagnetic mode), while the other one is in coupling with itrather signi"cantly (magnetic mode). The sound velocity of the magnetic mode depends on the magnetic "eldthat leads to phase shift in modes. As a result of magnetic birefringence, a linearly polarized acoustic waveentering a crystal is transformed into an elliptically polarized wave while emerging from the crystal.Oscillations of ellipticity induced by variations in H lead to sound amplitude oscillations, which are detectedby a piezoelement. We named these sound oscillations in AF after the scientists who were the "rst to studythis e!ect, i.e., the Gakel'}Turov oscillations (GTO). According to Turov's model, the description of GTOrequires an analysis of magnetostrictive coupling of the acoustic wave and the lower branch of electron spinexcitations [2]. This approach proved to be e!ective in interpreting the results obtained in Ref. [1] formanganese carbonate.

Note that the theory [2] was completed by calculations of coupling of the sound and the second branch ofelectron spin excitations [3].

This paper is devoted to GTO investigations in iron borate (FeBO3), it being an AF and belonging to the

same space and magnetic symmetry group as MnCO3, but di!ering from it in much more strong magnetos-

trictive e!ects.

2. Experiment

The sample selected for the experiment was grown from the gaseous phase by the technique as in Ref. [4].It was in the form of a hexahedron plate of thickness h"120mm. The area of each of the two basal facesoriented at right angle to the C

3-axis was &20 mm2. These faces were subjected to mechanical treatment to

obtain optical quality of the surface. After the treatment the error in the orientation of the C3-axis relative to

a normal to a plate surface did not exceed 13.The sample was placed between two linearly polarized piezotransducers made of X-cut piezoelectric LiNbO

3of disk shape with diameter 2mm and thickness &12lm, set on massive brass disks of thickness &3mm(Fig. 1). The latter ones were the rear electrodes, which smoothed out the frequency characteristic and enlargedthe strip of working frequencies of the transducers through sound re#ection removing. One transducer wasconnected through a coaxial waveguide to a generator and was the emitter of sound, while the other one wasthe detector. The transducers set on basal faces of the crystal were arranged so that the polarizations of theemitted and detected sound were either crossed at 903, or mutually parallel. Measurements were made in thecontinuous generation mode. The detector was screened from direct electromagnetic signals, that were nottransformed into sound, by means of two diaphragms made of aluminium foil of thickness 5}7lm, which werethe second electrodes of the transducers. Acoustic contact in this junction was provided by using thin layersof organic siliceous liquid, which hardened at 150K through junction cooling. The magnetizing "eld H wascreated by Helmholtz coils and was parallel to the basal plane of the crystal. In the course of the experiments,the FeBO

3sample was immersed in liquid nitrogen. Control measurements were made on a thin FeBO

3plate (of thickness&50 lm) in pulsed generation mode by using a bu!er delay line.

For crossed polarizations of emitter and receiver, experimental magnetic "eld dependence of the amplitudeA

Mof transverse sound, passed through the FeBO

3plate, is shown in Fig. 2b. Curve 2b was obtained after

"ltration of small-period oscillations. Filtration was carried out by integrating the signal on the recorderentrance. These small-period oscillations are not a noise, since they disappear when the magnetic "eldvariation stopped, and the results of repeated recordings correlated with one another. So, we can state theexistence of a "ne structure of the curve. A comparison of oscillations in Fig. 2b with the oscillations observedby Gakel' [1] in MnCO

3shows that they are apparently of the same nature. In addition, as well as in Ref.

[1], we observed a half-period shift in the oscillation phase on transition from the crossed polarizations of theemitter and the receiver to their parallel orientation.

Y.N. Mitsay et al. / Journal of Magnetism and Magnetic Materials 219 (2000) 340}348 341

Fig. 1. Experimental head scheme: (1) plate of iron borate; (2) piezotransducers; (3) massive brass disks; (4) aluminum foil; (5) coaxialwaveguide.

3. GTO calculations for FeBO3 on the basis of Turov's theory

After transverse sound wave passing through a plate of thickness h, the amplitude of a component withpolarization orthogonal to polarization of incident wave is determined as follows [2]:

AM"J2 sint

0cost

0J1!cos(*k ) h). (1)

Here t0

is the angle between polarization vectors of incident wave and its magnetic mode, *k is the di!erencein wave vector values of magnetic and nonmagnetic modes. The amplitude of the incident wave is taken as a unit.

Using Eq. (1) for iron borate, we calculated the dependence AM(H), which is shown in Fig. 2a. A compari-

son of the experimental (Fig. 2b) and theoretical GTO curves demonstrates their signi"cant di!erence.According to the theory, the amplitude of GTO must be independent of H and their period must beconsiderably smaller than in the given experiment. In the case of FeBO

3the physical nature of this

phenomenon is apparently more complicated than that predicted by Turov's theory [2].In this paper we suggest a theoretical model that takes into consideration the peculiarities of FeBO

3magnetoelastic coupling and describes the experiment satisfactorily.

4. Model

Iron borate (FeBO3), is an easy-plane AF crystal with weak ferromagnetism. Its crystal lattice may be

described by D63$

spatial symmetry group. Small de#ections of magnetic moments from basal plane beingneglected, its magnetic energy in homogeneous state is given by

Fm"1

2Em2#D(l

xm

y!l

ym

x)!2M

0Hm, (2)

where E is the exchange parameter, D the Dzyaloshinsky constant, m and l the ferro and antiferromagneticvectors, associated with sublattice magnetizations in standard forms

m"(M1#M

2)/2M

0,

l"(M1!M

2)/2M

0,

342 Y.N. Mitsay et al. / Journal of Magnetism and Magnetic Materials 219 (2000) 340}348

Fig. 2. Curves of magnetic "eld dependence for transverse sound amplitude AM: (a) theory for isotropic model (Turov); (b) experiment; (c)

theory for unisotropic model taking into account uniaxial magnetic anisotropy in the basal plane of the crystal.

l<m, l+1,

M0"DM

1D"DM

2D.

Here the z-axis is selected orthogonal to the basal plane.Experiments with the use of magneto-optical e!ects (see, for example, Ref. [5]) show that mechanical

action on iron borate monocrystal results in a signi"cant magnetic anisotropy in the basal plane. Suchanisotropy was observed, in particular, by one of the authors studying (with Zubov) the stressed crystalthrough re#ective magneto-optical Kerr e!ects. This anisotropy is apparently caused by deformations of the


Fig. 3. Axes and vectors orientation in a crystal of iron borate.

crystal through magnetoelastic coupling. In our experiment, organic siliceous liquid hardened duringmagnetoacoustic junction cooling, that led to deformations in the basal plane of the crystal. The deforma-tions decreased deep into the sample. These considerations give a possibility to formulate a simple physicalmodel of the phenomenon: mechanical boundary conditions result in the appearance of uniaxial magneticanisotropy in the basal plane. Anisotropy constant depends on the z coordinate along the C

3-axis. Its value is

minimal in the center of the crystal. Inhomogeneity of the anisotropy is formed by deformations, which areinhomogeneous throughout the crystal. Considering that crystal growth tension and other defects in ironborate crystals usually do not lead to perceptible magnetic anisotropy, we will not take them into account.However, if the contribution of these defects to magnetic anisotropy existed, it would not distort the principalcharacter of the suggested model. Note also that the homogeneous pressure in#uence on the magnetic andresonance properties of a weak ferromagnet was investigated in detail in Ref. [6]. So, the magnetic energy ofthe crystal in our model may be represented as

Fm"1

2Em2#D(l

xm

y!l

ym

x)!2M

0Hm!al2m . (3)

Here the last addend describes the magnetic anisotropy, associated with the mechanical boundary conditions(m is parallel to easy axis for l in the basal plane, see Fig. 3), and the anisotropy constant a'0 depends on thez-coordinate. In Eq. (3) inhomogeneous exchange energy is absent, since we assume that the coordinatedependence of anisotropy is weak enough, and consider comparatively long-wave oscillations. Expression (3)may be reduced to the following form:

Fm"1

2Em2!Dm!2M

0Hm sin(a!h)!a cos2 h. (4)

Angles a and h in this formula are shown in Fig. 3. Elastic and magnetoelastic energies for FeBO3

crystal aredetermined by its spatial symmetry group and have the following forms:

F%"1

2C

11(u2

xx#u2

yy)#C

12uxx

uyy#2C

66u2xy#1

2C

33u2zz#2C

44(u2

xz#u2

yz)

#C13

(uxx#u

yy)u

zz#2C

14(u

xxuyz!u

yyuyz#2u

xyuxz

), (5)


Fm%

"B11

(uxx

l2x#u

yyl2y)#B

12(u

yyl2x#u

xxl2y)#2B

66uxy

lxly#2B

14[2u

xzlxly#u

yz(l2x!l2

y)]

#B31

uzz

(l2x#l2

y). (6)

Here x-axis is in coincidence with 2-fold symmetry axis C2. Minimizing F

%#F

.%we obtain expressions for

static deformations

u0xx!u0

yy"(B

66C

44!2B

14C

14) cos[2(a!b!h)]/[2(C2

14!C

44C

66)],

u0xy"(B

66C

44!2B

14C

14) sin[2(a!b!h)]/[2(C2

14!C

44C

66)],

u0xz"(2B

14C

66!B

66C

14) sin[2(a!b!h)]/[2(C2

14!C

44C

66)],

u0yz"(2B

14C

66!B

66C

14) cos[2(a!b!h)]/[2(C2

14!C

44C

66)]. (7)

Investigating transverse sound propagation along the C3-axis, we may limit consideration by studying the

l and m vector oscillations in the basal plane. Let the angle u describe the rotation of l from its equilibriumposition. In this case (see Fig. 3) we have the following expressions for vector l components:

lx"cos(a!b!h#u),

ly"sin(a!b!h#u).

Picking out static and dynamic components in the deformation tensor, we may decompose the crystalenergy to the C

2in dynamic variables u and u

ik. (Here inafter u

ikmeans not the whole deformation but the

dynamic part of deformation tensor only.) Taking into account the low frequency of the acoustic oscillationsinvestigated, we may as in Ref. [2] consider oscillations of vector l as following adiabatic crystal deforma-tions, caused by sound wave propagation. In this case, by minimizing the energy in u we obtain

u"

2B14

Muyz

sin[2(a!b!h)]!uxz

cos[2(a!b!h)]NM

0[H

.%-#Hm sin(a!h)]#a cos 2h

, (8)

where H.%-

"(B266

C44

!4B14

B66

C14

#4B214

C66

)/[2M0(C

44C

66!C2

14)] is the e!ective magnetoelastic

"eld caused by spontaneous magnetostrictive deformations. For equilibrium ferromagnetic vector m we have

m"[d#2M0H sin(a!h)]/E. (9)

To diagonalise quadratic in dynamic variables uik

form of crystal energy, it is necessary to make coordinatetransformations as follows:

r"x cos[2(a!b!h)]!y sin[2(a!b!h)],

q"y cos[2(a!b!h)]#x sin[2(a!b!h)],

z"z. (10)

In these coordinates the quadratic part of the sum F%#F

.%is transformed into the form

F%&&.%

"2u2qz

C44

#2u2rz

(C44

!*C%&&). (11)

Here

*C%&&"4B2

14H

%9M

0M2H

%9H

.%-#H sin(a!h)[H sin(a!h)#H

D]N#2aH

%9cos 2h

, (12)

where H%9"E/4M

0is the e!ective exchange "eld, and H

D"D/2M

0is the Dzyaloshinsky "eld.


Using expression (11) for e!ective elastic energy we may write phase velocities and wave vectors fornonmagnetic and magnetic modes as

vq"JC

44/o, k

q"u/v

q,

vr"J(C

44!*C%&&)/o, k

r"u/v

r,

(13)

where o is the crystal density and u the sound frequency. The polarization vector of the nonmagnetic mode isparallel to the q-axis and of the magnetic mode to the r-axis.

The angle h in formulas (10) and (12) depends not only on "eld H value, but on coordinate z as well. So, theorientation of the r and q axes is changed #uently along the z-axis in the crystal, therefore, the energy will beredistributed between nonmagnetic and magnetic modes, during the sound wave propagation. The velocityvrdepends on z as well. All this makes the problem of calculation of amplitude of the wave passing through

the crystal much more complex than in the case of homogeneously magnetized sample.

5. Amplitude calculation

First of all, it is necessary to determine h(H, z) dependence. Minimizing the magnetic part of energy (4) weobtain the following equation:

H"!

a(z)H%9

sin 2hM

0H

Dcos(a!h)

. (14)

Setting up coordinate z dependence of magnetic anisotropy constant a(z), we may determine h(H, z) from (14)numerically. Then let us introduce angle c between the r and C

2axis. As it follows from Fig. 2 and formulas

(10), this angle is determined by the expression c"2(a!b!h). For amplitude calculation Jones matrixmethod known in optics [7}9] is used here. One may imagine the crystal divided into n homogeneouslymagnetized layers with uniform anisotropy in each layer. In this case association between incident andpassed through mth layer waves is given by

Au065rm

u065qmB"¹

mAu*/rm

u*/qmB. (15)

Here the matrix ¹m

is determined by the expression

¹m"A

exp(!ikrm

h/n) 0

0 exp(!ikqh/n)BA

cos*cm

sin*cm

!sin *cm

cos*cmB, (16)

where *cm"c

m`1!c

m"!2(h

m`1!h

m) is the r-axis rotation on transition from the mth to (m#1)th

layer.Applying Eq. (15) sequentially to all the n layers, we obtain the following dependence between incident and

passed through the crystal waves:

Au065r

u065qB"¹

n¹

n~12¹

2¹

1Au*/r

u*/qB"¹A

u*/r

u*/qB. (17)

It is suitable to represent matrix ¹ in the form

¹"AA

1B1

C1

D1B#iA

A2

B2

C2

D2B. (18)


Let t be the mean angle between polarization vectors of incident wave and its magnetic mode. It is notdi$cult to show that

t(H)"t(0)!2h(H, z"!h/2). (19)

Here inafter, an origin of coordinates is selected at the center of the crystal. On the basis of (r, q) the incidentlinearly polarized wave is described by

Au*/r

u*/qB"A

u*/ cost

u*/ sintB. (20)

Here the incident wave is u*/"exp(iut). Using Eqs. (17) and (18) we determine the component u065M

of thewave passed through the crystal with polarization orthogonal to polarization of the incident wave u*/

u065M

"u065r

cos(p/2#t)#u065q

sin(p/2#t)

"[C1cos2t#(D

1!A

1) sint cost!B

1sin2t] cosut

#[C2cos2 t#(D

2!A

2) sint cost!B

2sint] sinut. (21)

Hence, we obtain the required expression for the amplitude

AM"JR2

1#R2

2, (22)

where

R1"C

1cos2t#(D

1!A

1) sint cost!B

1sin2 t,

R2"C

2cos2t#(D

2!A

2) sint cost!B

2sin2 t.

Formula (22) gives a possibility to determine AM(H) if the dependence a(z) for anisotropy in the crystal is

given earlier. Let us suppose

a(z)"k1#k

2DzD. (23)

Parameters k1, k

2, a and t(0) optimization leads to the dependence shown in Fig. 2c. It is found that k

1and

k2

correspond to "elds +200Oe on the surface and +130Oe at the center of the crystal.

6. Conclusion

The oscillation period in Fig. 2c is much larger than it is in isotropic (Turov's) model (Fig. 2a). Thisdi!erence may be explained by addend 2aH

%9cos 2h appearing in the denominator of the expression *C%&&

(see Eq. (12)). This addend is caused by uniaxial magnetic anisotropy in the basal plane of the crystal inducedby the deformation. The di!erence in height of maximums is the result of magnetic "eld dependence of the r-and q-axis orientations. The last circumstance may be seen in formula (22), since t is a function of H.

The calculations show that parameters a and t(0) are responsible mainly for the height of the maximums,and parameters k

1and k

2determine monotonic intervals of the curve. Adapting theoretical curve to the

experimental one, we found that the selection of parameters k1, k

2, a and t(0) values is practically unique.

Note that the curve in Fig. 2c is represents "eld values greater than +50 Oe, when the domain structure isalready absent in iron borate.


References

[1] V.R. Gakel', Pis'ma Zh. Eksp. Teor. Fiz. 9 (1969) 590 [JETP Lett. 9 (1969) 360].[2] E.A. Turov, Zh. Eksp. Teor. Fiz. 96 (1989) 2140 [Sov. Phys. JETP 69 (1989) 1211].[3] I.F. Mirsaev, Fiz. Tverd. Tela 36 (1994) 2430 [Phys. Solid State 36 (1994) 1321].[4] M.B. Strugatsky, Cand. Sc. Thesis, Simferopol State University, Simferopol, USSR, 1988.[5] G.B. Scott, J. Phys. D 7 (1974) 1574.[6] I.E. Dikshtein, V.V. Tarasenko, V.G. Shavrov, Zh. Eksp. Teor. Fiz. 67 (1974) 816 [Sov. Phys. JETP 40 (1975) 404].[7] R.C. Jones, J. Opt. Soc. Amer. 31 (1941) 488.[8] R.C. Jones, J. Opt. Soc. Amer. 38 (1948) 671.[9] R.C. Jones, J. Opt. Soc. Amer. 46 (1956) 126.


gakel’–turov oscillations in iron borate

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