Observation of Ferroelectric Domain in Boracites

Article

Reference

Observation of Ferroelectric Domain in Boracites

ZIMMERMANN, Anne-Marie, BOLLMANN, Walter, SCHMID, Hans

Abstract

Ni-Br, Ni-Cl, Mn-Cl, and Zn-Cl boracites were studied by electron and polarization microscopy.

Among others, 180° head-head or tail-tail domains were found in the orthorhombic phase of Ni-Br, Ni-Cl, Mn-Cl, and Zn-Cl boracites. They may be due to growth stacking faults on {100}

planes. The orientations of strain-free domain walls were calcd. by the 0-lattice method.

ZIMMERMANN, Anne-Marie, BOLLMANN, Walter, SCHMID, Hans. Observation of Ferroelectric Domain in Boracites. Physica Status Solidi. A, Applied Research , 1970, vol. 3, no. 3, p.

707-720

Available at:

http://archive-ouverte.unige.ch/unige:32933

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

1 / 1

A. Zn\lntEJUTll'N et. aJ.: Ferroelcct,ric Domains iu Borncit.cs phys. stat. sol. (a) 3, 707 (1 970)

Subject classification: 14.4.2; 4

Bultelle institute, Ad~:a.ncecl Btuclies Center, Geneva

Observation of Ferroelectri c Domain in Boracite ·

By

A. ZI.MME.ln1ANN1 ), W. Boi.L~[i\1\.K, and

rr .

SCli~UD

707

.Ni-Br. Ni-CI. Mn-CI. and Zn-CI horacitcs l1a.vc bccu studic<l wit.ll electron and polari- zation microscopy. Among others 180° hMd-hcad or· tail-t·.ail domains are found in the orthorhombic phttse of Ni- Br . .1\Ji-01. }ln-Cl, ann Zn-CJ boracik>. '!'hey may be due to growth sta<'king fl\ults on {100} planes. The orientatio.us of titmin-free domain walls are calculated by mc~tns of the 0-Llttice method.

Ni-Br-, Ki-CI-, i\ln-CI-und Zn-CI-llorazitkristalle wurden im Elcktroncnmikroskop und im PoiMisa.tionsmikroskop unt.ersucht. In dcr orthorhombischen Phase von Ji-Br-, Ni-Cl-. Mn-Cl- tmd Zn-CI-Borazit wurden neben gewohnlichen. l80°- und 90°-0omanc.n auch 180° -Kopf-Kopf- oder Schwanz-, chwa,nz-Domanen gcfunden. 'ie worden Stapel- fohlern auf {LOO}-Ebencn zugeschricben. Die Orienticrung ven.crmngsfreicr Domiinen- wi:i.nde 'vu.rdc mit Hilfe der 0-Gittertheorie bcrechnet.

L Iotroducfion

From t.be mineral boracite, Mg~B7⁰13^Cl, are derived the homologucs Me₃B₇0₁₃X where Me stands for a bivalent metal ion and X for 01, Br, or I. Most of them have ~~t least one ferroelectric phase [1 to 3]. 'rhis paper reports on obser- vations of ferroelectric domains in Ni-Br, Ni-0.1, l\1n-Ol, and Zn-Ol boracites by electron and polarized light micr·oscopy.

All known boracites except. Or-Br and Ur-I borac:ito undergo polymorphic transitions [4, 5). By order of decreasing temperature the phase sequence is

T~-O~v-monoclinic (x)-0~,. fm· Zn-01 boracitc [2] and T~-C~v for Ni-01, Ni-Br, and Mn-Cl boracitc [5, lJ. Table 1 gives the pseudocubic lattice constants, the transition temperatUl'es, and the. pace groups of these boracite ·.

Table l

Lattice constan

Space

I

^{T '1.'}

I

^{SpttCC '1' Spaoo}

(t\l 300 °K) pace

(A) gro11p (oK) group (oK) group (o I() group

Ni-Br 12.0:35 Ts 398 C~v

~J

II

Ni- 'I 12.019 T~ 610 C8v

Mn--CI 12.2+8

' l'S

^{6, 0} O~v

Zn-CI J2.0u5 T<~ 723 ^{C~,. X 4(j} C'3v

In the orth01·bombic phase ibe polarization vcctot· .lies along a. cttbic (100) direction. in t.be trigonal phase along a cubic (111) direction wit.h t.he restriction that 110 polarizat.ion reversal is possible )n the o~;,. phase because of the sym- metry rclat.ionship T~jq,, (for details see [2]). 'l'bc Rpace group x is not yet

1) Prcl:icnl oddrcss: Lu.boralorium fiir Fcsvkiirpcrphysik clcr E.T. H .. CH-8049-Ziirioh.

708

determined. Obscrvatjons with the polarization microstopc indi(·ate that the struct11rc may be monocliJ1ic [5].

Tbc tmnsitions to t.he ferroelectric phases involve a spontaneous strain of tbe cu hie cell: 1 n the trigonal phn.sc of Zn-01 bomcite t.lw V· eudocubic pace diagonal along the polarization a)\is is longer tJJan the three other ones. in the orthorhombic phase the face diagonals of the cube plane pcrpPndicuJar to th • polarization veclor arc no longer of equal length. TJ1ese spontaneous strain·

are very small and increase for one kind of metal ion with decreasing diameter of the anion from I over Br to 01. Since by electron microscop,v the ferro- electric domains are mostly revealed by means of diffraction contrast, t.be compounds with 1"hc largest stra.in, i.P. the largest ditlraotion effects, namely 1\i-Cl, Mn-01, and Zn-01 boracitcs, were chosen for t;he im•c.·tigation.

2. aH•J•Ic Pt'l'j):lTatiou

inglc cryslal. of boracitcs were obtained by chemical transport n·actions [6].

For the electron microscopy the crystals were ct•ushed in a mortar and suspended in alcohol. A drop of the solution was put. on a specimen grid covered by a thin earhon film. Then the specimen was Jnounted in a goniometer and heating stage in a 1 'iemens EJmiskop I. Tbe temperature of the sample was determined by the heating power, gauged before with an iron-const.antan thermocouple.

For observations under the polarizing microscope. plates were grt)und pamllcl to natw·aJ facet. and polished with diamond paste of 0.3 p.m. 'T'ransl'arent, gold electrodes of about 300

A

t.hickncss were vapour-deposited.

3. Experimental Rc ults

3.1 Sh·uctm·e fllltl <li}'lmction po.tfm·us~)

Diffraution vattcrns were taken in the different phases of Ni-01, l\ifn-01, ~ and Zn-01 boracite. The stnteture of '1'~ sho\\'s ext:ioct.ion nt.les due to the face- centered cubic st.ructure and extinction rules dne to mirror glide ]_Jh'Wles and screw axe . The latter ext.inction rules arc not always ob~crved on the diffrac- tion pattems because of dynamical interactions of the diJJel'ent c:Jif[ra.ctc<.l beam:>. Spots wllich should not appear in the reciprocal lattice ·how less intcm;ity than allowed spots. The f.c.c. Clo'll of T~ is transformed lJy a pha ·e t.ransition to the ort.horbombic cell of G~,. as given in Fig. J. 'l'hcrcforc, four translation· arc lost and the extinction ruJes of tmrrt.ixcd jndicel'! in the para- clcctrie phase a.re replaced in ihe ·~ ,. pha.sc by e)..--tinction n1les due to mi1Tor gUde planes and screw axes. On a diffraction pattern of a s.inglc domain it is po. siblc to define the orientation of the c-ax:is of the ortb01·homblc cell. Oo the other l1and.

U3,.

is an equi-b·anslation subgroup of T.'i and Rhows therefore the same extinction rule· as the f.c.c. structure.

The fenoclectric domains ?5,·e rise to two striking featw·cs in cleetron <.Lif(rn.c- tion, namely doubled Kikuchi linel; and spikes. Doubled Kikuchi lines appear on dHfraction pattern& of the <.:~" phase in Ni-01, Mn-Ci, ancl Zn-01 horacitc aml on d itlracLi0JI patterns of the trigona.! phase of Zn-CI bora cite (cf. ll'ig. 2). The jnh•nsity distribution of two paeallel pairs of Kikuchi lines suggests t.bat

~) On a<·counl of the small defol'mation of lh<' unit cell in the fel'l'Oclcrlric phases. aU indiccsunl<'ss otherwise stated a1·c cubic or pscudocnbic.

Oosern1 t ions o! lt'erroelcctric Domains in ~ora cites

J•'i~. 1. 'fl•r<,rl.llfJrh•>mhk unit cell, Lho indicatrix. '""t

tltt,; jWinrizaliun n.r~ ~h·1·H iu lilc fram(!work of the Cll l>iC uuit ('Cii

Fl(!. 2. 'mct·l.ron tllffrucllon vottern of Ni-CI. honwil;r showlug o111ullled Klku('hl liue• (l.nuo zone<; of (100)

••·ctlon~) X

zz•

A f - - - + - - - + - - - ' - (8

8 / A

/ / /

709

Llw_v arise from !he same (hkl) plane in two dificrent domains. A minimum tilting angle of the corrcsponcljng planes of about. 1.5' fot· Xi-Cl, of 5.5' for Mn-C'L of 5' in Cbe orthorhombic phast'. and of 4 . .:-· in the trigonal phase of Zn-CI boracite wa.s ealt-ulatccl. The value found for Zn-Cl bomcite agree wit.h that found by X-ray: in the lri~ona.l pha:::c [2j. Xo X-ray a.ualys:is has been done on Mn-CI horacite. \\Tith the clcdt~tion

or

^{1 .5'} from it right angle \\'C find a dif- ference between the orthol'}lo.rnbic a- and b-axcs in Nj-Q'J borac.ite of 2 to 3°/_{00 .} Tbe. mallnes::; ofthi. strain explains why t.ho contrast between diffe1·ent domains

710 .A. Zr~ullm.'rM .. ~, \\'.l3or..r.JILL'-~. <tnd H. St:Jnrm

:Fig. 3. Spikes of l 0° hcncl honrl domains nr~ •ccn on " diffr:tdion p;t~tern of 1'\i-CI IJorncltc

is so poor. As was demonstrated for Fe-I boracite by Kobayashi [13, 14J, strains I""

of even smaller magnitude can now be readily resolved by special X-ray methods.

According to Gevers et al. [7J ferroelectric domain walls may gi\'C ciJ c to spikes in the reciprocal lattice. 'l'he. e spikes are seen on Fig. 3. The distance of the satellite spots from the matrix spot increases with increa. ing distance from the Laue zone and the sateUite spots lie in opposite directions on different sides of the Laue zone. In tho kinematical approximntion the spikes arc ortho- gonal to the domain wall and thus allow one to determine the orientation of the domain wall from the diffraction pattern nnd the corresponding electTon micrograph. In connection with pola1·ization microscopic studies it is possible to determine the configuration of the polarization in tbc different domains.

Electron trt.icroscopy of boracitcs js complicated as electron::; are sti'Ongly absorbed and the cxti11ction distance is rather long. Therefore, the only crystnl flakes wl1ich can lJc observed in lrnnsmiss.ion have a t.hickness less than about lOOO

A,

i.e. about one extinction di. tam•c. The electron micrographs ::;how walls due to fcnoelcctric l.witming and wal1s d11e to stacking faults [8]. The latter ones will be described in the next. section. The fin;t kind of domain walls is observed in Ki-CJ and Mn-Cl boracito. In Mn-Ul boracite they can be moved

Obscr\·ations of Ferroelectric Domains in l3oracitcs

Fig. 4. Corresponding dark flchl ulicrogrui>h or 180"

head-head domains In Yl-CI bornrite.

The light and dark holes sre due to the carbon gri<l used for supporting the specimen

lfit:. 5. Fcrroclcotric domoin walls in i\ln-01 bor!tt'iLe moved l>y the electron hcarn

7ll

712

faidy easily by beating the crystal with tho electron beam (.Fig. 5). No move- ment of domain walls was seen in Xi-Cl bora.cite probably because of the low contrast between di:Cferen t domains.

Ln Lhe orthorhombic phase of Ni--Cl, :Mn-C1, and Zn-Cl boracite and in the trigonal phase of Zn-01 boraoitc the electron micrographs show mostly domain walls of the {100} t}'IJe. They oonespond to walls between domains with anti- varll.llel ))Olarization in the orthorhombic plH:t. e and to a head-tail configumtion , in t.hc trigonal structure [1, 2].

The {110} domain walls arc rare. They cones pond to 90° head-head domains in the orthorbom blc phase and to 109° head-head domain. in the trigonal phase [I, 2].

The 90° head-tail domain ·wall of the orthorhombic phase observed ·with the polarization microscope [1] was never found by electron microscopy.

lng. 6 shows an 1\'In-01 boracite crystal with a ww of clislocations (D) and ferroelectric (001) domain wall due to tacking faults ( ). Dislocations were observed only in Mn-01 boracitc. The strain field shown on the micrograph indicates a large Burgel's vectOI'. The a-bsolute \71;L)ue of a BoTgers vector of a perfect dislocation of the !ypc +CUO} is .6G A.

Fig. U. lllio·mgr.lllh of )[u-CI IJorn~ll~ wllh n rvw of olislvc·•Ltluns (D) and lS<l" hcud-he:ul olou1nlrli! (~)

Obsen·ations of Fcrroclcclric Domains in Boracitcs 713

:J.3 1 0° 111wd-heacl 01• tail-tail (/onwills

3.3. 7 Ob.se1·vations 1.cith llu~ electron ?nicro.scope

Domains with anliparallel polarization do not give ri ·c to disbnguisbable rcciproca.l lattices. o diffraction pattcms of such domains sironJate a single domain. The cLiifraction pattern (Fig. B) shows three Lane zones of the (172) r sec bon

o r

^{the rcc:i proc11l} lattice of Ki-01 bot·acitc. , atellitc spot from spikes along [01 OJ, 1.001 ], and [11 0

I

are seen, the spikes due to orLhorhombic (001) walls correspond to the domain shown on the micrograph (Fig. 4). The cliffraction pattern may be indexed to get the (712) section of the reciprocal lattice. But the [OOlj di1·oction of the (1.72) section turns out to become the [OOJ] di1'ection in the (712) section. The·e Lwo sections are tho only compatible ones in the ort.horbombic structw·e. Thu. the domain wall.· ·con on the dark-field pictm·e mu t be l 0° head-head or tail-tail domain walls. , i.e., the polarizations meet head to head or t<til to 1·ail at t.he domain walP) The domo.ins have a width of -WO to 3000

A.

r

The . ame domain configuration wa. found in the orthorhombic phase of Mn- 'l and Zn-01 bomcite. rl'hc domain walls of Zn-01 l>oracite stayed fixed on decreasing the tern perature to the monoclinic and trigonal phase. This kind of domain always giYcs a lamellar structure, the width of the domains va1·ying between about 100 and 3000

k

The 1 0° head-head or tail-tail domain waUs do not sho'v any strain as can be seen from the micrographs (Fig. 4 and 6).

3.3.2 Obserralion o{l\ 0° luad-head domains u;ith the polcm:ziny mic1·o.scope The l 0° head-head or tail-tail domains wel'e observed in the electron micro- sc:ope for .Ki-01, ~1n-Cl, and Zn-CI boraeite and in the polarizing microscope for r\i-CJ and 'i~Br boracite. For the ophical obset·vation (001) plates were chosen wit.h

P sl l

[001]. Observation of orthorhombic 180° head-head or tail-tail domains i possible uccause t.he indicatrix turns

u y

90° around the polar axis if the polarization jumps from [0011 to [001] (cf. Fig. 7). In a e1·ystal plate of thickness d. superimposed anti parallel domains of thickness cl-Ll and Ll, respec- tively giYC a total vath diffCL'CilCC of

T = (d-2Ll) (ny - n.,),

~---~A

I I I I I I

A) - - - -

8:~---, A I

A~--~~---~8 _I

I

I I I I I

AJ.:.--- - - -

/ n.,

-.__>< ^!Ia

/

-

^~

a~---~Y }'ig. 7. t.:orrl'latinn or 11\r pol:lrbutiou vrttor anti the jndicatrix

3) The head of t.ho polurj~r.ation corresponds to the positive end of tl1o polarization vector.

714 _-\. Znt~LERMAXX. W. BoLr.~JAN!\, and B. Scmrm

l>ig. s. ¹ o• hcn<'l-ltcud dOnlllins in Jli-Jlr horal'it.c IJJJd~r lhc polttrizltf.iQU mi,(trQS('(IP•' (hright stripes). The blrc fringe nee of the :siuglc. domain is ~Om]H~mmted (dark regions). f11 tbc right bnnd lower part of the. p11otogra.ph

tlw cor>lro~t i~ rc\'crscd tluo 1:<• .>lightlr different tlilckocss of the plutc

while a single domain, penetrating the wlwle thickness of the slab, gives

T

0

=

d (n, - n") •

These path cliffereJ1ces allow to produce contrast between unswitched regions and such ones with superposed 180° head-head domains (Fig. 8). From the path differences T₀ and

r ,

^{the thickness of the slab (d = 100 fLm), and the}

intrinsic birefringence n, - n .. , the thickness L1 of the switched stripe domains was found to he 5 fJ.lU foT a particular crystal. The stripe domains appea.r neal' the natural face of the plate for au applied electric field of 20 kV/cm. If the field is increased, more domains appear and the old ones become larger, but the thickness L1 docs not increase. The 180° head-head or tail-tail domains in Ni-Br boracite exist only as long as the field is applied. The domain boundaries oriented perpendicularly to the (001) head-head interface are of the ordinary 180° domain type with appro:>..'imately (llO)o.r. as composition p.lane [1.]. Their traces on the (001) plane enclose an angle of about 85° with each other - the deviation oftbese tra.ces from the cubic <100) directions is too big to be due to the spontaneous st.rain of the pseudocubic cell, but may be ascribed to elast.ic interactions of the domains among themselves.

In Ni-Cl boracite (Fig. 9) the 1 0° head-head oc tail-tail domains began to appear for an applied field of 6.9 kV/cm. It ·was possible to polarize the whole layer· uniformly with a field of 19 kV/cm. The domains persisted even without an applied field.

After cooling from above the Curie point both Ni-Br and Ki-Cl boracite plates contained j Llxtaposed domains with antipa.rallel polarization pene·trating the entire thiclmess and separated by ( 11 0)0 . r. walls. By applyjng a field it was possible to create the 180° head-head or tail-tail domains only on the naturally g.rown side of the plate and only in those single domains which couJd provide the tail-tail eonfigmation. In some samples the field strength necessary to

Ob '<'I'\' at ion~ of Ferroelectric Domains in Bomcitcs 715

.f'lg. o. 1 o• head-head domaius of Nl Cl l>orncite In polnrl.zed light (001) i>lalc

induce the tail-tail domn.ins was also capable of displacing the spontaneously formed (11 0)0 • r. walls Ia tcrally, in other samples the coercive field for the lateral movement was in excess over that fot· the tail-tail domain formation.

Observations of 180° head-bead or tail-tail domains with the electron micro- scope have been reported before 011 the tetragonal phase of BaTi0_3. But these domain walls had relaxed either to a wedge-shaped bounda1·y or to a ruierotwi.n [9, J 0). A direct optical observation of these domains in Ba1'i0₃ is not possible because the indicatrix section of anti parallel domains is identical

3.3.3 Discu,~sion of180° head-head domm:n 1./)alls

'l'be obson1ations in the polarizing microscope and the electron microscope prove the existence of J 80° head-head or tail-ta'il domains. The observations under the polarizing microscope are done on bulk material, tho~e in the electron mjcroscopc on thin crystal flakes. At the moment it is not possible to give one single explanation for t.hcir existence.

The J 80° head-head or tail-tail domains observed on Ni-Dr and Ni-CI botacite may be interpl'eted phenomenologically by a symmetric hysteresis for the bulk material and a biased hysteresis for the surface layer. The coercive field and the bia vary from one crystal to another.

The 1 0° head-head domains arc observed in the electron microscope on crushed crystaJs not subjected to an elcct.1·ic field. Therefore, lhere must be some reason why the energy is lo,-.;•erecl enough to keep this configmation stable.

There is strong evidence that the 180° he~d-head domains observed in the electron microscope aJ'e due to growth stacking fauJts. 'l'his hypothesis of the origin of the 1 0° bead-head domains is supported by the following facts:

a) (100) twinning and com posit.ion planes of the cubic boron-oxygen skeleton are known to occ:m· jn natural bomcite as well a· in Mn-Cl boracite [1

J.

716 1\. ZmilmRiiiAN".N, \V. BOLLMA~!\. aud H. Scmrm

b) It was not possible to move this kind of domains by heating the crystal with the elecLron beam. A stacking fa.ul t involves a change of t.be boron-oxygen skeleton not TC'ponsible for the fcrroe!ectricit.y. Therefore, the composition planes cannot be moved by means of the electric field.

c) The formation of the 180° head-head domains induced by ~>tacking faults is also supported by the 0-lattice theory as shown below.

1'hoso stacking faults madring the basis of the surface layer al'C probably formed due to a change i11 supcrsatmation shortly after the onset of Lhe cooling phase of the ci·ystaiJization.

4. Application of tho 0-La.tticc 'l¹11Cory to Domain Walls

.J..1 lnh·mluclion

The 0-lattice theory which is a general geometrical theory of crystalline inter'- faces is given in [11

J .

The basic idea is ilie following: The two crystal meeting at the bounda1y (r1ys-1aJ J a11d <'lystal 2) are idealized as t.wo interpenetrating point lattices which arc related by a linear homogeneous non-degenerate trans- formation A:

a;(2) = A a:;(l)'

I A l

=l= 0' (I)

where A, in tJ1e stnTounclings of the origin, relat.cs the c'losc-t neighbour points in both lattices. The 0-poinls as coincidences of points which are in equivalent positions in both lat.tices (whether occupied by atoms or not) at·e the solutions of

(1 - A -1) :x;(O)

=

bO·l ' (2)

where 1 is the unit tra.nsfonnation (identity) and tho vectors bfL> are Lranslation vectors of lattice 1 and as such lattice vectors of the b-latt.ice. The 0-points al'e considered a.s points of best fit between the two lattices. Depending on the eank of the matrix (J - A -¹⁾ =

'r

tho 0-points can form a point lattice (eank('.L') = 3), a lattice consi ting or parallel lines (rank('l') = 2), or of parallel planes (L'ank('l')=

=

I). These 0-points, -lines or -planes are called 0-elcments. They have to be separated uy cell walls. Then, a boundary may be chosen as far as possible through 0-clcments. On the one side of the cho,-cn boundary the atoms are placed into the positions of latt.ice 1 and on the other into those of lattice 2.

The ]j11c of intersection of a bounda:ry with a cell wall is a dislocation. 'l'he most favorable boundarie arc tho o which contain no dislocations, i.e. Lhose which in the case of rank('J')

=

l are placed thmugh an 0-plane. The 0-lattice theory shall now be applied to ferroelectric domain ·walls. (In a difJcrent way the orien-

tations of ferroElectric domain wa]Js were calculated by Fousek and Janovec [12).)

The face-centered cubic tinit cell of the boracites can be interpreted as hcing composed of right cubes, foUl' of type A and four of type B (Fig. 10 (S<l1>)). The A-and the B-type cubes differ in their oxygen lattice. In o1·der to set up a rule for determining the 01'ie11 ta tion of the fcnoelcctric polal'ization (l] we may imag- ine the four centres of the A-cubes to be connected. In the ort.horhomhic phase the polariuttion is coupled with a spontaneous shear of the u1:Ut cell aJong one of its edgf'S by about 0.2%. By this Rhcar one of the connections be- tween centres of A-cubes becomes Jonger than the opposite one and the two others change hy equal amounts [J4). The 1}0la,rization vector points in the (100) dirPCtion j1·om the longer to the . horter connection. For every direction of polarization two equivalent dil'ections of shear ex:i t (Fig. J 0).

Observations of .i"crroelcctric DomuinR in Bora-cites 717 z

s

⁽¹²⁾

c-{21)

, )

f'lg. 10. Representation of the fonr shear trao,rorulnticm~. ln Cul the;\· anrl lite 13-c·lllJCS nrc marked. The In·

variant platlCS nrc indlcatrd uy hutcllillg

-1.2 90° d(}'IH(Iill '1-t:UI/S

We first discuss the 90" domain walls, i.e. those configurations in which the orientations of the polarization vectors include an angle of 90°. We start with the orthogonal unit cell and take tbe unit equal to the Jattice constant. This ort.bogonaJ unit cellls deformed (sheared) by the transformation S<ti) (i

=

J, 2) in order to produce lattice 1 and by 8(²1:) (lc = l, 2) for producing la-ttice 2:

:x;(ll

=

^{s<t i)} x<orth) ^-> (S<H>J-1 x<'>

=

x<orihl ,

a;(2)

=

8(21') :x;(orth) •

In the approximat.ion sin a

=

tg a

=

a, cos a

=

l the transformat,ions (li'·ig. 10) are

s<n>~ G

^{a 0 1}

D · ^S "" ~ G

^{0 1 0}

D·

W"'

~ G

⁰

D· ^{S '"' ~ ( :}

0

D

l l

0 ,0 a

718 A. Znli\IERJJ.AS~, W. BoLuuN::;, and H. SCllMID

After elimina,ting x<orth) we obtain

x<2>

=

^{S(2kl(S(l •>)-1 x(ll}

=

A :x;(l> • Then

(l - _A-1) x(O) = b(L)

becomes

([ _ S(l i)( (2.1:))-1) x<O) = b(l.) •

In the four possible combinations (i, k) = (1, 1), (1, 2), (2, 1), and (2, 2) the matrices (I - A -1 ) = 'l'(i,k) are

T'"' ~ G

^{- (j}

" ' ) c

^{- q}

D ·

0 _{(j '} ^'J.'(l2)

=

0 0

0 0 0 ^(j

~, ~ G ^{0 0}

^{0 0 - (j 0}

⁾

^{• 1'(22)}

^{= (}

^-~

^{0 0}

^{(j 0}

D ·

The rank of the matTices T<¹¹> and '1'(^22> is 2 and that of 'f (L2l and '1'<²¹l is l. These latter cases will be favotued as here domain walls aTe possible without dis.lo- cations, i.e. without strain. In the case of T<¹²l the basic equation (2) becomes

'f(l2) x(O) = b(L) ,

I.e. - (j ~0)

=

b<fl •

(J :t~O)

=

b~L) ,

The solutions are independent of

xi

^{0> and x~0>} which means that they are planes perpendict1lar to the y-axis. On eliminating x~O) follows b\Ll = -b~L>, the equa- tion of tho b-subspace which is important for attribut,ing the BUl'gers vector to

z z z

1

z z

1 2' 1

~

^{0 0}

j

^®

_~

y y y

a b c

z z z

z

¹

1

X X X

d e r

Fig. Jl. DiiTrr~nl klmls of •tr:tin-free 90" rlomnin w:llli<. a) and b) polarization ,-ectors pnrnllcl to the wall, d) hend-hc;td, e) tnil-l:<il wall, c) anti f) walls nrislog from the chRnge of•ign of the polarizatlon 2

Observations of Ferroelectric Domains in Boracites 719

the di:;Jocations iJ the boundary is placed arbitrarily and hence intersects cell walls (see [11]). In the present case the optimum boundary is the (:r, z)-plane. Both polaz:.ization vectors arc parallel to that boundary (Fig. lla, b, c). For

'1'(21) the equation is

i.e.

a x~O) _ a :l.~o)

=

^{b~l,) ,}

b(l.) x~O>

=

x~O) - ~.

a

Here the bow1clary plane is parallel to the y-axis and inclined by 45° to the x-axis (Fig. 11 d, e, f). Depending on wh.ich side of the boundary lattice 1 (lat-

tice 2, •·cspcctivcly) is realized the two polarization vectors meet either head- head OJ" tail-taiL

It is interesting to note tha.t if e.g. the polal'izat.ion of lattice 2 is inverted this boundary i · rotated by 90° aroru1d the y-axis so that again the polarization vectors meet head-bead (tail-ta.il, respectively) but not head-tall. Hence, this calculation disqualifies a bom1dary inclined by 45° as a dislocation-free head-tail boundary, a fact which is in agreement with the observations. On the other hand, t,Jw boundary which i · parallel to both polaJ'iza.tion vectors persists on changing the sign of ono polarization.

'l'(l1) and

'I<

²²> Jead to 0-line lattices, which shall be shown for the case of '1.'<²²>. Tho basic equation hero becomes

- a x~⁰J = bfj·> ,

(J X~

=

b~l,) ,

w]lich ntcan:s that. t.hc 0-lines are lines parallel to the z-axis and the b-subspace is t.!tc (b_2• b3)-planc.

1'hc J·clativo or.ientations of htttice I and 2 can be changed by applying a ro- taLion to crystal 2. llowcver, tho 0-latt.ices calculated above arc the optimal ones.

4.3 1 0° domain tntl/ . .,

There arc two possibilities for determining the nature of the 180° walls: a) '1' i ·formed from S<¹¹>(a) and S<ll>( - a) which leads to a rank ('l') = 1 case

·with the (x, z)-plane as a 0-planc and the b₁-a:d. as the b-subspaee. Bot.h pola- rization vectors arc parallel to t.he domain waH (antiparallel boundary).

b) 'J' is formed from S<Ul(a) and s(J~J( -a), which leads to a rank ('l') = 2 case wl1ich is a 0-line laWce with 0-lines parallel to tho z-axis and the (b₁, b₂)-plane as b-subspace, and hence cannot lead to a disloea.t.ion-f1·ce boundary.

180° head-head domain walls \\"ero discussed :in Section 3.3.3. Such domain wa..IJs in the ped"eet crystal being pcl'pcndicular to the polarization vector would in any case have to be a dislocation boundary. In addition the situation is elec- tr.ica.lly unfa,ourablc. HoweYet·, a,swning the existence of a stacking fault on a {100} plane, sucJ1 that t'vo A-(two B-cubes, respectively) wouJd follow on top of each other, the same shea•· in both domains wouJd produce opposite polari- zation at such a face. Hence the gl'omctrical fit at such a wall would be strain- free.

Analogous considerations on domain walls can be appHcd in the trigonal pha.e.

i20 ^,\. ZDt)IERJL\K).I et ul.: .l<~errveleclric-Domains in Boracites

11 dmou;le<lymnenls

The authors woulu like t.o thank Prof. vV. Kanzig as well as Dr. E. Asd10r for \7CJ'y hclpf11l discussions, Mi~;s U. Ludwig and lVh·. J>. ]fontaine for the execotion of the dra.wings and the Battelle Institute for the financial support of

this work.

Refcl'ence

[1) ll. ¹ cmno. Rosl Kl'istallov 7. 32 (1967); 'oviet l'hys. - Gro~~ih of Cryst-als 7, 25 (196\J) (COilSultants Bure<~u).

[2] ll. ¹ 'mann, phys. stat. sol. 37, 20\J (HJ70).

(3] J. Ko1u YMnrr, H .. omrw. :.mcllil. AsCJrun, phys. Rtnt. sol. 26. 277 (HJ68).

[4) '1'. ITO, N. liTOilDIOTO, and R. ¹ AU,\NA<M. Acta. <:rysL 4. 310 (1951).

(5) H:. Scmuo, C. I<LIBOL, und J. :K.oUL\Y.\!S.III.. Helv. phys. Acta-E~. 5\J9 (l!J(i!J).

[6] H. cmno, J. Phys. Chcm. 1 olids !!G. 9i3 (l96!l).

[7) R. Q}.:vl':ns. J. VAN LANI>UYT, and •· . .A:unL'(CK .. '\, phys. slaL. sol. JS. 3-1-:~ (HH>6);

1 , 363 (1966); !!G. 577 (1968).

[8] A. Znr:IIER)u\NN, H. CTIMID, and\\'. BoLlillJA.'>N. Hclv. phys. Acta 42, 597 (I9G9).

[9) 1\J. 'fAN AKA and G. HONJA. J. Phys. , 'oc. Japan 19, 954 (I !)(i±). \ [IOJ R. GEv·tms, lf. BL.lKK. and S. A~1 ELIN"CKX. phys. stat. ~;ol. 13. 449 (1960).

[11] W.J30J,L111:AX~, Urystal Defect.s tmd 01·yshLWne Jntcrf'acf's, Springer-Verlag, Bcrlin- Heidclbol'g 1970, in the press.

[12] J. Fous'F:K and v. J.\NO\'BC. J. appl. rhys. 40. 135 (1969).

L13] J. KonAYASW, N. Y~MADA. and T. Azmu. Rc\>. sci. 1nstmm. 39. HH7 (1968).

[14) J. J<on.n·,, 'Ill, . '. YA)IADA. H. HAllA. J. )l1ZO'£A:\"T, 0. ·.IIL\.DA. A. Ku~IADA, and H. Scmuv. Proc. TT. T ntemaL. 1\lcciing x'erroelectricity. Kyoto 19()9; ,J. J?hys. oc.

Japan 2 (• uppl.), 67 (1970). ·

( Heceiuerl Se1Jiember 7. 1970)