.I. Phys. Chem. Solids, 1974, Vol. 35, pp. 16451655.
Pergamon Press.
Printed in Great Britain
DIPOLAR INTERACTIONS IN RARE EARTH ORTHOFERRITESI. YFeO3 AND HoFeO3 R. BIDAUX, J. E. BOUREE and J'. HAMMANN Service de Physique du Solide et de R6sonance Magn6tique, Centre d'Etudes Nucl6aires de Saclay, BP NO. 2, 91190 GifsurYvette, France (Receioed 10 January 1974; in revised form 18 March 1974) AbstractThe part of dipolar interactions in the magnetic properties of YFeO3 and HoFeO3 is examined in detail. In YFeO3, one finds that the contribution of these interactions to the anisotropy in the xOz plane (easy magnetization plane for the iron moments) has the same order of magnitude as the crystalline anisotropy. In the case of HoFeO3, the molecular field formalism is used in order to interpret the existence of a single, ferromagnetic ordered structure for temperatures below a rearrangement temperature, TR. The physical parameters introduced within the framework of this formalism are fitted by comparison with the available experimental results.
INTRODUCTION A great number of papers dealing with magnetic properties of rare earth orthoferrites have been published during the past few years (see, e.g. [13]). The main concern of these papers is to provide a reasonable description of the mutual influence of iron and rare earth ions, which gives rise to the observed successive magnetic reordering processes [410]. But whereas all the authors agree about the fact that isotropic exchange is responsible for nearly the whole of the magnetic energy, discordance arises as to the description of the c o m p l e m e n t a r y anisotropic contributions, among which the dipolar one should figure as an important term and can be calculated exactly. Our main scope in the present paper is to determine the part taken by dipolar interactions in the stability of magnetic configurations for some typical orthoferrites, with or without an applied field at 0 K. But as will be seen, this study will also lead us to a coherent description of the temperature dependent behaviour of these orthoferrites. Special attention has recently been paid to yttrium orthoferrite (YFeO3) [l l]. The interest of this material lies in the fact that only the iron ions are magnetic, so that a good understanding of the magnetic behaviour of these ions may be a first approach to the more intricate situations involving the presence of rare earths. We will thus start our paper by considering this compound (part A ) . An
idea of the importance of dipolar interactions in YFeO3 may be given by the experimental result of Ref. [12]. A 105 oersted applied field at 4.2 K is sufficient to rotate the antiferromagnetic direction from the x to the z axis of the orthorhombic unit cell. With this field acting on the weak ferromagnetic component (  3 0 0 e.m.u./mole), the energy involved in the rotation is less than 1 K. Such a fact suggests that dipolar interactions may be responsible for the anisotropy of the iron system, at least with the same order of magnitude as the crystal field effects on the S state of Fe 3+. The next step of our investigation wi/l be the case of HoFeO3 (part B), in which ironholmium interactions are routed to be greater than holmiumholmium interactions[13] and thus govern the only observed rearrangement ( T R  50 K) [14], favouring a ferromagnetic rare earth structure. The stability of this structure will be discussed within the f r a m e w o r k of the molecular field approximation, in terms of the twosinglet model appropriate to holmium ions, taking into account dipolar interactions together with isotropic exchange between ironholmium pairs as well as between holmiumholmium pairs. HoFeO3 has been chosen for the numerous experimental results[13, 15] which are available and for the fact that its lattice parameters [16] are very similar to those of YFeO3. Moreover, the various magnetic configurations described in our
1645
1646
R. BIDAUX et al.
paper are common to a class of orthoferrites which feature a high local crystalfield anisotropy in the xoy plane at each rare earth site; thus we readily study the case of TbFeO3 as an other representative of this class, in which terbiumterbium interactions are strong enough to provide an extra ordering at lower temperatures. A good understanding of this behaviour consists in a stability comparison between the "HoFeO3 type ferromagnetic" structure and the new antiferromagnetic arrangement of the terbium ions. Due to the small difference between the corresponding energies, this comparison in TbFeO3 brings forward the question concerning the shape of the volume in which dipolar sums have to be calculated. Some preliminary theoretical results were already reported[17], but the detailed discussion of this problem will be presented in Part II of our publication. AYFeOa
I. Description of the magnetic interactions in YFeO3 The energy of the system can be written as: E = Ejso + EoM + FAN + Eow + Ez where: E , s o =  ~I ~ ,.
Z j ~ . t~j is the isotropic exchange
with EDM, SO that the location of the iron moments in the xoz plane is taken for granted in all the situations considered below [12, 2022]. For the same reason the canting angle fixed by the D M interaction will be almost insensitive to the supplementary anisotropic terms, but will be able to display small variations in the presence of an applied field. At this point, we bring into account the dipolar interactions because they give rise to an anisotropy for which an exact expression is available without introducing supplementary empirical parameters. This point of view can be supported by the fact that the Fe 3÷ ions are in a S ground state and thus should present a small sensitivity to the crystal field, as indicated by the isotropic behaviour of the susceptibility just above TN [23]. II. Molecular field treatment of the dipoledipole interactions The Fe ions are split into four sublattices constructed on the four inequivalent sites of the orthorhombic cell, which are labelled as follows: 1
1
1
(1) ~,0,0;
(2/ ~,0,~;
11 (3) 0,~,~;
10. (4) 0,~,
energy. 1 ~ D~ ( ~ ^ t ~ )
is
the
(anisotropic)
DzyaloshinskiMoriya exchange energy. EAN which we do not write explicitly, is the crystalline and magnetocrystalline anisotropy contribution to the energy. EoiP is the dipolar energy. Ez is the Zeeman energy. We make the following assumptions: (1) The isotropic exchange provides the dominant part of the magnetic energy. Thus it determines the tendency to antiferromagnetism and the ordering temperature (TN ~ 645 K)[18, 19]. (2) At the temperatures considered, the magnetic moments of the iron ions are practically saturated to their maximum value, /z = 5 ~B where/~B is the Bohr magneton. (3) The 'DzyaloshinskiMoriya (DM) term [2, 15], which corresponds to a small energy correction to the previous one, brings about the canting angle between neighbouring spins, and their location in a xoz plane [20]. (4) The direction taken by each magnetic moment in a xoz plane depends on supplementary anisotropic terms in the energy. Whatever their physical nature might be, the contribution of these terms to the energy is very small compared
The magnetic moment on any point of a given sublattice a is constant and equal to ~ . We deal with the dipolar energy as in [24]: 1
Eolp = ~ ~ tLi. CLj. tli. Indices i and j cover all the N iron sites in the crystal, and "r~j is the dipoledipole interaction tensor of rank 2 defined as: 1
rii rii
where fi is the unit tensor and r~ = rj  r~. We also define
for i E a, where subscripts a and /3 name any of the (1), (2), (3), (4) sublattices. With these conventions the total dipolar energy becomes
1647
Dipolar interactions in rare earth orthoferritesI
Eo,e = ~ ~ 'a3 ~ I~,, . g£,,~. I*#.
/~, =/~3 = (/~,, 0, e ) ; / L 2 =/~4 = (  / z , , 0, e);/~x2 + E 2 = 2 /~ , and find f o r the d i p o l a r energy:
The T,~'s follow the relations given by the symmetry properties of the lattice (Table 1).
N (1) E,o,, = ~. 102' [7.195 p2 _ 69.791 e 2] ergs
IN
Table 1. FeFe interaction constants in YFeO3: symmetry relations ~r:_= ~=, = %, = "L, all these tensors are diagonal ],, = j~: = j . = j,~ = j, J,= = J.., = L, = J,~ = J.J,, = A, = A. = J,~ = L J,, = L, = ar=~= A~ = J, Dt2 =  D 2 , = D34 = D43 = D, Dr, =  D . . = D32 = D23 = D~ F o r o b v i o u s r e a s o n s all t h e o t h e r D . ~ ' s a r e z e r o
D, + D.. = D = (0, D, 0)
We now account for the fact that p.~ = p.3;/x~_= /x4[1,25]. In a first step, we assume the net ferromagnetic component to be so weak as not to give any significant contribution to Eo~e, so that at the same time we are relieved from making any assumption about the volume of summation in which the T,o's are calculated. Taking
N (2) E2or, = ~ 
102~[7195/*'+4789 e 2] ergs.
Expression (1) is obtained by assuming the sample to be cylindrical (needle shaped) with a (+z) direction of magnetization, so that the bulk demagnetizing factor comes to be 0 (situation which could also be met in a sample of any shape by formation of alternate domains). In expression (2) we have assumed the whole sample, which has been given a spherical shape, to be a uniformly magnetized monodomain. Although situation (1) might be closest to physical reality, one can see that in both cases the correction brought by the ferromagnetic component is of second order in E, and can be neglected if one recalls that ~[/x < 10212, 19]. In conclusion we have shown that dipolar interactions stabilize the spontaneous magnetic configuration GxFz (in Turov[26] and Bertaut's notation[27]), according to the e x p e r i m e n t a l data obtained from YFeO3, LaFeO3, LuFeO3[19]. In temperature units, the dipolar energy amounts to
E,ote
=
N E'.ote =  f kBTote
with ~LILI= ~[L3:
~I,L2 = I.L4 = ( ~ x , [~y, ~Ll~z)
Tore = 0'112 K. we find N 1021 [7.195/*;'+4.592/zy~ Eote = ~" + 2.603 tL~] ergs where /z.,,/z,/zz are expressed in e.m.u, and /z,2+/z~+/z=2 = / z 2 = 2 5 / ~ . This shows EDte to be minimal for /~., =/z,/zy =/z~ = 0. Thus, dipolar interactions favour strongly an x direction of antiferromagnetism. Note that, in agreement with Section I, we ought to have set ~y = 0, which does not affect our conclusion; our result points out that, in presence of dipolar interactions only, the y direction is a hard direction of antiferromagnetism, as it already is when D M interactions are taken into account. Evaluating the correction due to the weak ferromagnetic component, we now set
III. Influence o f an external field applied in the x direction We wish now to discuss the results published by [21,22] and [12], in the light of the previous arguments. First, we introduce the effective field due to isotropic and anisotropic exchange, acting on any iron moment of a a sublattice (a = 1, 2, 3, 4): H a exch
J~ i~l[,i FJ a 2 ~ 2
+ J~3P'3 + J~j/,t4 
Dr I^/xl  D~2^ p2  D . 3 ^ p 3  D.4^p4
where Jo#=~J~i D.o = ~ D ~ jE¢
forafixedi~a forafixediE~.
1648
R. BIDAUX et al.
S y m m e t r y properties of the lattice ensure the relations given in Table 1 for the J ' s and the D's. Next step is to assume the "two sublattice" approximation, so that the exchange effective field becomes, according to the site: H, °*~h= H F '~h = (J, + J3)~, + (./2 + J 4 ) ~ 2  (D^ ~t2) H2 ~°h = 1"14"xCh= (J2 + ./4)It i + ( i l + J3)/x., + ( D ^ ~ 0
One can thus see that, when a molecular field is being used, one can formally introduce the contribution of each sublattice to the exchange constants without any further assumption. H o w e v e r , if the exchange interactions are supposed to be
Ha, = (T~'i'+ T~'~/Z
HA,= (T~ + TI~)/Z
HA~ = ( T ~ + T~'43/Z
HA` = (T~ + T~)/Z.
All the dipolar sums are calculated in a spherical volume since we expect the applied field to d e s t r o y any type of domains which could exist in its absence. Minimizing this energy with r e s p e c t to q3 and 0, we obtain the following results:
~o
H cos 0 + Ho 2He
for any H, as long as q, is small, and also:
[Hr.  HA, + HA,+ HA3  HA`  2Hr.,] cos 0 + 2 H ~ cos ~ 0 q~=
"
H
confined between nearest neighbours, [Yz+./4[>>
for H <~HcR with Hc~ defined by HcR= lim (H(O)).
Is, + s,I In the presence of an applied field along the x axis (Fig. 1), the total energy for a pair of inequivalent ions becomes (using when possible Cinader's notations [28]): E =  H a p . cos 2q~  Ho/Z sin 2q~ +½Ix [ H ~  Ha, + HA~ H~] c o s 2 0 cos2~o +½/Z [HA3 HA`] cos 20  ½/Z [HA~+ HA,] COS 2~o +I/ZHr~ c o s 4 0 cos4~o  2 / Z H cos 0 sin~0 + const. with
The angle 0 is determined for any field H ~< HcR by 4Hr.,H~ cos 3 0 + [2H~ ( H K :  HA, + Ha,.+ H a ,  HA`  2HK,)  H 2] cos O  H o H = O.
The critical field is given by He. = }[Ho + (no ~
+ 8H~(HK2 HA, + HA2+ HA~ HA,)) I/z]
and the initial twoion susceptibility by: HD 2
Xx =/Z 2H~(HK2  HA, + HA,+ HA3  HA`  2Ha)"
Hu = (Y2 + £)/Z, Ho = D/z, K., = HK2/Z [12], K4 = Hr.,/Z [12], LZ
_ _ _ _
//z X
Fig. 1. Intermediate configuration of the Fe system, for an H field applied along the x axis.
N o w  H a , + HA,+ HA3 HA` = 455 Oe is obviously the difference of the dipolar fields seen by an iron ion along its own direction, according to whether the antiferromagnetic axis is the x or the z axis. F r o m the experimental results obtained by Jacobs et al.[12], we deduce the values of HK, and H ~ , namely HK,= 745 Oe_ and He., = 260 Oe. At this stage, we want to point out briefly the importance of the exact experimental value of the weak ferromagnetic moment /zz(0) in the determination of HE, HK, and H~. In Ref. [12] one finds /zz(0) = 300 e.m.u, mole leading to HE = 6.4 x 106 Oe compared with /zz(0)=264 e.m.u./mole commonly found in the literature (see, e.g. [19, 29]), which gives HE = 7.3 x 106 Oe, HK2 = 600 Oe and Hx, = 230 Oe. But, in the sequel, we shall keep the values given by [12] and assume the conclusions of this section
1649
Dipolar interactions in rare earth orthoferritesI to be valid for the F e  F e interactions in HoFeO3, which exhibits the same cell parameters [16] and the same internal positions for the oxygen sites as YFeO3. BHoFe03
The substitution of holmium for yttrium introduces new terms in the energy of the whole system. F o r convenience' sake we shall discriminate, among these new terms, between "purely magnetic" contributions due to i r o n  r a r e earth ( F e  R E ) and rare e a r t h  r a r e earth ( R E  R E ) interactions on one hand, and "magnetoelectrostatic" contributions arising from the twosinglet origin of the induced moments responsible for the rare earth magnetism, on the other hand. Therefore we write the total energy U=E+z,
where E is the " m a g n e t i c " part and ~ the "magnetoelectrostatic" part of the energy, and analyse successively these two terms. First, E = E~*+ E RE + E F'~E,where E Fe
1 =
~
.
g
K~
2
•
K~
4
/ i # j
is the magnetic energy of the F e system alone;
interactions, whereas the T's are attached to d i p o l e  d i p o l e interactions as in (A,II). N o t e that the two terms in the expression of E F~RE do not contain a 1/2 factor, as a consequence of the fact that the double summation runs on two disjoint families of indices. Detailed account of the ~ term will not be given here, and the interested reader is referred to [30], where a complete study of magnetism in twosinglet systems is given. Roughly speaking, one can say that the birth of an induced moment in each twosinglet subsystem is achieved at the cost of a mixing of the crystalfieldonly lower levels, which requires an expense of electrostatic energy. E P° has been discussed in S'ection A. Calculation of the supplementary terms E RE, E r~RE and I can be done in the molecular field approximation provided some basic assumptions concerning the rareearth ions and their associated induced magnetism. I. G e n e r a l a s s u m p t i o n s
The trivalent evenelectron rare earth ions H o 3÷ are localized in each cell at the position 4(c) of the Pbnm space group[31]. Each of them displays two singlet ground states lying nearby each other (A/kB <~ 4.2K) [32], but far from the nearest excited states[15,32]. In the following, we shall suppose that these two singlets are the only populated ones: consequently each rare earth moment can only be polarized along a well defined direction with respect to the crystallographic axes. According to the experimental results[13, 15, 32, 33], the rare earth ions are distributed over four inequivalent sublattices generated from the four "4(c)" positions of the orthorhombic unit cell, and labelled as follows:
E R E =  ~1 ~.l Lkl m , . ml + ~1 ~1 mk. ,~ERE . ml Sublattice label
~'~ mk.H
Relative position of the generating site in the cell
Anisotropy direction in the x0y plane
k
is the magnetic energy of the R E E FERE =  ~__, Iiktt, " m~ + ~ i,k
system alone;
I' 2' 3' 4'
~,Y,I '+~,'y,l :~,y,l ~,   X , ~, l ' y , 'ai
~r~ a '1rOr
I~,. T~k~RE . m~
i,k
is the magnetic coupling energy between F e and R E systems. Indexes i and j cover all the N iron sites in the crystal, k and l the N rareearth sites. ~ and m refer respectively to the iron and the r a r e  e a r t h magnetic moments. Lk~ and Ilk represent respectively the isotropic R E  R E and F e  R E exchange
The algebraic value a is measured from Oy, a positive value indicating a clockwise departure from Oy so that ( O y ~ O x ) defines the positive angular direction. It is well understood that the axis of anisotropy may be characterized at each site either by ~ or by ~r + a. L e t subscripts t~,/3 . . . . name any of the 1, 2, 3, 4 iron sublattices, and subscripts a ' , / 3 ' , . . , name any
R. BIDAUX et al.
1650
of the 1', 2', Y, 4' rare earth sublattices. C a l c u l a t e d within the m e a n field a p p r o x i m a t i o n , the c o n t r i b u tions to the total e n e r g y b e c o m e :
+~
'v~ T,,.,, ~T,,. . . . T .3 2..  . T34. . . =. T.z . . =. T44. := = T~. T 2'~r   T2y [ T ~ . = T'{~. T 2~ ,y._ T23.x , _ _ T3,..x ~ , _ _ T34.xy__ T 4 x yz__ T44. ~, y: _ T ~ ,:=, _. _ T y = _ . , 3 ,   ~~ x3: _z   ~~3y : ~~ ,,: , ' r '~, ~ , r  ~,~, [TIT. T,3. [T~. = T~3, =  r ~ . =  T ~ . = T~. = T~. = TI~. =  T 3 . YY fT[~',  Ti3.
 ~ ~'~ L ¢ 0 , m ¢ . m ~ , a'dl'
1
IT~, = TT~, T ~  T x~  T ~  "r~  "r~., 21'
N(1
E =~
Table 2. F e  R E and R E  R E interaction constants in HoFeO3: symmetry relations
~1,~.~.~
,~j# m . , . . . .
1
ma,~H.
l~I~.,/t..m.,+l
(~ff m.,)
.~. ~ . . 1 . . ,
w h e r e the i n t e r a c t i o n c o n s t a n t s /, L tensors h a v e b e e n defined as f o l l o w s :
and the
L.,a, = ~ , Lk~
forafixed k ~ a'
Io~,= ~ , I,~
for a fixed i ~ a
,E. %,
23'

32'

,t34'

,,t , 4 2 ' ~
.t44'
YY
.m¢~
2 a,a'
"L,~, =

YY Y2v  T 2Y 3Y ,__ T 3Y 2Y . T 3Y 4Y . T ,Y, Y2 .__
I
Y~. = Tt'~, = ~T!~, = T,~, = I T~. = [T,~. =
T~. = T~;. = Y~l. = T~. = TI~. = TI~. = Y~. Tw : T_~2 ~, = TL~. = T~. : T~~, = T~I~, = T~~, T~;. = T~. = T~;. = T~I. = T~, = TIT, = T;~. Y , 4 ,  T 2 2 ,  T 2 4 ,  Y31,  T 3 y  Y , . .  T43. T~',~,=  T ~ . = T~,, =T~':. =T~';, = T~:, = T~;. TI~. =  T ; ; . =  T ~ . = T~. = T~. =  T ; ; . =  T ~ . xy

xy
__
xy

xy

xy

xy

xy
I,,. = I , y = ~2,' = Izy =/'32' = I3,. = L2. = L~. = I1 I,... = L,' =/2... = 1..4. = I3,. = I33. = L I ' = L 3 ' = [2
for a fixed k ~ a ' f o r a fixed i ~ a.
k~a,
R e l a t i o n s b e t w e e n the 1"s, the L ' s and t h e I's, b a s e d on s y m m e t r y p r o p e r t i e s of the lattice, are g i v e n in T a b l e 2. O n the o t h e r hand, n u m e r i c a l v a l u e s of the T.,~, ~  ~ and T .~. , ~ c o m p o n e n t s h a v e b e e n c a l c u l a t e d for H o F e O 3 using the cell p a r a m e ters of YFeO~[16]; ~ =0.018 and ~ = 0060115, 31]. T h e " m a g n e t o  e l e c t r o s t a t i c " term, calculated within the m o l e c u l a r field f o r m a l i s m , is written:
N A , (1 r =~ [ ~ ~
a
a'(ho,)~]
~tanh
2kBT ]J
with A'(h~,) = A ~ w h e r e a = 2 m J A , m, standing for the single n o n  v a n i s h i n g c o m p o n e n t of t h e m a g n e t i c m o m e n t o p e r a t o r b e t w e e n the t w o crystalfieldonly g r o u n d states of e a c h R E ion, and h., is the c o m p o n e n t of the internal m e a n field on the local a n i s o t r o p y axis. It m u s t be n o t e d that, w h e n A goes to zero, ~" v a n i s h e s as e x p e c t e d and o n e is left with a p u r e l y K r a m e r s  d o u b l e t e n e r g y term. A t T = OK, ~ is r e d u c e d to[30]: NA
w h e r e m o is the a ' sublattice m a g n e t i c m o m e n t at 0K, as g i v e n by [34, 351:
m Oe= ms ~
ah°, .
W e n o w l o o k f o r the v a r i o u s m a g n e t i c structures
All these tensors are diagonal and I
xx  xx T r y  T3,r =
T ~  T x~ ~T,.3. Ty,. Tg.~.= T~.~. l T ~ , . = T~.~,. TI.~,.= TI;. x, _ Try ~,  T 2 .~Y , .   T ~ .~" r try3, YY
__
2'4'

4'2'
YY
All the other components are zero Lvv= Lvz= Lr3 , = Lv,,=
L2,2.= Lr3 . = L.,.,= L2.v = L3,,, = L,,3,= L3,v = L z , , = L.,2 ,= L , , v = Lz3,= L3,2,=
L, L2
L3 L,
w h i c h m a y t a k e p l a c e s p o n t a n e o u s l y in z e r o applied field. If we agree to satisfy the c o n d i t i o n that the c o r r e s p o n d i n g m a g n e t i c space g r o u p s should b e l o n g to the f a m i l y Pb.m in O p e c h o w s k i ' s notation, we find that the a b o v e defined sublattice partition r e d u c e s the number, of p o s s i b l e m a g n e t i c configurations to eight, as s h o w n in T a b l e 3. T h e arrangem e n t s of the F e ions w h i c h are still s u p p o s e d to h a v e a c a n t e d a n t i f e r r o m a g n e t i c s t r u c t u r e in the x o z plane as justified in S e c t i o n A are d e t e r m i n e d by the d i r e c t i o n of their w e a k ferromagnetic c o m p o n e n t w h i c h m a y lie either along x or z. T h e directions of the R E sublattice m o m e n t s m., along their r e s p e c t i v e a n i s o t r o p y axis are r e f e r r e d to the o r i e n t a t i o n of t h e s e a x e s as g i v e n in Fig. 2 r e p r e s e n t i n g configuration L T h e s a m e table indicates also t h e t h r e e t e r m s i n v o l v e d in the total
1651
Dipolar interactions in rare earth orthoferritesI Table 3. Magnetic configurations and associated magnetic energies Fe system
R E system 1' 2' 3' 4'
E F~
E~
E ~°~ E, E ~" E,
E; 0
x z x z
++++ ++++
+ +  
E w~ E~v
0
V
x
+ 

+
E F%
Evt
0
VI
z
+ 

+
E Fe:
Evf
0
VII VIII
x z
+ +++
E v~ E~c,
Evm Evm
0 0
+  
E Fc~
Ely
Y
E~
I // III IV
+
0
~2~~ m4
~"
"
0
Energies are written for a group of four ions (two inequivalent Fe plus two inequivalent RE ions).
Fig. 2. Configuration I of the Ho system in the x0y plane.
E ~c, and E ~°: are the energies written in Section A.III with 0 = 0 and 0 = ~'12 respectively. E~ = [(72.3629 + 17"8048N) x 102' (L, + L.~ + L3 + L,)] m 2 sin" a + [52.4344 x 102.  (L,  L.. + L~  L,)] m2 cos 2 a + 0.9561 x 102~m 2 sin a cos a  2 m H~ sin ,~ E,v = [7.4665 x 10~ + (Lt + L2  L3  L,)] m 2 sin 2 a [86.513 x 102~+ (L~  Lz  L3 + L,)] m" cos 2 0.9561 x 102' m 2 sin a cos ct Ev, =  [81.096 x 10:' + (L~  L2  L3 + L,)] m 2 sin2 a + [ 13532 x I 02~  (L, + L..  L,  L,)] m 2 cos 2 a  09561 x 10z' m" sin t~ cos a E w , = [36.2242 × 102'  (L,  L2 + L3  L,)] m ~ sin2 a  [(78.9426  178048 N) × 10"' + (L, + L_~+ L~ + L,)] m 2 cos 2 t~ + 09561 x 10~'' m 2 sin a cos a  2 m / 4 , cos a E~ = 4m/z [{(39.1461  8.9024 W) x 102' sin a + (I, +/2) sin a + 2.1506 x 102' cos ct} sin ~o +{8.6273 x 102~ sin a +29.4109 x 10~~ cos ~} cos ~o]. T, = 2727.330 cos t~ + 800.027 sin T2 = (3630.094  825.54 W) sin a + 199.433 cos a
b y its o w n i n t e r a c t i o n s . C o n s e q u e n t l y t h e F e s y s t e m s h o u l d stabilize in t h e YFeO3 p a t t e r n w i t h t h e s a m e v a l u e o f ~0 in c o n f i g u r a t i o n s II, I V , VI, V I I I w h e r e a s c o n f i g u r a t i o n s III, V, V I I c a n n o t b e t h e s t a b l e o n e s at OK. B e s i d e s , e x c e p t in c o n f i g u r a tion L t h e m a g n e t i c e n e r g i e s are i n v a r i a n t b y c h a n g e o f ~ into 7r + ~o and t~ into ~" + a. (b) C o n f i g u r a t i o n I is f a v o u r e d b y a s t r o n g F e  R E e x c h a n g e c o u p l i n g w h i c h in f a c t is e x p e c t e d to b e m o r e i m p o r t a n t t h a n R E  R E exc h a n g e c o u p l i n g , d u e to t h e d i s t a n c e s b e t w e e n t h e i n t e r a c t i n g ions. But as long as this c o u p l i n g is i s o t r o p i c , as s u p p o s e d h e r e , it o n l y a c t s t h r o u g h t h e weak ferromagnetic c o m p o n e n t of the Fe ions, m a k i n g e v e n t u a l l y its e f f e c t c o m p e t i t i v e w i t h RERE e x c h a n g e or w i t h d i p o l a r i n t e r a c t i o n s . (c) D i p o l a r energies are strongly shaped e p e n d e n t in all c o n f i g u r a t i o n s w h i c h e x h i b i t a nonnegligible net magnetic moment. L e t us n o w turn to t h e s i t u a t i o n at n o n z e r o temperature. An entropy consistent with the m o l e c u l a r field a p p r o x i m a t i o n c a n b e a t t a c h e d t o e a c h c o n f i g u r a t i o n in the f o l l o w i n g m a n n e r : let pO a n d p~' b e t h e p o p u l a t i o n f a c t o r s o f t h e t w o g r o u n d s i n g l e t s o f t h e k t h R E ion; t h u s pkO+pk 1= 1, a n d
T3 = m~ [(3355.177  825.54 ~r) sin2 a
( in Oersteds
/
2431"175 cos 2 44.333 sin a c o s a ] W stands for the demagnetizing factor along the direction of the net R E ferromagnetic moment. energy which describe respectively the F e  F e , FeRE and RERE interactions. For a given configuration, the four inequivalent R E m a g n e t i c m o m e n t s ms. ( a ' = 1', 2', 3', 4') h a v e the s a m e m a g n i t u d e ( m , , = rn), as well as t h e c o r r e s p o n d i n g i n t e r n a l fields (h,, = h). T h e m a g n e tic e n e r g i e s are easily c a l c u l a t e d , as f u n c t i o n s o f t h e e x c h a n g e i n t e r a c t i o n s Ij, I2 and L~, L,., L3, L4 (defined in T a b l e 2) a n d are g i v e n in T a b l e 3. I n s p e c t i o n o f t h e s e e n e r g i e s s h o w s t h a t , at OK f o r H =0: (a) E x c e p t in c o n f i g u r a t i o n L n o m a g n e t i c c o u p l ing e x i s t s b e t w e e n F e i o n s a n d R E i o n s , so t h a t t h e F e s y s t e m a n d t h e R E s y s t e m will t a k e an e v e n t u a l magnetic order separately, each system being ruled
p•.•=kfiT/
e x p (  A'(hk)'~
where A ' ( h E ) = A ~ is t h e v a l u e at t e m p e r a t u r e T o f t h e g r o u n d s i n g l e t s splitting in p r e s e n c e o f a n e f f e c t i v e field c o m p o n e n t h, a l o n g t h e a n i s o t r o p y axis o f t h e k t h R E ion. T h e thermal average of the magnitude of RE magnetic m o m e n t at site k is g i v e n by:
m k = m ~ ° ( p ~  pk ~)
with
ahk mE° = m~ ~ / ~ .
R. BIDAUX et al.
1652
The corresponding entropy of the R E system is written: S R~ =  ks ~ (p~ In pk° + pk t In p t). Making use of the foursublattice assumption, these expressions now become:
Finally, the free energies FI, F , . . . . . Fvm and FpARA, at given temperature T, are expressed in terms of the angular p a r a m e t e r q~ and have to be minimized separately with respect to this parameter. This minimization is easily achieved using an expansion of the F ' s to second order in q~ (q~ being of the order of 102rad, terms of order ~p~ and higher will be considered as negligible in the energies).
mk(k E or') = m., = m~,,° ( p . , °  p,,, I) N ~ (p.,O In p~O + p.,, In po,'), SRe = _ k a4" where we keep in mind the fact that m,, °, p,0, p ? , etc . . . . for a given temperature, take different values according to which configuration is being considered. In a similar line, the entropy of the F e system can be calculated as a function of the population factors relating to the energy levels at each inequivalent F e site. However, experimental evidence shows no measurable difference, in all R E orthoferrites, between the respective magnitudes of ~ at room temperature and at OK; for our purpose, i n the range of temperatures considered which will never exceed 5060 K, this means that the F e system is practically completely ordered and we shall consequently assume S ~ = 0. F r e e energies for all configurations are now available, but at nonzero temperature the situation where the R E magnetic moments are disordered becomes competitive; this situation will be referred t o   o r indexed by the s u f f i x   P A R A , and features the following characteristics: When no external field is applied, the iron system behaves as in YFeO3 (the R E system provides no effective interactions). h,, = 0 for all R E sublattices (at every R E site, effective field due to the F e system is directed along the z axis and consequently can induce no R E moment), so that Im°,,I remains equal to zero. The ground singlets splitting at every R E site is now equal to A and the entropy is given by: S PARA
:

kBN (t90 In p0 + p ~In p ')
where
p! p°+p'=l
and
~=exp(
[email protected]),
while the "magnetoelectrostatic" contribution is:
II. Comparison with experimental data on HoFeO3 The values of the parameters which describe the magnetic behaviour of the H o 3÷ ions can be obtained from various experimental data [13, 15, 33]. The angle a of the anisotropy axis with the y direction is equal to +27 ° and the magnetic moment at saturation ms is approx. 7.5 ~B as obtained from magnetization curves at high fields. The crystalfield splitting A will be kept adjustable near the value of 4 K, since optical measurements[32] only give a maximum value of this parameter (A/kB <42 K). The most salient magnetic properties which we propose to analyze now, are: (1) The existence of a unique, ferromagnetic structure corresponding to configuration I[13, 33] from 50 K down to 0.3 K. (2) The value at zero applied field of the m.,(T) component of the H o magnetic moment as given by neutron diffraction data[36] and the variation of m , ( T ) + ~ ( T ) proportional to the bulk magnetization of the sample, extrapolated at zero applied field from constant temperature magnetization curves [13,141. These experimental results are shown on Fig. 3 and have been fitted with the expression of F~ in order to get an estimate of the various exchange parameters. Consideration of the respective distances between interacting ions, leads to the conclusion that L a n d / 2 must be much larger than L3 and L4 which in turn must be larger than L, and L2. In the following, these last parameters will be assumed negligible. The problem is reduced to the determination of two terms which are W = 2p.(L + I2) and L = ( L 3  L , cos 2c~)ms. These terms describe the exchange contributions of the F e and of the rareearth systems to the mean field which acts on a H o 3÷ ion and whose c o m p o n e n t along the anisotropy axis of this ion writes: h = T, cos q~ + (T2 + W sin a ) sin q~ + (7"3 + L)x, where x = re~ms. (Tt cos q~ + T., sin ~0) and T3x are respectively the dipolar contributions of the F e
Dipolar interactions in rare earth orthoferritesI
o NAG.  v NEU.
v,
" 0
"
I
I
10
20
well defined configurations leading then to a first order transition between these configurations, rather than minimizing the free energies with respect to all parameters as for instance the angle between the weak ferromagnetic component and the x axis. Taking this angle as variable would have allowed to account for a possible progressive rotation of the Fe moments at the vicinity of the transition temperature which is perhaps the observed behaviour. Nevertheless our main interest was to understand the existence of this only reordering temperature rather than its exact description which would have lead to many important
EXPERIMENTAL
TOT. CALCULATED R.E.
t
.....
~
o I 30
o
o o o
20
T(K)~
Fig. 3. Temperature dependence of the experimental and calculated spontaneous magnetization along [100], for HoFeO~. Bulk magnetization measurements are represented by circles; black triangles refer to neutron diffraction data which provide the x component of the Ho moment only. system and the R E system (for numerical values of T,, T~., T3, see Table 3). The minimization of the expansion of Ft with respect to tO, leads to:
m,(T,.+ W sin a) tO ~ tOo +
21xH~
1653
Ho x ; tOo = 2 H e
1.094 x 10~ rad as in Section A. Thus h = T , + ( T 2 + W sin a ) ~o [me(T,. + W sin a ) ' t T3+ L ] = hc + h~x. +x 2~HE IL hc can be adjusted using the high temperature part of Fig. 3 where x remains small, ho is then obtained by fitting the whole curve. The results are shown in the following table, where spherical dipolar sums have been used and where A/kB has been put equal to 4.2 K
difficulties.
The theoretical value of this reordering temperature TR, in our model, is found to be quite sensitive to the x axis demagnetizing factor 3 c involved in the dipolar sums. For example: N = 41r N = 4"rr/3 N =0
TR = 36"5 K TR = 38 K TR = 3 9 K .
Figure 4 represents the temperature dependences of F~  FpARA in the case W = 41r/3 and F~v  F p ~ A ; it shows the stability down to 0 K of configuration I compared with configuration I V which could have been favoured by the high antiferromagnetic value of L. The other structures have not been considered, for, relative to the dipolar interactions, they are much less stable than I V , and relative to the exchange interactions, their energies are not defined by the same, above determined, linear combinations of L3 and L4. It is also clear from Fig. 4 that a slight variation
~xkOel 0"
10
20
30
40 ' T(K)

5
h,
ho
W
L
6"8x10 ~0e
2.8x1030e
8 x l 0 t0e
975x1030e
FI
 F PARA
(,/J~4"X/3)
............. FIV FpARA Our theoretical curve compares well with the experimental one only up to 40 K approximately (Fig. 3). There are two main reasons which make this obvious. The first one is due to the crude approximation of the molecular field near a transition temperature; the second one is that in our assumptions we are only comparing energies of two
Fig. 4. Variations of FIFpAR.~ (for JV'=4~r[3) and F , , ,  Fp~,R~,, plotted in KOe x p. units as functions of temperature. The different parameters have been given the following values W = 8 x 105 Oe, L =  975 x 103 Oe and A/ka = 42 K.
1654
R. BIDAUX et al.
of the various terms involved in the expressions of the energies, m a y lead to a situation where a second reordering temperature exists, the F e system being again decoupled from the R E one and returning to the YFeO3 pattern, and the R E system taking now a structure defined by its own interactions. This situation occurs in TbFeO3 and will be discussed in P a r t II. Figure 5 shows that it may also be the case in HoFeO3 for the highest value of the demagnetizing factor due only to the different values of the dipolar interactions as a function of the shape of the sample.
~xkOe 2.5
1 ,
2
3
,
/
,,./
T (K)~
c
%6000
I 11
 ?,5920
b[s,sg°° I [[
~,~,5850 F

O/
5,280(
/
5
._////':
L5,2?OC
!
I
I[
100
I gonio I Ho Fe 03  debye
7.5
200
300 T(K)
10
Fig. 6. Lattice c o n s t a n t s a, b and c in HoFeO3 vs temperature (R. Kleinberger).
12.5
small temperature d e p e n d e n c e of~.the lattice parameters below about 100 K. This means that the second order crystal field effect on the S = 5/2 state may be of the form DS~ 2 with a constant D at experimental temperatures. The term K2 which issues from this expression varies from 0 to its saturation value between Tu and about 0.2 TN (see reference [37]) which is equal to 130 K in our case.
/
zoj,
Fig. 5. Variations o f F~  F p ~ for different values of x axis demagnetizing factor J¢" and F , v  F ~ A as functions of temperature. CONCLUSION
In this paper we have pointed out the part taken b y dipolar interactions in the stabilization of magnetic configurations in two typical orthoferrites. A simple formulation has allowed us to rate these interactions at about 40% of the second order anisotropy energy in YFeO3. Extension of this formulation to the case of HoFeO3 yielded non negligible values of the F e  H o and H o  H o isotropic exchange so as to correspond to the observed temperature behaviour. The existence of a reordering temperature is well understood only b y the presence of F e  H o interactions, the second order anisotropy term K2 of the F e ions being assumed constant in the temperature range considered. This assumption has further been justified b y Xray measurements on HoFeO3 which have been done b y Kleinberger et al. Their results which are represented in Fig. 6, show the regular and very
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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
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10 (1960), Naish V. E. and Turov E. A., Fiz metal. metalloved. 11, 161 (1961). 27. Bertaut E. F., In Magnetism III (Edited by G. T. Rado and H. Suhl), p. 149 (1963). 28. Cinader G., Phys. Reo. 155, 453 (1967). 29. Gorodetsky G. and Treves D., Phys. Rev. 135, A97 (1964), Gorodetsky G. and Treves D. in Proc. of Intern. Conf. Magnetism Nottingham, p. 606 (1965). 30. Bidaux R., GavignetTillard A., Hammann J., 3. Phys. 34, 19 (1973). 31. Geller S., 3. Chem. Phys. 24, 1236 (1956). 32. Malozemott A. P. and White R. L., Solid State Commun. 8, 665 (1970). 33. Koehler W. C., Wollan E. O. and Wilkinson M. K., Phys. Rev. 118, 58 (1960). 34. Hammann J., Thesis, Rapport CEA No. 3866 (1969). 35. GavignetTillard A., Hammann J., Proc. 12th Conf. Low Temp. Phys. p. 697. Academic press of Japan (1971). 36. Abdalian A. T., Thesis, Orsay (1972). 37. Yosida K., Prog. Theor. Phys. 6, 691 (1951).