Magnetoelectric effect and magnetic torque of chromium chlorine boracite Cr₃B₇O₁₃Cl

Article

Reference

Magnetoelectric effect and magnetic torque of chromium chlorine boracite Cr

₃

B

₇

O

₁₃

Cl

YE, Zuo-Guang, et al.

Abstract

The magnetoelectricity of the boracite Cr3B7O13Cl has been studied by quasi-static and dynamic magnetoelec. (ME)H measurements on (001)cub platelets with a ferroelastic/ferroelec. single domain state, obtained by poling with an elec. field of E - 45 kV.cm-1, applied upon cooling through the 42mL' -> mm21' phase transition at 160 K. The magnetoelec. effect has been found to be essentially quadratic with typical values of the ME tensor components b333 = 1.5 x 10-18 [s/A] and b322 = 4 x 10-19 [s/A] at 4.2 K, whereas the linear ME coeff. a33 shows strangely small and fluctuating values of the order of 5 x 10-14 [s/m]. The temp. dependence of the ME effect has been measured between 1.8 and 40 K, indicating a spread max. of b333 near 11 K. The magnetic torque showed a not completely quadratic dependence vs. magnetic field at 4.2 K, thus suggesting the contribution of weakly ferromagnetic ordering. The information obtained at this stage of study permits one to derive the magnetic symmetry of Cr3B7O13Cl at 4.2 K only tentatively to be the orthorhombic with point group m'm'2, implying a very weak spontaneous magnetization parallel [...]

YE, Zuo-Guang, et al . Magnetoelectric effect and magnetic torque of chromium chlorine boracite Cr

₃

B

₇

O

₁₃

Cl. Ferroelectrics , 1994, vol. 161, no. 1-4, p. 99-110

Available at:

http://archive-ouverte.unige.ch/unige:31343

Disclaimer: layout of this document may differ from the published version.

1 / 1

Ferroelectrics. 1994. Vol. 161. pp. 99-110 Reprints available directly from the publisher Photocopying permitted by license only

© 1994 OPA (Overseas Publishers Association) Amsterdam B.V.

Published under license by Gordon and Breach Science Publishers SA Printed in the United States of America

MAGNETOELECTRIC EFFECT AND MAGNETIC TORQUE OF CHROMIUM CHLORINE

BORACITE Cr

₃

B

₇

0

₁₃

Cl

Z.-G. YE, J.-P. RIVERA and H. SCHMID

Department of Mineral, Analytical and Applied Chemistry, University of Geneva, CH-1211 Geneva 4, Switzerland

and

M. HAIDA and K. KOHN

Department of Physics and Advanced Research Center, Waseda University, Shinjuku-ku, Tokyo 169, Japan

(Received December 1, 1993; in final form January 21, 1994)

The magnetoelectricity of the boracite Cr3B7013CI has been studied by quasistatic and dynamic mag- netoelectric (ME)H measurements on (OOI),ub platelets with a ferroelastic/ferroelectric single domain

~tate, obtained by poling with an electric field of E = 45 kV ·em ^{- I,} applied upon cooling through the 42ml'-> mm21' phase transition at 160 K. The magnetoelectric effect has been found to be essentially quadratic with typical values of the ME tensor components (3₃₃₃ = 1.5 x JO-ts [s/A) and (3₃₂₂ = 4 x I0-¹⁹ [s/A) at 4.2 K, whereas the linear ME coefficient a33 shows strangely small and fluctuating values of the order of 5 X 10-¹⁴ r s/m I. The temperature dependence of the ME effect has been measured between 1.8 and 40 K, indicating a spread maximum of J3333 near II K. The magnetic torque showed a not completely quadratic dependence versus magnetic field at 4.2 K, thus suggesting the contribution of a weakly ferromagnetic ordering. The information obtained at this stage of study permits to derive the magnetic symmetry of Cr3B7013CI at 4.2 K only tentatively to be the orthorhombic with point group m'm'2, implying a very weak spontaneous magnetization parallel to the twofold polar axis. The tem- perature dependence of the spontaneous polarization P, has been investigated down to 4 K.

Keywords: Magnetoelectric effect, magnetic torque, ferroelastic/ferroelectric domains, weak ferromagnetism, boracite Cr3B7013Cl.

1. INTRODUCTION

The crystal family of boracites includes more than 20 isomorphous compounds with the general formula M3B70₁₃X (M-X), where M stands for divalent metal ions (Mg, Cr, Mn, Fe, Co, Ni, Cu, Zn or Cd) and X for halogen ions (Cl, Br or I).

They have a cubic paraelectric and diamagnetic or paramagnetic high temperature phase with space group F'43c (Shubnikov point group 43ml'). Upon cooling most of them exhibit a phase transition from the cubic prototype phase to a fully fer- roelectric/fully ferroelastic orthorhombic mm21' phase, or to a sequence of phases with point groups mm21', ml' and 3ml' (Zn-Cl, Fe-CI, Fe-I and Co-CI).l-3 The trigonal phase was also found in Fe-Br.⁴ The boracites with paramagnetic 3d- transition metal ions of Fe, Co, Ni or Cu appear particularly interesting because they become magnetically ordered at low temperatures (probably with the excep- tion of Cu-I), showing thus the simultaneous presence of ferroelectricity and weak

99

100 Z.-G. YE eta/.

ferromagnetism. ⁵ In such a multi-ferroic phase the magnetoelectric effect, which consists in the induction of a polarization by applying a magnetic field ((ME)H- effect) or the induction of a magnetization by an electric field ((ME)E-effect), is of FM liFE 1-type according to the nomenclature of Reference 6.

Chromium-chlorine boracite Cr₃B₇0₁₃Cl (Cr-Cl) is known to undergo the fol- lowing sequence of structural phase transitions^7:

T1

=

264 K T2 = 160 K

43ml' 42ml' mm21',

where the tetragonal42ml' phase (space group P42₁c⁸⁾ is the first and so far unique one in the crystal family of boracites. Interesting properties related to the presence of this unusual non-polar 42ml' phase have been disclosed recently in Cr-Cl:⁹·10 i) in the temperature interval of the tetragonal phase (160 K < T < 264 K), application of an electric field E//(001)cub ( E > 8 ^X 10⁴ V ·em -¹⁾ can induce a polar mm21' phase, giving rise to double hysteresis loops of antiferroelectric-like behaviour; ii) during the first order transition from the cubic to the tetragonal phase at T₁ = 264 K, domains of the mm21' phase were found to be induced by internal stress gen- erated at the mechanically non-matching 43ml'/42ml' phase boundary. At low temperatures, anomalies of the spontaneous polarization and birefringence were also reported and ascribed to a magnetic ordering origin.¹¹ The magnetic suscep- tibility (x) of Cr-Cl measured between 4 and 300 K indicated an antiferromagnetic character at low temperatures with a spread maximum of

x

near 30 K. ¹² The magnetic moment determined thereby corresponded to the moment of the spin alone calculated for the Cr²+ ions, no weak ferromagnetism being present. The purpose of the present work was to study the magnetoelectric effect and the mag- netic torque of the Cr-Cl crystals with an attempt at determining the magnetoelectric coefficients, magnetic phase transitions and the related magnetic point group at low temperatures.

2. EXPERIMENTAL PROCEDURE

2.1 Sample Preparation

Single crystals of Cr-Cl boracite were grown by the chemical vapour transport technique.¹³•14 Platelets of (lOO)cub-cut with thickness of 30-60 1-Lm and area of about 2 mm² were carefully polished and electroded by deposition of semi-trans- parent Au/Cr layers, which permitted the simultaneous optical control of the fer- roelastic/ferroelectric domain states during the electric field poling and to minimize the mechanically induced strain due to surface contacts. Gold wires were fixed at the centre of the platelets.

2.2 Ferroelastic! Ferroelectric Poling

Ferroelastic/ferroelectric poling of the Cr-Cl samples was realized by means of a special optical He-flow cryostat adapted to a polarized light microscope. The ferroic species 43ml'Fmm21' allows six fully ferroelectric and fully ferroelastic domain

MAGNETOELECfRIC EFFECf OF Cr3B7013Cl 101

states for the mm21' phase, ¹⁵ with the spontaneous polarization Ps parallel to the six equivalent (lOO)cub directions.^1•7 In order to avoid the contribution of domains with different orientation, the magnetoelectric measurements have to be made on a ferroelastic/ferroelectric single domain. Since the spontaneous deformation of the ferroelastic domains is "fully" coupled with the sense of the spontaneous po- larization vector, an electric field poling with E//(001)cub will be appropriate to obtain such a single domain state.

According to the electric field strength/temperature phase diagram established previously for Cr-CI,⁹ the coercive field (Ec) for the induction (in the 42ml' phase) or for the poling (in the mm21' phase) of a single domain of the mm21' phase presents a minimum value around the tetragonal ~ orthorhombic phase transition temperature at T₂ = 160 K and it increases rapidly upon further cooling below T2 •

10 Therefore an electric field (E//(001)cub) of E = 45 kV ·em_, (with a margin of safety compared with Ec) was applied at 180 K and kept upon cooling through T2 down to 100 K. Optical domain control indicated that a single domain state of the mm21' phase was obtained from the 42ml' phase at temperatures closely above T2 , with extinction directions along (llO)cub' i.e., turned by 45° with respect to those of the tetragonal domains, as shown in Figure 1. Once established, the single domain state subsisted at 100 K after removal of the poling field and remained stable upon further cooling down to 10 K at zero electric field. Note that in order to destroy such a poled state at 100 Kit would be necessary to apply a field strength higher than 95 kV ·em -^{1. 10} This poling procedure was reproducible and applied in the cryostat for the ME measurements.

2.3 Magnetoelectric Measurements

The magnetic field induced magnetoelectric effect (ME)H was measured in a vertical He bath cryostat by using both the quasistatic method and the low frequency dynamic method. The quasistatic method consists in measuring the electric charges induced by slowly increasing or decreasing the magnetic field at constant temper- ature by using an electromagnet with a linear sweep of H versus time. The dynamic method consists in detecting the charges induced by a low frequency ac magnetic field of about 10 Oe and 133 Hz under a static magnetic bias field of at least 100 Oe, superposed parallel to the ac field. This technique permits to measure contin- uously the ME effect as a function of temperature.

A Keithley electrometer (K642, special low noise) was used in the charge mode, so that the calibration was not necessary for quasistatic measurements. This is an advantage of the (ME)weffect compared with the less convenient inverse effect, i.e., the electric field induced magnetization measurement (ME)£, which requires calibration. The temperature was measured with a calibrated carbon glass resistor with He as exchange gas.

The (001)cub platelets of Cr-Cl were fixed on a special rod in such a way that the magnetic field can be applied in directions between the axes 3 and

3

by going through the axis 2, i.e., the angle <p varying from 0 to 180°, as indicated in Figure 2. The samples with ferroelastic and ferroelectric single domain state underwent a cooling from 50 K down to 4.2 K in a magnetic field of 10 kOe parallel to the axis

102 Z.-G. YE eta/.

FIGURE l Photograph showing the domain transforma_!ion in a Cr-Cl (OOI)cub plate (thickness = 28 11m, area = 1.35 mm²⁾ from domains of the tetragonal 42ml' phase (a) to a ferroelectric/ferroelastic single domain state of the orthorhombic mm21' phase (b), performed by an electric field poling with

£//(OOI}cub (45 kV·cm-•) applied at 180 K (a) and kept upon cooling down to 100 K (b). See Color Plate I.

3 or 2 in order to align the possible magnetic domains at low temperatures. The induced charges were measured on the planes perpendicular to the axis 3.

2.4 Magnetic Torque Measurements

Magnetic torque measurements were performed at low temperatures on a (OOl)cub platelet of single domain (wt. = 1.2 mg) by means of an automatic recording torquemeter in a magnetic field up to 18 kOe. The specimens were fixed with the c-axis parallel or perpendicular to the axis of rotation of the measuring rod, so that the magnetic field can be rotated in the orthorhombic ab plane or in a plane containing the c-axis. A thermocouple of Au-Co vs. Cu was used for temperature measurement.

MAGNETOELECTRJC EFFECT OF Cr3B7013CI 103

C(3)

~ 'pME _H

;f:f

(110)cub

-

b(2)

FIGURE 2 Correlation between crystallographic axes (mm21' phase), optical indicatrix and orien- tation of the magnetic field H for the magnetoelectric measurements of Cr-CI. H can be turned in the 2-3 plane from plus to the minus direction of the 3 axis (0,;;; q> ,;;; 180°). The ME charges are detected on the (001 )cub surfaces.

3. RESULTS AND DISCUSSION

3.1 Quasistatic ME Measurements

Figure 3 gives the recorded variation of the magnetic field induced charges as a function of time at 4.2 K with different orientations of H, uncorrected for the drift_

The magnetic field was programmed so as to increase linearly with time from 0 to 10 kOe in 2.5 min. It can be seen that the (ME)H signal is much more important with H parallel to the spontaneous polarization (H/131/Ps) than with H parallel to the axis 2, i.e., perpendicular to P5 , in spite of a larger drift in the former case.

Figure 4 shows the sinusoidal form of the angular dependence of the magnetoelectric charges at 4.2 K for various magnetic field strengths.

In order to calculate the magnetoelectric coefficients, the recorded curves were corrected by taking into account the drift with time, which can be determined by joining the beginning and the end of the measurements at zero field (dashed lines in Figure 3) and subtracted from the measured values. At 4.2 K the drift for various measurements shows a linear variation with time within the limits of a measuring cycle. After correction for the drift, the measured data can be fitted with a poly- nomial of the form Y;

=

ax; + bx~, where ^X;

=

1, 2, . .. , 10 [kOe] andY; stands for the related induced polarization. By matrix calculus, ¹⁶ one can obtain the coefficients a and 13 of the linear and quadratic ME effects, respectively. In the case of Cr-Cl, the induced polarization is fitted according to the following forms:

H//3: P3 = a33H3

+

^~13mH~, H//2: P3 = a32H2

+

^!13322H~.

Figure 5 gives the fitting results of one of the measurements at 4.2 K for H//3.

It can be seen that the coefficient 13₃₃₃ brings a large contribution to the ME effect in Cr-CI, as one could expect from the two-fold shape of the polarization curves (see Figure 3). Therefore the magnetoelectric effect is essentially of second order

104 Z.-G. YE eta/.

0 2 3 4 5 6

Time (mn)

FIGURE 3 Variation (without correction of drift) of the magnetoelectric charges of Cr-Cl as a function of time, i.e., of magnetic field strength (which increases linearly with time) at different orientations of H (T = 4.2 K, thickness = 47 fLm, area = 2.28 mm²).

with a typical value of the ME tensor component ~333 = 1.5 ^X

w-

¹⁸ [s/A] at 4.2 K. However, a slight linear effect was found to be present and the linear ME coefficient a₃₃ shows very small and fluctuating values of the order of 5 x 10-¹⁴ [s/m] (T = 4.2 K). Figure 6 shows the angular dependence of the coefficients at 4.2 K, where ~322 (4 X I0-¹⁹ [s/A]) is smaller than ~333 and ^a32 smaller than I0-¹⁴ [s/m]. This reflects the angular variation of the induced charges in Figure 4.

In order to study the temperature dependence of the ME coefficients the variation of Ps was measured continuously down to 4.2 K (Figure 7). Upon cooling a wide spread minimum of Ps was found near 30 K, followed by a sharp peak at 15 K and a minimum near 12 K. Ps increases thereafter before reaching a plateau near 4 K. With such a variation of Ps, which lies parallel to the measured magnetoelectric polarization, a slight instability in temperature may give rise to charges of pyro- electric origin, much more important than the magnetic field induced charges.

Several measurements at the same temperature were therefore necessary in order to obtain recorded curves with linear drift and a mean value of the coefficients. Figure 8 gives the thermal variation of the ME coefficient ~333 between 1.8 and 40 K. It increases upon heating and shows a spread maximum near T m = 11 K with a value of 2.6 ^X I0-¹⁸ [s/A], indicating a magnetic ordering.

3.2 Dynamic ME Measurements

A special dynamic method, as described in Reference 17, was used for continuous measurements of the ME coefficients. Owing to very weak linear ME effect, only

1/) Q)

2' 0 r. (._)

·;;: u

u

_Q)

Q) 0 (ij

c 0>

0

~

105

3

-

(Ps_~, l ^-._/ _H

"

^\

~o-_.L¹ -2

H= 10 ke£

Angle({)

FIGURE 4 Angular dependence of the magnetoelectric charges_of Cr-Cl at magnetic field strengths of 5, 8 and 10 kOe: cp = 0°, H/13; cp = 90°, H/12; cp = 180°, H/13. (T = 4.2 K, thickness = 47 fim, area = 2.28 mm²).

~

N

'E 50

<-!

~ u

% 40

'/

w

z

0....~

/

c 0 30

""§ N '<''!> ^'!-

/

·;: 0 X~'!>'!>'!>

~

0 0.... 20 '<''!>

'/~'!>

(.) (,}-!>'!>

·;::

«.:

~'!>'!>..,

ti /"

~

Q) 10

_g _~

Q) c

a3^~H~

0>

~ 0

- - -

00 2 4 6 8 10

Magnetic Field [k~]

FIGURE 5 Dependence of the magnetoelectric polarization upon magnetic field (H//3), fitted ac- cording to the function P3 = o.33H3 + ~I3333^H~, showing a strong contribution of the second order ME effect and a weak contribution due to the linear term (T = 4.2 K).

106 Z.-G. YE eta/.

...J -

16

{3333

l(J=Oo: H//3//~

...J -

lfJ =90°: HI! 2..L~

10

~ _IJl 12 _E

...

"' IJl

~Q

...

><

a33

~Q

OQ...

8 ^{...__ ><}

I ^'

⁴

_'

⁵ ~

! ----:

...

4

... .

- _- - · ^- -

^a3z

·

0 0

0 45 90

Angle l(J (degree)

FIGURE 6 Angular dependence of the magnetoelectric coefficients a and 13 of Cr-Cl boracite (T = 4.2 K).

2400

N _'E 2475

u

u

.=:-

~ 12K

2.470

2465

0 10 20 30 40 50

T(K)

FIGURE 7 Temperature dependence of the spontaneous polarization Ps of Cr-Cl at low temperatures measured along the (OOI)cub direction of a ferroelectric/ferroelastic single domain, showing unusual anomalies due to magnetic ordering.

MAGNETOELECTRIC EFFECT OF Cr3B7013Cl

30.---.

20

. ~

.1. I

;. I

I -

/ I

..

^{/ .}

: /

.

.

^{:/. :}

.

^{• / .}

/

..

^:

I

10 I

I

\(

20 Temperature ( K)

_j

30 40

107

FIGURE 8 Temperature dependence of the coefficient of the quadratic magnetoelectric effect j3₃₃₃ between 1.8 and 40 K determined by quasistatic measurements, where the spread maximum near 11 K indicates the onset of a magnetic ordering.

the coefficient of the quadratic ME effect was measured. The temperature de- pendence of 13333 showed two anomalies at low temperatures, one at about 10 K and the other one, less pronounced, at about 14 K. The former is close to the temperature of the maximum of 13333 measured quasistatically and the latter to the temperature of the peak of the spontaneous polarization P5(T) (Figure 7) and the jump of the spontaneous birefringence at about 15 K, ¹¹ suggesting two magnetic phase transitions.

3.3 Magnetic Torque

The angular dependence of the magnetic torque curves showed an entirely twofold type shape when the magnetic field is applied in the orthorhombic ab plane, in- dicating an ordinary paramagnetic behaviour. In fact, the magnetic torque Ton the unit volume (or the unit mass) of the sample can be expressed as T = -~J.-omH sin a, ¹⁸ where a is the angle between the magnetic moment m and the magnetic field Hand "- " indicates that the torque is in the direction opposite to that of increasing a. In the paramagnetic case m

=

xH, therefore T

=

-IJ.-oXH² sin a. On the other hand, when the magnetic field is applied in the plane containing the c-axis, the magnetic torque curves at 4.2 K were slightly distorted from a twofold form. The amplitudes of the torque showed a not purely quadratique dependence versus the applied field, as shown in Figure 9. This behaviour suggests a linear component of the magnetic torque due to the contribution of a weak spontaneous magnetization along the c-axis, as is the case for Cu-Cl boracite, in which a sharp jump of the torque to an opposite sign was measured when the applied field is perpendicular to the c-axis (in the ab plane), resulting from an actual switching of the spontaneous

MAGNETOELECTRIC EFFECT OF Cr3B7013Cl 109

K¹⁶). A linear component of the ME effect has been found to exist with the coefficient a₃₃ of the order of 5 x 10-¹⁴ [ s/m], strangely small compared with those of other boracites, e.g, <X33 = 1.2 X 10-¹³ [s/m] for Cu-CJI⁶ and <X23 = 3.4 X 10-¹² [s/m] for Ni-CI at 4.2 K.²⁰

The spread minimum of the spontaneous polarization Ps and the broadened maximum of the spontaneous birefringence (Lln₅⁾¹⁰ near 30 K would result from the precursory effect of a magnetic ordering at lower temperatures. The sharp peak of Ps (Figure 7) and the jump of Lln5 at 15 K, ¹¹ as well as the anomaly of the ME coefficient !3333 measured dynamically, ¹⁷ suggest, however, an antiferromagnetic- type ordering. The only antiferromagnetic point group permitted in the ortho- rhombic phase of Cr-CI is mm2, which allows the linear ME coefficients a₁₂ and a_{21 .} The maximum of !3333 measured by the quasistatic method near 11 K and the anomaly of !3₃₃₃ measured dynamically at about 10 K can be attributed to the onset of a long range magnetic ordering, which seems to be of the weakly ferromagnetic ::ype according to the magnetic torque measurements. It should be noticed that the measurement of the temperature dependence of the magnetoelectric coefficients

by quasistatic method was very delicate since the temperature dependence of the

spontaneous polarization Ps presents a complicated form with sharp variation and anomalies (Figure 7). Therefore it gave only discrete values and a spread maximum of !3333, the magnetic event at 15 K being not revealed thereby.

Two magnetic point groups are possible for the orthorhombic ferromagnetic phase, i.e., m'm'2 or m'm2'. The former one allows the non-zero diagonal com- ponents a₁₁, a₂₂ and a₃₃ for the coefficients of the linear ME effect, and the latter one gives the coefficients a₃₂ and a₂₃. The results obtained in this work showed that the coefficient a₃₃ at 4.2 K, though very small and submerged in the strong quadratic ME effect, is predominant over a₃₂ which tends to be zero. This would be in agreement with the magnetic torque measurements, which did not show a purely quadratic dependence of the magnetic torque versus magnetic field, sug- gesting thus a linear component of ferromagnetic origin along (001)cub directions.

On the basis of these considerations, the magnetic point group of Cr-CI at 4.2 K can only be tentatively suggested, at this stage of study, to be m'm'2, with a (very weak) spontaneous magnetization parallel to the spontaneous polarization M,/1 P,/12, as is also the case for Cu-CI boracite.¹⁶·19 The measurement of the other ME coefficients like a_{11 ,} a₂₂, a₁₂ and a₂₁ turns out to be difficult because of the mul- tidomain state of the orthorhombic phase without electric field poling along (001)cub· Therefore, with a view to specifying the magnetic phase transitions and the related magnetic symmetry of Cr-CI at low temperatures more reliably, further investi- gations, including the measurement of the anisotropy of magnetic susceptibility and of the spontaneous magnetization by very sensitive techniques like SQUID on a single domain state and the magnetic structure determination by neutron dif- fraction, appear obviously to be necessary.

ACKNOWLEDGEMENTS

The authors would like to thank E. Burkhardt, R. Cros and R. Boutellier for technical help and the Swiss National Science Foundation for support.

110 Z.-G. YE eta/.

REFERENCES

1. H. Schmid, Growth of Crystals, 1, 25 (1969).

2. R. J. Nelmes and F. R. Thornley, J. Phys. C: Solid State Phys., 1, 3840 (1974).

3. H. Schmid and H. Tippmann, Ferroelectrics, 20, 21 (1978). 4. H. Schmid, phys. stat. sol., 37, 209 (1970).

5. P. Toledano, H. Schmid, M. Clin and J.-P. Rivera, Phys. Rev., 832, 6006 (1985). 6. H. Schmid, Int. J. Magnetism, 4, 337 (1973).

7. Z.-G. Ye, J.-P. Rivera and H. Schmid, Ferroelectrics, 106, 87 (1990). 8. S.-Y. Mao, F. Kubel, H. Schmid and K. Yvon, Acw Cryst., 847, 692 (1991).

9. Z.-G. Ye, J.-P. Rivera and H. Schmid, Ferroe/ectrics, 116, 251 (1991).

10. Z.-G. Ye, J.-P. Rivera and H. Schmid, Phase Transitions, 33,43 (1991).

II. Z.-G. Ye, J.-P. Rivera, E. Burkhardt and H. Schmid, Phase Transitions, 36, 281 (1991).

12. G. Quezel et H. Schmid, Solid State Comm., 6, 447 (1968).

13. H. Schmid, J. Phys. Chern. Solids, 26, 973 (1965).

14. H. Schmid and H. Tippmann, J. Crystal Growth, 46, 723 (1979). 15. K. Aizu, Phys. Rev., 82, 754 (1970).

16. J.-P. Rivera and H. Schmid, Journal de Physique, Suppl. au n"2, Tome 49, C8-849 (1988).

17. J.-P. Rivera, (these proceedings).

18. (see e.g.) D. Jiles, "Introduction to Magnetism and Magnetic Materials," Chapman and Hall, London, 1991, pp. 60-61.

19. M. Haida, K. Kohn and J. Kobayashi, J. Phys. Soc. Japan, 39, 1625 (1975).

20. J.-P. Rivera and H. Schmid, J. Appl. Phys., 70, 6410 (1991).

COLOR PLATE I. See Z.-G. Ye eta!., Figure I.

FERROELECTRICS, Volume 161(1-4).

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