Structural transitions in boracites: one or two order-parameters?

Article

Reference

Structural transitions in boracites: one or two order-parameters?

TOLEDANO, Pierre, et al.

Abstract

In light of the structural and dielec. data available for 23 boracites, a phenomenol. approach of the sequences of ferroelec.-ferroelastic phases obsd. in this family of compds. is proposed. It makes use of a single 6-dimensional order-parameter, in agreement with the theory of V.

Dvorak and J. Petzelt (1971) and Yu. M. Gufan and V. P. Sakhnenko (1973) and in contrast to the 2-order-parameter model claimed by A. P. Levanyuk and D. G. Sannikov (1974, 5). A new interpretation is suggested for the temp. dependences of the polarization and dielec.

permittivity, which involves 2 coupling terms, linear in the polarization components, and resp.

quadratic and cubic in the order-parameter components.

TOLEDANO, Pierre, et al . Structural transitions in boracites: one or two order-parameters?

Japanese Journal of Applied Physics , 1985, vol. 24, p. 347-349

DOI : 10.7567/jjaps.24s2.347

Available at:

http://archive-ouverte.unige.ch/unige:32143

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

1 / 1

Proceedings of the Sixth International Meeting on Ferroelectricity, Kobe 1985 Japanese Journal of Applied Physics, Vol. 24 (1985) Supplement 24-2, pp. 347-349

Structural Transitions in Boracites: One or Two Order-Parameters?

P. TOLEDANO*l, H. SCHMID*, M. CLIN*t and J, P, RIVERA

* Departement de Chimie minerale, analytique et appliquee Universite de Gem!ve, CH-1211, Genive 4, Suisse tGroupe de Physique theorique, Facutte des Sciences 33,

rue Saint-Leu, 80039 A miens Cedex-France

At the light of the structural and dielectric data available for 23 boracites, a phenomenological approach of the se- quences of ferroelectric-ferroelastic phases observed in this family of compounds, is proposed. It makes use of a single six-dimensional order-parameter, in agreement with the theory of Dvorak and Petzelt, and Gufan and Sakhnenko and in contrast to the two-order-parameter model claimed by Levanyuk and Sannikov. A new interpretation is suggested for the temperature dependences of the polarization and dielectric permittivity, which involves two coupling terms, linear in the polarization components, and respectively quadratic and cubic in the order-parameter components.

§1. Introduction

Halide boracites with general formula M1B7013X (with M=Mg, Cr, Mn, Fe, Co, Ni, Cu, Zn, Cd, and X=Cl, Br, I) have been the subject of a large number of ex- perimental studies which have specified their structural ,.-...,_and physical characteristics'·" Thus, some distinctive features common to a large fraction of compositions were verified:

I) a cubic-to-orthorhombic first-order transition

F43c~ Pca2,, was found in 23 members of the boracite family.

2) in four materials (Fe-CI, Co-Cl, Zn-Cl and Fe-1), the more complete sequence of phases was identified:^1>

F43c--> Pca2, ~(m)--> R3c ( 1.1) The trigonal phase was also found in Fe-Br ," while a monoclinic modification was evidenced in Cr-CJ.5l Two distinct theoretical interpretations have been proposed for the preceding sequence of phases, namely i) a single order-parameter model'-'1 transforming as a six-dimen- sional irreducible representation (IR) at the X point (k=O, 0, n/a) of the cubic face-centered Brillouin-zone (BZ) surface(> ii) a model involving two order- parameters, ¹⁰•111 in which the zone-boundary instability triggers a zone-center instability. Here, the trigonal phase is assumed to be a proper ferroelectric phase, whereas the

~ onhorhombic a~d ?1onoclinic phases result from the Simultaneous act1vat10n of the two irreducible degrees of - freedom. This latter proposition relies on the symmetries reported for the trigonal and monoclinic structures, ¹²•13

>

and on the unusual form found for the dielectric cons- tant in certain boracites^14•151 at the cubic-to-orthorhombic modification.

In this paper we examine the symmetry (§2) and dielec- tric (§3) properties of boracites. We show that, when tak- ing into account the complete set of structural and dielec- tric data, and some theoretical arguments that were overlooked in the two theories, one can indeed explain the ferroelectric-ferroelastic transitions in boracites, in the framework of a single order-parameter model. A more comprehensive study of the structural transitions in boracites will be published elsewhere.">

§2. Symmetry Analysis or the Strnctnral Transitions in Boracites

The structural modifications occuring for 23 ferroelec- tric-ferroelastic boracites, are reported in Table I. Ni-l boracite does not figure in this table, as its low- temperature cubic-to-monoclinic magnetostructural modification requires, in our opinion, a specific inter- pretation.17·181 As shown in Table II, the orthorhombic, rhombohedral and monoclinic phases observed in

boracites can be a priori related to two distinct IR's of the F43c high-symmetry group (i.e. a three-dimensional IR r, and a six-dimensional IR labelled r 1) associated respectively with the center rand the surface point X, of the cubic face-centered BZ. The order-parameter expan- sions F, and F, corresponding to r4 and r 1 are listed in Table III. One can note that a cubic invariant figures in the two expansions (i.e. r, and r, violate the Landau con- dition191) in agreement with the first-order character, verified experimentally for the cubic-to-orthorhombic transitions. A careful examination of the experimental data at the light of the theoretical results summarized in Tables II and Ill, allows the following conclusions:

I) The F43c--+ Pca2, transition can be unambiguously connected with the six-dimensional IR r1 in agreement with the theoretical model proposed in refs. 6-8. Actual- ly, the structural data confirm the breaking of transla- tional symmetry induced by r, with a two-fold increase of the primitive cubic ccl1.201 Besides, the improper character of the transition is attested by the whole set of dielectric and elastic data. 21 For example the magnitude of the spon- taneous polarization (given in Table I) vary from 3 X 10-⁶ cm-² (Cr-Cl), to about 3 x 10-² cm-² (Fe- I, Cu- Br or Co-l), which are typical values for an improper fer- roelectric. Levanyuk and Sannikov, ¹⁰•11' have objected that the drop of the dielectric permittivity observed on cooling in most boracites (see Table I) should contradict a single-order -parameter model. In § 3 we show, on the contrary that such a model is coherent with the temperature dependence observed for the spontaneous polarization and dielectric permittivity.

347

2) The rhombohedral symmetry R3c has been deduced from X-ray measurement³¹ and was confirmed by a thorough structure determination. BJ Mossbaiier211 and EPR221 measurements are consistent with that structure.

The primitive rhombohedral cell was assumed to be isotranslational with the primitive cubic cell and thus sup- posed to be a proper ferroelectric phase connected with the zone-center IR r,. Along this line, Levanyuk and San- nikov10·111 have suggested a mechanism of deactivation of the six-dimensional IR r 1 when going from the or- thorhombic to the rhombohedral phase. However, there are some major objections to consider the polarization as the transition order-parameter for the R3c phase, name- ly: i) Passing from an improper-to-proper ferroelectric regime should give rise to a significant increase in magnitude of the spontaneous polarization P,. This is not observed in Co-Cl and Fe-Br boracites where the same order of magnitude is found for P, for both the or- thorhombic and rhombohedral phases231 (see the values in Table !). ii) The dielectric properties of a proper fer- roelectric are very sensitive to the influence of an applied

348 P. TOLEDANO, H. SCHMID, .M. CLIN and J.P. RIVERA

Table I. Structural, dielectric and calorimetric data in the boracite family. In column 4, Down and Up mean respectively that a downard or upward jump is observed on cooling for the dielectric permittivity.

Compound

~-lg-CI

C:r-CI

!l:[n-Cl Fe-CI

Co-Cl l'<i-CI Cu-CI Zn-Cl Cd-C!

Mg-Br Mn-I3r re-Br

Co-Br Ni-Br Cu-Br Zn-Br Cd-Br Cr-1 Mn-1 Fe- I

Co-!

Zn-1 Cd-1

Sequence'\ of transit 10m helow the cubic phase (T, in Kelvins) Pca2,

Pca2.

Pca2, Po;;a2, Pca21

P.:a2, Pca2, Pca2, Pca2,

Peal, Pca2₁ Pca2, Pca2, Pca21

Pca2₁ Pca2₁ 1-'ca2,

Pca2 Pca2 Pca2

{m)

(m)

(m)

(m)

R3c

(m) ~ _i91

("I)

R3c

RJ..:

R3c

R3c

Table II. Landau symmetry analysis of some of the phase transitions induced by theIR's -r₄(I) and r₁(X). Columns (b) and (c): low sym- metry groups and corresponding primitive translations with respect to the translations t_1, t_{2 ,} 1₁ of the F43c unit-cell. (d) Equilibrium values for the order-parameter components corresponding to each phase.

(a) (b)

T(r,)

F4J~-'

i_X(r,l

~)ba2

_! R3c

kc

.RJc )Cc

(c)

fl,,l,,l, (V) /. 1,.1, r, r:,-r,(2VJ

(d)

1}, ,<0, /}• q, 0

1}, IJ~ ll;*o

1}, '1~*0,17> 0 p,-.<(l,lf.l, O,p, -p,-0

'

, Pi-P!-pi *0.'1/,-0, .::.,

·-r1: : : : 1,-= z,,

lr,-1~1 -t. ^(4\') 1 2 : 2 ~ ~

,P,*Pc p,'/=0,(1.1,- 4,

4

electric field, whereas no substantial alteration of these properties (except the ferroelectric domain switching) has been verified in the rhombohedral phase. Consequently this latter phase should be more liably related to the r,(X) IR, with a fourfold increase of the cubic primitive cell.

So far, no superstructure reflections corresponding to the breaking of translational symmetry has been found in trigonal boracites (i.e. in Fe-Cl,³•23

l Fe-B,'' Fe-1 121 Zn- Cl3·231). However, as they have not been looked intentional- ly by using long exposure time expcrimcut they may have escaped detection.

3) No detailed structural investigation of the monoclinic phase has yet been made. The two models proposing a Pc(2V)¹²' and Pb(4V)²⁴' symmetry for this phase,are respectively deduced from X-ray121 and EPR²⁵¹ analysis.Here also,the value obtained for the spon- taneous polarization in the monoclinic phase of Co-Cl²³¹ (see Table I) speaks in favour of an improper phase. No experimental data presently confirm the elaborated trig- gering mechanism proposed in ref. 11 for the onset of the monoclinic phase. Thus,although confirmations are still needed,it seems reasonable to relate the entire sequence of phases (1.1), to a single IR (r1(X)) of the F43c group,

7.6X 10 J

";c: UH 10--~

(

(mm2)

~2.4>; 10 ² (3m)

;:;1.8XIO '(m) 1.8XI0

2.6 < 10 ' 2 ._.., 10 _, J (mm2) l3.5XIO '(3m) (1-3)/ 10-;

(0.4-3Jx w-'

'J.2 ... 10 (2.3-

],(,I) X lU

~1.5'\10-l

Down Down

Down Down

Down Down Lp Dvwn

Up-Down Up Down Down Down

Up

Down Down

.-1(2(.1-mole ¹⁾

341Hl

1

.2145 {43m ... mm2) . 120 (mm2 ... m)

5527

4900 1220

J 196

2845

11\69

3100 1500

Table III. Order-parameter expansions F₁ and F_{2 ,} respectively associated to theIR's •iF) and r₁(X). FE expresses the couplings bet- ween the polarization compOnents and the six-dimensional order- parameter transforming as r1(X).

- - - -

- (1 " ' l p, ' ' ^-1 _!!~

F1^{~F,,--:; LJ} I},+Rl/ 1'/c.'/)-- L..'l· L; rJ;rl}

"-···I.; 4 t=l,; 2 ,, 11

f~=f;,-~ 2:: p: Bp,p,p,(co~ ^(1.1, co~ w1(co> \(/_,+~in iJ!J)

"', ,_,

+~in ^!JI1 ~in ^(\'11 + \'1;)} +~ :Z p;-6. 2.:: p: ^{co-, 4 IJI,}

4 ·-1.1 4, 1.1

I p, 4(P,!h+ p,p,+p p,)-llp,p. 'im 2 ' ' , p, ' ' 1Jl1 sm"' -. w

+pit::isin21J11sin2 !f!,+pcpisin2 w,sin2w,l p, ' ' . "

+2[p,p;(sm2 'P -slll2 'I/J~p;p:;bm21fJ:

-sin 2 IJIJ) l P7P~(sin 2 ifJ, sin 2 I{J,l]

Fr=r51(P,.p: co.; 2 IJ/,-P Pi cos 2 ~pJ · ON\Pi cos 2 'Ill

+ P_.p~ cos 2 IJ/1^)-03(P,p~ l:O~ 2 w,+ P_p~ cos 2 1,111)

+.!i,(P,p~ l:O~ 2 '/11 I P,.p; cos 2 ~p, I P,p~ cos 2 IJI,) +v(P,-P .. -P,)p1p1p1[sin 'f/2 sin ('f/1 1,11,) cos lf./1 l'O'> w·

. 1 . ' > Ill . , • ' ' '

~(cos \'J,-sm If,)]+-(P~+P;+P:)+- (P~p;+P;pj+P:pn

- 2Xo ' 2 • •

---r-~[P~(p¹ + p~l+ P;(p~+ Pil+P~(p; + p~)}-E.P

l'!c"~, = P"~' cos 1Jf1,_ ,(P-O, 1,2); IJn·-p~ <;in 1f1 1.(P= 1 ,2,3)

-

as suggested by the symmetry analysis of Table II. We will now show that the exceptional variety of dielectric data, characterizing the boracite family strongly support this choice.

§3. Dielectric Properties of Boracites

The contribution FE to the Landau free-energy, ex- pressing the coupling between the six-dimensional order- parameter (p,, If!) (i= 1,3) and the polarization com- ponents P, (u=x, y, z) is given in Table Ill. It contains two invariants linear in Pu. corresponding respectively to faintness indexes²⁶¹ 2 and 3. The latter one, which was not considered in refs 6-8, will be shown to play a crucial

Structural Transitions in Boracites: One or Two Order-Parameters? 349

- = - - 1

""' "" "" "" "' .o<-,

Fig. I. Temperature dependence of the spontaneous polarization in boracites. (a) and (b): Theoretical variation of P,(T) corresponding respectively to (2J/v) > (B/ {J) and (B/ /f)> (20/ {J) (c) Experimental curves for Mg-Cl²⁸J and Ni-Br.m (d) Experimental curve for Cu- Br29l

role for the description of the dielectric features of boracites at the cubic-to-orthorhombic transition. As the orthorhombic phase corresponds to the equilibrium values:

P1*0, pj=pj=O, 'lf,=O, P,*O, P,=P,=O (3.1) thus, for a qualitative model, we can consider the simplified single order-parameter model:

01 , B

p

F=F₀+- p-+- p³+- p⁴+Jp²P+vp'P+- (3.2)

2 3 4 2xo

.-,with a=a(T- T,), B<O, P>O, J<O and v>O. The usual minimization procedure yields the following results:

1) The transition takes place discontinuously at T1 = T, +2B² /9pa. The order-parameter jumps from p'=O for T>T1 to p'(T1)=-(B/3p) at T=T,, with an equilibrium value: p'=(B+(B²-4ap)¹¹²/2P) for T< T,.

2) The spontaneous polarization has below T, the equilibrium value:

P;= -xoP'(J+vp) with a jump at T,:

2xoB' ( vB) Pi(T,)=-

9P' -o+

Two cases can be considered:

(3.2)

(3.3)

a) if (2J/v) > (B/ p) > 3J/v at T= h

P;

b) if (B/P}>(2J/v), P: increases below T,, then start decreasing for p(To)= -(2J/3v) (Fig 1(b)).

The two preceding situations are actually encountered in boracites. Thus, Mg-Cl, Zn-Cl and Ni-Br correspond

~.to the case a) while Cu-Br and Co-l illustrate the case b) on Figs. 1(c) and 1(d) we have reproduced the experimen- tal curves found for Ni-Br,²⁷¹ Mg-Cl²⁸¹ and Cu-Br.²⁹¹ Cr- CI, Fe-Br and Fe-! also correspond to the case b) but the inequality (B/3P) remains valid for all T < Tt.

3) The dielectric susceptibility x=lim ((JP/(JE) is given

by: ^E~•O

and

x=xo for T> T1

x=xu-xop(2J-3vp) ap for ap T,;;, T1

(3.4) (3.5) Thus, at T1 one should observe a discontinuity correspon- ding to:

xoB ( ap)

(Lix)T= T, =

3p, (2JP- vB) aP T= Tt (3.6) which corresponds to a drop or a jump of x(T) at T= Tt on cooling depending if the product (2 JP-vB)(apj (JP)r__{7 ,} is positive or negative.

The two corresponding situations are illustrated on Figs. 2(a) and 2(b). As shown on Table I, a drop for (Tt) on cooling can be verified for all boracites except for Cu-Cl, Fe-Br and Mn-1. For Mg-CI, Zn-CI, Ni-Br and Fe-Br, according to the temperature dependence of

r

c! ^{; - - -}

- T T,

I ,

L

1,

:]'

_.. '·~ _{. · L _}

'\_'·'~_f'---

L--=c--=·-~T('(J

100 200 300

Fig. 2. Temperature dependence of the dielectric susceptibility in boracites (a) and (b): Theoretical variation of x(T) corresponding respectively to (L1Xho=T₁ <0 and (L1Xh-T₁ >0. (c) Experimental curves for Mg-CIJOJ and Cu-C1.¹⁴¹

P,(T) one has ((!p((!P)r-r,>O while (ap;aP)r~r,<O for Cu-Br, Co-l and Fe-!. In Fig. 2(c) the temperature dependences of Mg-Cl³⁰l and Cu-Cl¹⁴¹ are reported. A more detailed discussion of the dielectric properties of boracites is given in ref. 16.

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