Theory of spin reorientation in rare-earth orthochromites and orthoferrites

Theory of spin reorientation in rare-earth orthochromites and orthoferrites

J. Phys. Chem. Solids. 1974, Vol. 35. pp. 479-500. Pergamon Press. Printed in Great Britain THEORY OF SPIN REORIENTATION RARE-EARTH ORTHOCHROMITES ...

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J. Phys. Chem. Solids. 1974, Vol. 35. pp. 479-500.

Pergamon Press.

Printed in Great Britain

THEORY OF SPIN REORIENTATION RARE-EARTH ORTHOCHROMITES AND ORTHOFERRITES”

IN

T. YAMAGUCHI Institute for Solid State Physics,University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan (Received

2 April

1973)

Abstract- Mechanisms of the temperature-induced spin-reorientation in rare-earth orthochromites (and orthoferrites) are examined. It is concluded that the anisotropic parts of the magnetic interactions

between Cr3+(or Fe3+)and rare-earthions, the antisymmetricand the anisotropic-symmetricexchange interactions, are generally responsiblefor both the rotational and the abrupt types of the spin-reorientations. These anisotropic exchange interactions produce an effective field for the CP up-spins in the

direction perpendicular to that of these spins and an effective field for the Cr3+down-spins in the direction opposite to the abbve. These effective fields favor rotation of the Cr3+spins,retaining their original antiferromagnetic configuration. Thus, as the temperature is lowered, this effective field increasesdue to the increaseof the rare-earth magnetization,and when the interaction energy of the Cr3+spinswith these effective fields exceedsthe anisotropy energy of the Cr3+ion, spin-reorientation. takes place. At the beginning and ending of the spin-reorientation a second-order phase-transition occurs. The first-order nature of the abrupt spin-reorientation is stressed.Anisotropic exchange

interactions between Ci” and rare-earth ions also play an important role in inducing the spin-reorientation.

1. INTRODUCTION The class of the rare-earth orthochromites and orthoferrites (RM03, R = rare-earth and M = Cr, Fe) has two kinds of magnetic ions, M3+ and R3+ ions. Then, there are three types of magnetic interactions, M3+-M3+ , M3+-R3+ and R3+-R3+, each of which generally consists of the isotropic, the antisymmetric and the anisotropic-symmetric superexchange interactions. This inevitably makes the magnetic properties of RMO, complex. However, the very variety of magnetic interactions results in the various interesting phenomena in RM03. The interaction between M3+ spins is the 180 deg superexchange. The isotropic M3+-M3+ exchange constant J[ 11 is of the magnitude between 10’ and lo2 cm-‘, and gives the first NCel temperature TN1 of the order of lo* “K. The 90 deg superexchange between M3+ and R3+ spins is rather weak compared with the interaction between M3+ spins. That is, the isotropic M3+-R3+ exchange constant j[2,3] is of the order of 10°cm-‘. The anisotropic parts of the magnetic interaction between M3+ and R3+ spins, the antisymmetric and the anisotropic-symmetric exchange interactions,

abrupt

determine the spin-reorientation temperature TSR which ranges from loo to 10Z”K. The weakest interaction is that between R3+ spins which is also the 90 deg superexchange. The isotropic R3+R3+ exchange constant J’ [2,4] is of the order of or smaller than 10-l cm-‘, which gives the second NCel temperature TlV2of the order of 1O” “K. One of the prominent phenomena in RM03 is the spin reorientation[5] (SR) induced by temperature and/or by an applied magnetic field, in which the direction of the easy axis of magnetization changes from one crystal axis to another with varying temperature and/or applied field. In this paper, we report that the antisymmetric and the aniostropicsymmetric exchange interactigns between M3+ and R3+ spins are’both responsible for the temperature-induced SR. RM03 is a weak ferromagnet resulting from a small canting[6] of the antiferromagnetic M3+ sublattices. The net moment of R3+ ions is polarized parallel or antiparallel to the net M3+ moment by the M3+-R3+ magnetic interactions. When they are polarized antiparallel, a compensation of the R3' and M3+ moments occurs. The observed spin configurations and the SR in RM03 are classified *Work partly supported by the BroadcastingScience into six categories; (I) Just below TIVI, the sublatResearchLaboratories of Nippon H&B KyBkai. tice moments of M3+ and the net moment lie, in

479

480

T. YAMAGUCHI

most of RM03, along the u-axis and the c-axis, respectively. The magnetic symmetry of this kind of configuration is described, following Bertaut [7], by the symbol I,(G,, A,, F,), where G, represents the basic antiferromagnetic spin-arrangement of the M3+ ions along the a-axis and F, the ferromagnetic spin-arrangement along the c-axis due to the canting of the G, spins. A, also represents the antiferromagnetic spin-arrangement along, the b-axis due to the hidden canting of the G, spins. AS the temperature is lowered, the easy axis begins to rotate at a temperature T$, and ceases to rotate when the rotation angle reaches 90 deg at another temperature T1, resulting in spin-configuration rP(Frr C,, G,). Here, C, represents another hidden canting along the b-axis. T, and T? define the spinreorientation temperature. Finally, below T.jv2,a new configuration of R3+ spins arises. This case includes RFeO,[S] where R = Nd[8], Sm, Tb, Ho, Er, Tm and Yb and RCr03 where R = Sm[ll and Gd[9]. (II) The spin configuration below Ts, is the same as in type I, I,(G,, A,, F,). As the temperature is lowered, however, the easy axis jumps abruptly from the u-axis to the b-axis and the weak ferromagnetic moment vanishes below the temperature T,. The resulting spin configuration is r,(A,, G,, C,) and this configuration remains the same down to Tsz. DyFeO,, [ 101 and ErCrO, [I I, 121 belong in this category. (III) In some RMOB containing a non-magnetic R3+ ion such as LaFeO,, EuMO%, LuFeO, and YM0,[5], the spin configuration remains 14(GS, A,, F,) in the whole temperature range below T,,,,. (IV) In some RCr03 where R=Tb[13], Dy[l], Ho[14] and Yb[l5], the spin configuration is I’, (F,, C,, G,) below T,y, and this configuration remains the same down to T.A?*although the crystals contain magnetic R:‘+ ions. (V) There is another type of SR in which the I‘* (F,, C’,, G,) configuration changes discontinuously into ]‘,(A,, G,, C,). This is the case for NdCrO,[l6, 171 and is similarly interpreted as in type II. (VI) Finally, in RMO, containing a nonmagnetic RJ+ ion, there is another possibility that the spin configuration remains I’,( F,, C,, G,) in the whole temperature range. LuCrO, [ 181 belongs to this case: the single-ion anisotropy of A$]+ ions determines the spin configuration just below T,,,,; I‘, (G,, A,,, F,) for cases (I) - (III) and I‘?( F,, C,,, G,) for cases (IV) - (VI). The SR phase-transition in RFeO, has been investigated phenomenologically [ 19,201 and microscopically [2 11, using only the Hamiltonian of M3+ ions. Following those studies, the freeenergy is given as

F=A,(T)+A~(T)~sin26’+A,(T)~sin40,

(1)

where 19is the rotation angle of the spin system. This free-energy has extrema at values of 0 given by 19= 0, sin2 0 = -A2/2A4 and 0 = rrl2. Levinson er al. [2 l] have made assumptions concerning the temperature-dependences of the dipolar interaction energy and of the second- and the fourth-order anisotropy energies of a Fe3+ ion in RFeOs and have predicted that the magnetization may change direction in either of two ways; (1) The easy axis gradually rotates by 90 deg in a finite temperature range between Tt and T,, or (2) the easy axis jumps abruptly, possibly exhibiting thermal hysteresis of the transition. Assuming a large anisotropy energy for R:‘+ ions, Sivardiere[22] and others[5,33] have proposed that, as the temperature is lowered, the sublattice moment of R3+ spins increases and magnetic interactions between iU3+ and R3+ spins rotate the fvf”+ spins by 90 deg. One may, however, point out some difficulties associated with these theories and also experimental facts suggesting a new explanation. (1) The former theory can not be applied to RCrO, where a Cr3+ ion with spin of 5 = 3/2 has no fourth-order anisotropy term, although RCrO, has been found to undergo SR [ I.91 quite similarly to RFeOB. The use of the temperature-dependence of the secondorder anisotropy term alone can explain only the SR where the easy axis jumps abruptly and not the SR where the easy axis rotates continuously. (2) The SR takes place at relatively high temperatures in most of RMO,[ I, 5,9-12, 161. Also GdCrO,, in which Gd3+ ion is in the S-state, has been found to show the SR [9]. These facts suggest that the singleion anisotropy energy of R3+ ion does not seem to play a crucial role in the SR. Therefore, in this paper, we develop a theory of the SR within a framework which neglects the R3+ anisotropy energy. (3) The experiments show that the temperature-induced SR occurs only in RMO, having a magnetic R3+ ion [ 1,5,9-12, 161. (4) From the view point of the phase-transition, the observed results show that the temperature-induced SR of the I, to I2 type is of second-order at T, and T2 in the sense that the rotation angle 0 is a continuous function of temperature but the derivative de/dT has discontinuities. Furthermore, the specific heat measurement has shown that the application of the external field has an influence on T, and T2 in YbFeO,[24]. When the field H, along the c-axis is present, there is a component of magnetization in the c-direction in the whole temperature range

Theory of spin reorientation in rare-earth orthochromites and orthofenites

481

effective field rotates the weak ferromagnetic between T,, and TS2 and the reorientation towards the a-axis is never completed. So, there is no lon- moment and the SR takes place. It should be reger a transition temperature T,, consequently no marked that Aring and Sievers [25] have calculated the dispersion of the spin wave modes using the second jump in the specific heat. In an entirely analogous manner, the application of a magnetic Hamiltonian including the antisymmetric exchange field H, along the a-axis leads to the disappear- interaction. However, they only have pointed out ance of the transition temperature T?, consequently the importance of this interaction and have given no physical explanation of the SR. Recently, no first jump in the specific heat. The above third fact suggests an important role White[26] has pointed out the possibility that the exchange interaction befor the magnetic interactions between M3+ and R3+ anisotropic-symmetric ions in the temperature-induced SR, and the last a tween M3+ and R3+ spins is equally important in the possibility for a physical explanation of the tem- SR. As will be shown in Section 3, this interaction also produces the same effective fields as those the perature-induced SR as a kind of magnetization process by the temperature-dependent effective antisymmetric exchange interaction does. These field which acts on M3+ spins and results from the fields tend to induce the SR. interactions between M3+ and R3+ spins. The magIn what follows, we investigate the roles of the netic interactions between M3+ and R3+ spins con- antisym letric and the anisotropic-symmetric sists of the isotropic, antisymmetric and aniso- exchange interactions between M3+ and R3+ spins tropic-symmetric exchange interactions. The weak played in the SR. We employ the molecular-field ferromagnetic moment F, of M3+ spins makes the approximation to calculate the free-energy and treat R3+ spins polarized (F,*) by the isotropic exchange M3+ and R3+ spins as the classical spins. Then, interaction. This isotropic exchange interaction, of minimizing the free-energy expression with respect course, plays no role in the SR, since the effective to various angles which define equilibrium direcfield acting on M3+ spins is along the weak ferro- tions of the sublattice spins, we obtain stable spin magnetic moment. However, the antisymmetric configurations. Here, minimization is performed exchange interaction tends to make the two spin also with respect to the rotation angle 8. This point systems of M3+ and R3+ ions perpendicular to each differs from the previous arguments[ 19,2 I]. The abrupt SR of type II and V and non-exisother. Then, as will be shown in Section 3, this tence of the SR in type III, IV and VI are also interaction induces a new antiferromagnetic configuration of R3+ spins CUR along the b-axis, interesting problems to be studied. In the present M3+ spins being in the u-c plane. Here, the net paper these are studied from a unified stand-point R3+ moment FzR polarized by the isotropic ex- with particular emphasis on the importance of the change interaction should be parallel or antiparalroles of antisymmetric and anisotropic-symmetric lel to the weak ferromagnetic moment of M3+ exchange interactions between M3+ and R3+ spins. spins which rotates in the a-c plane. Then, we 2. PRELIMINARY confine ourselves to the M3+ spin system. The antisymmetric exchange interaction produces a 2(a) Hamiltonian new effective field for the M3+ up-spins of the antiRMO, crystallizes in an orthorhombicallyferromagnetic G, configuration in the direction distorted perovskite structure belonging to the perpendicular to that of these spins (+ Hz) and an space group D 2,,16-Pbntn [27,28]. There are four effective field for the M3+ down-spins in the direc- M3+ ions whose spins are represented as S1, Sp,S3, tion opposite to the above (-Hz). These effective S, and four R3+ ions (S,, S,, S, S,) in a unit cell. fields favor rotation of the M3+ spins by 90 deg, M3+ ions are on the sites; retaining their original antiferromagnetic configuration. As the temperature is lowered, these W (0, 1/Z 01, (0, l/2, l/2), W,O, l/2), (l/2,0,0), effective fields increase due to the increase of the (24 R3+ magnetization. Finally, when the interaction energy of the M3+ spins with these effective fields and R3+ ions are on the sites; exceeds the anisotropy energy of M3+ ion, the SR takes place. In addition to the effective fields along (4c)(1/2-X, 1/2+~, 1/4),(1/2+x, l/2--,3/4), the c-axis, this antisymmetric exchange interaction (2b) t-x, -Y, 3/4), (x, Y, l/4). also produces an effective field H, always along the a-axis, that is, along the direction perpendicular to We assume that the magnetic and the parathe weak ferromagnetic moment F,. Then, this magnetic unit cells are the same. Figure 1 shows JPCS-Vol.

35. No. 4- Ei

T.

482

YAMAGUCHI

the unit cell and the inequivalent four M3+ and four R3+ ions. The parameters x and y of R3+ sites are actually small[27] and their deviations from zero are neglected in this figure for simplicity. The Hamiltonian of our system is given as follows; 2 = $$yM + $ffA-R + $JjoR > (3)

(p, V) and (i, j) label unit cells and sublattice spins, respectively. The term .%?“-‘~ consists of three parts, the isotropic, the antisymmetric and the anisotropic-symmetric exchange interactions between M3+ and R3+ ions;

where the first term represents the Hamiltonian for M3+ ions, the second that for the interactions between M3+ and R3+ ions and the third that for R3+ ions. The Hamiltonian for M3+ ions, P”, is written as 25” = 2?plso + Xp,“,, + A?wmm+ %a, = c Jij’“S,P.Sj”+ c Llpsifi Iri.uj

x Sj”

cli.vj

+ C

+ x [Dil’(S[*lr)*

Si”Ujjw”‘Sj”

Pi.vj

ui

+Ei"{(Sh")*-

(Siu')'l

+

p('(S{~'Si~'

+

4i'(SiyPSirP

+

+

rip

+

(Siz'Sizl'

+

Si,"S*")

Si*wSiu")

Sis'Sir')

(4)

] )

where the first, the second and the third terms, respectively, represent the isotropic, the antisymmetric and the anisotropic-symmetric exchange interactions and the fourth the anisotropy energy of M3+ ions whose site symmetry is C,.

7

Y

The energy of this M"+-R3+ interaction is rather small compared with that of the M3+-M3+ interaction, but larger than or of the order of the anisotropy energy of M3+ ion. The anisotropic parts of this interaction play crucial roles in the SR as will be shown in Section 3. Moriya[61 has derived a rigorous quantum-mechanical expressions for Dtj and uii of the antisymmetric and the anisotropicsymmetric exchange interactions between the same kind of magnetic ions whose ground state is orbitally non-degenerate. We extend Moriya’s theory to the interactions between the different kinds of magnetic ions, M3+ and R3+. For R3+ ions we regard SlrKflas the total angular momentum operator. We then write the antisymmetric and the anisotropic-symmetric exchange interactions between M3+ and R3+ spins, respectively, as the second and the third terms of equation (5). The term %“l consists of the isotropic, the antisymmetric and the anisotropic-symmetric exchange interactions between R3+ spins and the anisotropy energy of R3+ ions. However, the isotropic [2,4] and the anisotropic exchange interaction energies between R3+ ions are small compared with those of $Pf and JP-R. Furthermore, since in any RMO, the SR occurs at considerably high temperatures, we neglect the contribution of the anisotropy energy of R3+ ion to inducing the SR. Thus, we neglect the R3+ Hamiltonian, XH, completely in this paper. This enables us to see the importance of $P’-e most clearly. The microscopic spin Hamiltonians (3) - (5) can be written in terms of the average sublattice spins denoted by S, where S, = N-’ C Si”

(6)

P

Fig. 1. The transition metal positions(l-4) and rare-earth positions(5-8) in orthochromites and orthofetites.

and N is the number of unit cells in the crystal. Further, it is convenient to introduce the following linear combinations which form the bases of irreducible representations [7];

Theory of spinreorientation in rare-earthorthochromites and orthoferrites 2F = S, +&+&+Sd, 2G=S,-Sz+S3-S.,, 2c=s,+s?-sg-s~, 2A = S, -S, - S, + S,.

+ (terms where x 73 y) + (terms where x e z) +2(u,,+c,.r) (G.F,+G,Fz) + 2(&r - CA (CA, + CJA,) + 2(G,, + b,,,) (CX,, + C,f.A + 2(%.,,-b,,) (CP,, + GJJ ; 2(tr,, + 4x) (A ,,F, + A F,,) + 2(fl!,Z- 4,;) (CL’, + GA-,),

(74 0) (7c) Ud)

The corresponding representations for R3+ spins are expressed as F”, GH,C” and A” which have the same forms as in equation (7) except that the indices 1, 2, 3, 4 are replaced by 5, 6, 7, 8, respectively. Now we introduce new isotropic exchange interaction constants multiplied by the number of nearest neighbor ions as follows;

T,,,/ N = (J,., + JR + Jc + J,,) F’/2 +(J,,-JR+Jc-J,,)G*/2 + (J,, + JH - Jc -J&?/2 +(J,,-J”-Jc+J,,)AL/2, -~an,,/N = CD,, + B,,) (CT, - G,FJ +(D,,--B,) (Csl.r-CPA + CD, + C,) (C,F, - C,,F,) + CD, - CA (CA,, - CPA + (B, + C,) (AT, -A#‘,) + (B.r - CJ (G,,C: - GA-,,), ~w,,,lN

= (Q,, + b,, + c,, + d,,) Fr2 + (ass-brsfcm--d,,)G,’ + (GT+~sr--~~r-drr)Cr* + (hrk.r - cm + &,)A,2

Table I(a). Isotropic exchange interaction constantsbetween W+ spins

(SC)


J.., = 4JII;JH = 2J,,:J,, = 8J,,,;J,, = 4J,.,; (8a) S = 2.f,, = 2.i,,; .P = 2J;, = 2-i,,. (8b)

Similarly, the antisymmetric, and also the anisotropic-symmetric exchange interaction constants multiplied by the number of nearest neighbor ions are used below in this paper. Symmetry considerations show that these constants are given as those in Tables l(a)-(f). In Table l(g) we give the second-order anisotropy constants of the M:‘+ ion. Then, usin the irreducible bases F, G, C, A, the Hamiltonians (4) and (5) are written as follows;

483

.pi,,,/2N = (J+J’)F~F’~+(J-?)CCH, /7;,,,i/2N

i;,,,J4N @a)

6%)

(9e)

= (D,.+ D,.) (A,,F,” - A,F,,“) + (D, - d,.‘) (G,,C,” - G,C,“) + (d,, + b,,‘) (G,F,” - G,F,*) +(D !I -2, ‘) (A C.“-A C “) + (D; + d:‘) (C:F:,‘( - C;;Ffft) + (d, - D,‘) (F,C,,” - F,,C’,‘{),

(90

= (a,, + &,‘)F,F,” + (q,, + d,,‘)F,,F,” + (L?,, + &‘) F,F,” + (ii,, - &,’ 1 c,c I’( + ( CT,,,- C.#,,,’) x C,C,,R + ((7,, - ii*,‘) C,CzR + (C,, + a,,,‘) (A uFz’l+ A,F,‘q + ( (7.m- c7.#,, ‘) ( G ,,Cz’l + G,C,“) + ( (72, + ti,,‘) (G,F,” + G,F,‘q + (22, - ii,,.,.‘) (A,C,” + A,C;q + (C.,.,,- r7,.,“‘) (C’,F,,” + C,,FzH) + (G,, - &.,,‘I (F,C,” + F,,,C,‘?, (%I

where a,, = - (N,, + a,,), 6,; = - (b,, + b,,) , c,, = - (c,, + cl/,), 4; = - (d,, + d,J, Liz, = - (CL + ii,,) and Z,,’ = - (Z,,’ + nvU’). Although RFeO, has the fourth-order anisotropy term, it is small enough to be neglected [29]. Table I(b). Antisymmetric exchangeinteraction constants between kP+spins s, s, s2 s3 S4

S2

&I

S,

(B,, B”, 0) (Cm 0, C,) ( 0, D,, D,) (0, -D,, D,) C-C,, 0, CA (-Em B,, 0)

T. YAMAGUCHI Table I(c). Anisotropic-symmetric S,

exchange interaction constants between kP+ spins* S,

s3

S,

(

d .r.r

0 d,z d 111

od,,

0

C,,

- C*r 0 C*: 1

CUU ( b a-1 -L

0 b YY

(

8,* 1 -aru a,,

a,,

*al,

= - (a,,

+ au”),

b,, = - (b,,

+ b,,),

c,, = - (c,,

+ cv,)

and d,, = - (d,,

+ d,,)

-a,, au2 arr

.

Table l(d). Isotropic exchange interaction constants between W+ and R3+ spins

Table l(e). Antisymmetric

- - Q S, S* 0:) s3 Lju’* Q:‘, C-Q*‘, s, ( D.r’,-D,‘, -D,‘) Table I(f). Anisotropic-symmetric

exchange interaction constants between W+ and R3+ spins

((-Q,‘, 6,‘, <.-‘, LJ;‘, Q.r’, Q,,‘, (- Q,, & - 0:) (

D,,

-D,,

-DA

(

D,,

-D,,

exchange interaction constants between MS+ and R3+ spins*

-D,)

485

Theory of spin reorientation in rare-earthorthochromites and orthoferrites Table l(g). Second-ordersingle-ionanisotropy constantsof M3+ion

D.Sz2 E(Sr*-sSU2) PGJ” SSJ 4(s,~,+s~ss,) r(S.3,t.SJJ

s,

sz

s:,

s,

D E P 4

D E P

D E

E

r

-4

-r

-P -4

F,,(G,) = E

where thecree-energy CA?& is normalized by N and by the squared mean value of M3+ spins,
D -P

r

-r

Y

2(b) Free-energy Since the strongest interaction is the isotropic exchange interaction between M3+ spins and the observed basic antiferromagnetic arrangement is of G-type, the Hamiltonian (3) is also written as w=

sY~+2?po,

(10)

where xfJ = (JA -.le+Jc--Jo)G2/2

(12)

where the angular brackets denote the .th&mal average taken with respect to the density matrix p. = e&*a/Tr e-pro. It is evident that the angular dependence of the free-energy arises from the term . In order to calculate the free-energy, we adopt the molecular-field approximation and approximate the mean value of the product of spin operators by the product of the mean values of spin operators: c&S,> = . We furthermore take the classical-spin approximation. Following Bertaut[7], three types of antiferromagnetic spin configurations are allowed grouptheoretically, that is, T,(G,, A,, Fz:FzR), r1 (A,,

G,,

C,: CzR) and I-dFr,

C,, G,: FsR, CUR).

G and F refer, as previously mentioned, to antiferromagnetic and weakferromagnetic spin arrange ments, respectively, and C and A refer to two kinds of antiferromagnetic cantings of the sublattice moments, resulting in no spontaneous magnetization. We assume that both the overt and hidden canting angles of M3+ spins are small, that is, G + F, C and A. First of all, let us consider the anisotropy-energy of M3+ ions for the T,(G,) configuration. Taking thermal averages of equation (Sd), we obtain the expression for the anisotropyenergy as

F,,(GJ

= -15,

(13b)

F,(G,)

= D.

(I3c)

Then, the spin configuration just below TN1 is determined by the relationship of the two parameters, D and E, in the spin Hamiltonian for the M3+ single-ion anisotropy: the contribution of the canting energies are neglected for simplicity. [i] T,(G,) is stable for E < 0

(11)

is the unperturbed Hamiltonian, and Rp. is the remaining part of the Hamiltonian (9). The freeenergy <%> of the system is given, to first-order perturbation of Zp,, by [2 11 = - kT In Tr(em4*“) + ,

(13d

and D > E.

(144

This is realized in most of RM03, that is, in types I, II and III in Section 1. [ii] T,(G,) is obtained for E>O

and

D>-E.

(14b)

[iii] T,(G,) is stable for E
< E

or E >O,D

C-E,

(14~)

which is applied to RM03 of IV, V and VI types. These relations are used later. 3. ROLE

OF ANISOTROPIC EXCHANGE INTERACTIONS BETWEEN MJ+ AND R3+ SPINS

In this section, on the basis of the Hamiltonian in the previous section, we investigate the mechanisms of the SR of r, * TZ and the other types, emphasising the role of the anisotropic exchange interactions between M3+ and R3+ spins. Contrary to the previous models [19-211 which claim the single-ion anisotropy energy of the M3+ ion to be responsible for the SR, it is shown that the antisymmetric and the anisotropic-symmetric exchange interactions between M3+ and R3+ spins may induce various types of theSR. 3(a) Rotational spin reorientation (r, -+ r,) First, we investigate the mechanism of the rotational SR of the type I, where the easy axis rotates continuously from the a-axis to the c-axis in the finite temperature range between T2 and T,. Symmetry considerations show that, as is shown in Fig. 2(a), the spin configuration of the high tem-

T. YAMAGUCHI

486

perature phase in this case (T2 s T s TN,) is given as .rJG,, A,, F,: FLR). FzR configuration for R3+ spins due to the polarization by M3+ spins is consistent with the presence of a compensation point in some RMO,[l, 5,9-12, 161. On the other hand, the spin configuration of the low temperature phase (TN2 s T s T,) is r,(F,, C,, G,: FsR, CUR) as shown in Fig. 2(c). FzR and C,R configurations have been found in ErFeO, by analyzing the optical spectra of E?+ ions[30]. That is, the R3+ spins in the a-b plane are separated into two pairs along the b-axis and the net R3+ moment is along the a-axis. In the SR region, we expect that each spin of M3+ and R3+ ions rotates continuously and coherently from r., at T2 to r2 at T,. The spin configuration in this region is shown in Fig. 2(b). From the Hamiltonian (9), the normalized freeenergy which is compatible with the spin configuration in this SR region is found to be

TzSTSTNI

b

(a)

TI<_TLTz

b

F = / ‘.h’ = (J, -Js+Jc-Jo)

G,‘*/2

+ (J,,-JR-JC+JD)Ay’2/2 + (J.4+J‘q-Jc-J,))cy’2/2 + (J,+Js+Jc+Jn)Fz’72 - (D, + B,) G,‘F,’ + (D, - C,) G,‘A,’ cos e + (B, + C,)A,‘F,’ cos 13 - (D,+C,)C,‘F,’ sine+ (B,-CC,)G,‘C,’ X sin 0 + (LIesin* 8 + E.cos*~) Gz’2 + (D.cos2 0 + E3in* 0) Fz’z - E(A,‘? + C,‘p) + 2p (G,‘A,’ xcose+C,‘F,‘sinO)+2q(A,‘F,‘cose - G,‘C,,’ sin 0) + 2rG,‘F, cos 28 +2s.{(~+J’)F,‘FzR’f (j-p)CIIrCyR’ - (&+b,‘)G,‘FzR’+ (b,+‘&r)A,‘F,R’ XC0Se+[(D~-D*‘)G,‘C,,R’-(B,+D~‘) X C,‘F,“’ + (bz - tiI’)FZ’C’,R’] .sin 0}, (15) where the terms of the anisotropic-symmetric exchange interaction between M3+ and R3+ spins are omitted (see equation (17)). Here, (G,‘, A,‘, C,‘, F,‘) and (F,“‘, Cyff’), respectively, are ohe normalized basis vectors of the M3+ and R3+ spins, compatible with the spin configuration in the SR region, in the rotated coordinate system where the Y-axis is along the net moment of kf3+ ions. B is the rotation angle of the easy axis in the a-c plane. As mentioned previously, the mean value of the product of spin operators has been approximated by the product of the mean values of spin operators. s is the ratio of the mean values of the R3+ and M3+ Spins, /, which is the only parameter depending directly on the temperature. Other parameters in equation (I 5) such as the exchange constants J, D, 1 and ij and the anisotropy con-

a a’ (b)

-he I T<_TI

Fig. 2. Stable spin configurations for the rotational spinreorientation of the r,, + r2 type in orthochromites and orthoferrites. S,, S,, S, and S, represent the spinsof the transition metal ions and S,, S,, S, and Ss the polarized magnetic moments of the rare-earth ions. (a) The hightemperaturephase(Tz s 7’ s TN,)(b) The spin-reorientation region (T, c T s TJ (c) The low-temperature phase stants K = (D, E, p, q, r) of M3+ ion are assumed to be independent of temperature. In this equation we have omitted the terms of the anisotropicsymmetric exchange interaction between M3+ spins, since they are really small enough to be neglected for S-state ions such as M3+ ions [6].

Theory of spin reorientation

in rare-earth orthochromites

By using the Hamiltonian (9g), the free-energy of the rotating system due to the anisotropicsymmetric exchange interaction, which should be added to equation (15), is found to be P summM-R=4~.{[(ii,,+d,‘)sin28+(H,,+ii,,’) x COS?TJF2’FrR’ + (a,, - C7UU’)Cy’C”R’ + (7r,, + ii,,‘) Gr’FzR’ cos 20 + (iiur+ZUr’)Ay’FzR’cos 0 + [ - ((?,=- ciuz‘) G,‘CuR’ + (&, + ii,,‘) Cu’FzR’ + (ii,, - &,‘) x FZ’CUR’]sin 0). (16) The second and the last terms of this equation have the same forms as the corresponding terms of equation (15). In what follows, we will show that just these play a crucial role in inducing the SR. Then, based on the correspondences of the exchange constants to the anisotropic-symmetric ones,

minimizing equation (15) with respect to $, +,4’, @ and 0, we obtain the expression for the free-energy as a function of the temperature T. To simplify the calculation, it is necessary to know the order of magnitudes of parameters. In RCrO, they are given in Table 2, where three types of the isotropic exchange constants have already been given in Section 1. The overt and hidden canting angles of M3+ spins are both of the orders of D/J - IO+, which is represented as E. Generally, we assume that the anisotropy term K of the M3+ ion[3 1,321 is of the order of l 2 and that both the antisymmetric and the anisotropic-symmetric exchange constants, b and a, between M3+ and R3+ spins are one order smaller than the isotropic exchange j. Table 2. Orders of magnitudes of various exchange constants and of the single-ion anisotropy constants. Those of R3+ions are also given for use in the succeeding paper I

.7&T + 2(&,+ii,,‘) or2(~,,,-+ZyU’)or 2(&z+ Gi,,‘), 6,&D, + 2(&&a,,‘), &!f&’ + 2(&f&‘), D,kB,‘--* 2(ti su-+H IY‘),

(17a) (17b) (17c) (174

M3+-M3+

Jbl

8

E D[bl

Kldl $-I

M3+-R3+

R3+

8

&I

M3+

B, ii

J','"D', a’

R3+-R3+

we can generally conclude that the anisotropicsymmetric exchange interaction between AP and R3+ spins also contributes to the SR. Inclusion of the other terms of equation (16) leads to a somewhat complex calculation but to essentially similar conclusions. The correspondences of equation (17) hold also in the SR of the other types such as l’., --, Il. Thus, we omit, for simplicity, the terms of the anisotropic-symmetric exchange interaction ii below in this paper. In the classical-spin approximation, equilibrium directions of the sublattice magnetizations are defined in terms of $, 4 + 4’ and Q’, where $ and 4 2 4’ are the overt and hidden canting angles of the M3+ spins, respectively, and 20 is the angle between two sublattices of the R3+ spins (S,, &) and (S,, S,). Then, the normalized basis vectors are expressed in terms of I& +,4’ and Cpas follows;

487

and orthoferrites

K’[Rl

K’[hl

IalSee Ref. [ 11. YSee Refs. [l] and [6]. Wee Ref. [6]. rdlM3+single-ion anisotropy. See Refs. [29], [3 l] and

1321.

Wee Refs. [2] and [3]. %ee Refs. [2] and [4]. rgrDy3+ single-ion anisotropy. See Ref. [4]. “‘rGd3+ single-ion anisotropy. See Ref. [29].

We use the order-estimations given in Table 2 to calculate the free-energy, where terms higher than e2 are neglected. Minimizing the free-energy with respect to +, 4, 4’, @ and 8, we obtain the following five equations; J,= [(D,+B,)

-2s(f+S’)

[email protected]]/2(J,+J,), (19)

G,’ = cos a,hcos 4 cos cp’ A,’ =cosJ,sin+cos4’ C,’ = cos * cos 4 sin 4’ FZ’ = sin * F R’[email protected], CiR’ = sin a.

1, 4, $‘, $9

UW (18b) (18~) (184 (18e) (18f)

Inserting equation (18) into equation (15) and

cb=-[(D,-C,)/2(J,-Jc)lcos8,

(20)

4’=-[(B,-CJsin8+2s(J-JI)[email protected]]

/~(JB-JCL

(21)

(JB-Jc) (~+~)2(s~+s~cos~)s~sin~+ (JD+JB) X(J-P)‘(s,’ sin o--s-sin cP)scos @ = 0, (22)

T. YAMAGUCHI

488 cos e {[2(D--E) + (Dz-Cc,)2/2(JD-Jc) - (B, - C,) */2 (JR --Jc) ] sin 0 +2s,‘[ (~-?)2/(JB-JC)]ssin Q} = 0,

(23)

This equation has the following two solutions (I. 1) and (1.2). (I. 1) sin Q = 0. In this case, equations (19)-(2 1) and (32) determine the stable angles to be

where so= [2(8,+&J /2 (J+.v, so’=

(J”-t-JR)

- (s+P)

a =o, 8 =o,

(D,+B,)l

(24)

(33a) (33b)

tj/ = [(D,+B,) -2s(J+j’)]/2(J,+J,), cb =- (Dz--c,)P(J,>---Jc),

r#J’= 0.

[2(o,-~6,‘)(Jg-Jc)-(j-jl~(B:-C,)] /2(.7-J’)‘.

(25)

The angles Q and 0 which are assumed to be generally of the order of 1 are determined by equations (22) and (23). The stability conditions with respect to $I, I$ and $’

This gives the I-.,(G,, A,, F,: FzR) configuration of the high-temperature phase ( Tz c T s TJ. The stability conditions of equations (29) and (30) are written as a2F/aQ2 = 2[ (J+J’)2/(JD+JB)]~(~+~g)

aZF/a+* = 2(JD+Js) > 0,

(26)

a*[email protected] = 2(JD--Jc) > 0,

(27)

aZFlaq2 = 2 (.JB-Jc)

> 0

(28)

are shown to be always satisfied, by using the parameters given in Ref. [I]. (J.d= 8.0cm-I, .JB= 34.8 cm-‘, Jc = 16.0 cm-’ and JD = 69.6 cm-l.) Further, the following equations should be positive for a stable spin configuration; PF/aW

= 2{[ (.f+.f’)2/(JD+JB)] xs’cos~-[(Y-5;)*/(JB-Jc)]

(S~+S~COS q3)

X (so’ sin f3-s .sin @)s*sin Q}, a2Flae2 = 2(D -E)

(33c)

(334 We)

d2F/M2 = 2(0-E)

WI

+ [(D, - C,)*/2(J, -Jc)] cos2 0 + [(B, - CJ2/2(J8 -Jd] sin2 e -2~~’ [(~-J’)2/(J8--JC)]S.sin Cpsin 8. (30)

+ (D,-C,)V2(JD-JC)

S'COS @= scl,

A = -2s) [(J-.7)2/(5, -.I,)] / LW -E) + (D, - CNWD

-(Bz-

CZNWB

-Jc)l,

--Jc) (31)

SC1= -s”[(J+.7)2/(JD+JB)]/ [(J+J’w(.lD+JB) +(As,l’ - 1) (J-J’)’

(I)

sin e = Assin @.

(32)

Then, equation (22) is written as ssin @{[Us-Jc) (J+J’)?+(J,+JB) (I-J’)2 x (As,‘-l)][email protected]+ (Js--Jc)(~+J;)2S”} =o. (22’)

(35)

The equilibrium values of various angles are easily obtained as Q = sin-’ [(s2 -s~,~)~~~/s], 0 = sin-‘[A(s2 -sC12)112], JI= KD,+B,)-2s,,(J’+J’)]/2(51,+Jd, 4 = - [ND, - CA/XJ~ -Jdl

then equation (23) has a solution

(34)

where

/(JL3-Jdl. Equation (22) and (23) give the following three sets of solutions, Cl), (1.2) and (II). If we define

> 0. (30’)

From equation (26), equation (29’) tells us that, if so < 0, the configuration is unstable for s s IS,,] and is stable for s > [so]. However, since s is small in the high-temperature phase, the configuration is unstable for this case. On the other hand, if so > 0, the configuration is stable for any value of s. Then, the fact that this configuration is realized in the high-temperature phase determines s,, to be positive. From equations (14a) and (27), equation (30’) is shown to be always satisfied.

(29)

cos 28

> 0, (29’)

W-4 (36b) (36~)

[(s,l’+ l//P)

- s2]112, 4 =-{[A(B,-CC,)-22(J-J’)1/2(5,--c)} x (s?- &,2)“2.

(364 WeI

From equation (36b), 0 is found to be zero for the critical value of S, sC1.As s increases, e increases. This implies the rotation of the spin system. The rotation angle 0 is positive or negative, respectively,

Theory of spin reorientation in rare-earth orthochromites and orthoferrites for a positive or negative value of A. However, the two corresponding spin configurations are physically equivalent. Finally, for another critical value of s, (s,,~+ l/A 2) ‘12,0 is n-/2. Thus, equation (36) is realized in the range of s, s,, C s g (s,,‘+ l/A?)“* and corresponds to the SR region (T, s T s Td. Furthermore, equation (34) determines the angle Q to be smaller or larger than 7r/2, respectively, for a positive or negative value of s,,. Then, following equation (18e), the net R:‘+ moment F,“’ is parallel or antiparallel to the weak-ferromagnetic moment F,’ of W+ ions, when the constant s,, is positive or negative, respectively. Equations (14a), (26) and (27) together with the range of s, s,, c s G (s,,?+ l/A*)l’*, show that the stability conditions a*Flaw = 2[ (J+s’)‘/

(29’1 (30’3

are always satisfied if s,, is larger than -so. Finally, equation (23) has another solution; (II) cos 0 = 0. The angle Q is determined by equation (22): (JB-.Ic) (.7+J’)*s,s .sin @ + (Jn-tJJ (J-Y)* x S,‘S’[email protected] +[ (JB-JC) (J+J’)*(JD+JB) (.J-.F)*]s* X sin Cpcos Cp= 0.

Wa)

Using the angle QIthus to be determined, the equilibrium values of other angles are obtained as follows; (37b) -2s(J+J’)

9 =o, +‘=-[(B,--C,)+2s(.Y-.P)

cos @]/2(5D+J*), (37c) (374 [email protected]]/2(J,-J,). We)

This corresponds to the T,(F,, C,, G,: FIR, CvA) configuration of the low-temperature phase (Tm c T c T,). Also in this case the stability conditions a*F/aQ*= 2{[(J+~)~/(J,+J”)](so+s~cos XS’COSa- [ (J-.F)‘/(.J,-Jr)] X (So’- s.sin @)s.sin a} a2F/ae2=-2(D-E)+(B,-C,)2/2(JB-JC) -2s,‘[ (I-.P)*/(JB--JC)][email protected] should be satisfied.

Up to the order of E*, the free-energies F( I-,), F(l’,,) and F(l’,) of the above three cases (I.]), (1.2) and (II), respectively, are given as F(r,) = F”+E-[4(JD+J,q) - (J,,+J~+Jc+JD)] x (D,+B,,)‘/~(JD+JB)*-[~(JL,-Jc)

- (J,,--J,,--J~+JD)~(~,--C,)*/~(JD-JC)* -2[(~+J’)2/(J,fJB)]S(S+Sg),

(384

F(l’,,) = [(38a) where s is replaced by s,,] +({(D-n+ [4(5,--Jc)

-VA -Jo-Jc+Jdl

x (D,-C,)*/~(JD-J~)*-[~(~~-JC)

--(J.,+JB-JC-JD)l x (B,-C,)*/8(J,--Jc)‘}.A*-2[(3-J’)*

/(JB-Jc)l*(Aso’+1))(s2-sc1*), (3W

(Jo +JB) 1 x (1 + sols,*)s~> 0,

#F/at?* = (LIZ-C,)*/2(JD-JC) +2(0-E)A* x [(s,,‘+ l/A’) -s2] > 0,

e = 712, (I, = [(D,+B,)

489

@) (29”‘) (30”‘)

F(r,f

= Fe+D- [4(Jo+Js)-(J.,+JB+Jc+Jo)] x (D, + B3*/8(JD +JB)’ - [4(JB-Jc)-(JA+JB-Jc-JrJ)l x (B, - C,)‘/8(Js -Jc)* -2[(J+J’)%JD+JB)] x(s,+s~cos aJ)S~COS a-2[(J--j’)” /(JB -Jc)l(so’ +s*sin @)s*sin a,

(38~)

where F, corresponds to the first term of equation (12). As long as s s s,,, the free-energy F(rJ is lower than those of F (IY,,) and F (r,) . Thus, in the high-temperature phase the M3+ single-ion anisotropy energy favors the r,( G,, A,, F,: FzR) configuration and 8 remains zero until s reaches s,~ where the temperature is TP. In this range of s, the spin configuration, therefore, remains r4, although the sublattice moments of M3+ and R3+ ions increase as the temperature decreases. When s reaches the critical value scI, the free-energy F(l’,J crosses F(T,). Then, 0 and @ take non-zero values and the easy axis begins to rotate. At the same time, the R3+ spins which are directed along the c-axis above T2 begin to split into two pairs, (S,, S,) and (S,, SB), along the-b-axis. The free energy F(l’,J is the lowest until s reaches (sc12+ l/A*) l’*, where the temperature is T1. The temperature dependences of Q and f3 are shown by equations (36a) and (36b). At T,, 0 turns out to be n/2. In the SR region, the net R3+ moment (S,) cos @ to be observed is constant, since s cos @ = se, and the M3+ sublattice moment ( Snr) is nearly constant at low temperatures where the SR takes place, and following equation (36c), the overt canting angle J, of M3+ spins is also constant. These are consistent with the observed results of GdCr03 [9]. Furthermore, as represented by equations (36d)

490

T. YAMAGUCHI

and (36e), the hidden canting C, of M3+ spins appears .at T? and the other hidden canting A,, vanishes at T,. In the SR region these two hidden cantings coexist. Finally, when s exceeds another critical value (s,,? + l/A’)“‘, the free-energy F( 1-J ’ is the lowest, and the r&F,, C,, G,: FzR, CUR) configuration is realized. Now, the derivatives of the angles @ and 0 with respect to the temperature T are obtaiped from equations (36a) and (36b) as follows; &D/a-r = (as/aT)s,,/s(s’So,‘)“‘, at3/aT = (as/aT)s/(s’-s,,‘)“‘,[(s,,“+

(394 l/A’)

-.Tj”?

(39b)

Then, [email protected]/aT and M/aT have infinite discontinuities at T:, and at T, and T?, respectively. This is characteristic of landau’s second-order phasetransition [33] and consistent with the experimental results mentioned in Section 1. The nature of the second-order phase-transition is also examined from the view point of the symmetry in the SR region. In Table 3(a), we list the symmetry operations for three spin configurations under the assumption that the magnetic and the paramagnetic unit cells are the same. The magnetic symmetry group of the SR region is the subgroup of the higher symmetry groups for T.,!? s T c T, and T, G T s T,,,,. At T, and T, the number of symmetry operations is reduced to a half of that in the higher symmetry regions below T, and above T.,, respectively. This is characteristic of Landau’s second-order phase-transition [33]. The rotational SR, therefore, is proved to be characterized by two temperatures, T, and T,, where the secondorder phase-transition takes place. 3(b) Spin reorientation

in terms of effective

system and shown by the shaded arrows in Fig. 3(a), where all spins are projected on the a-c plane for simplicity. The suffixes I, 2, 3 and 4 indicate that the effective field acts on the S,, Sp, S, and S, spins, respectively. An effective field

gt.d,wz

= 2( S,) (J+J’)

cos Q,

Wa)

is produced along the c’-axis and then it interacts with the M3+ spins of the configuration F,’ (S,,,)$. Effective fields gpBH,s=’

= - gpBHzd=’

= - 2( S,3) (L%, + b,,‘) cos @ (40b)

act along the a’-axis and interact with M3+ spins of the configuration G,’ - (S,,,). Effective fields gt.~~H,.,“’ = -,qt~~H~~~’

= 2(S,) (B,+b,‘) x cos @ cos e

(4Oc)

act along the b- (or 6’-) axis and interact with M3+

field

model

In the previous subsection we have derived analytically the rotational SR, by assuming the magnetic interactions between M3+ and R3+ ions. In this subsection we discuss the physical origin of the SR on the basis of the effective field model. First of all, let us confine ourselves to the system of M”+ spins and express the interaction between M”+ and R3+ spins in terms of the effective fields acting on the M3+ system. The effective fields due to the interactions with R3+ spins are classified into two types; the isotropic and the anisotropic ones. The isotropic parts of the effective fields consist of H,‘, H,’ and H,’ which originate from the terms of equation (15), (J+J’), (d,+B,‘) and (d, + B,‘), respectively. These fields are produced along the axes in the rotated coordinate

‘5 ‘\ A HL24

%#S4

a

ii

H:3

(b) Fig. 3. Effective fields acting on W+ spins which are projected on the u-c plane for simplicity. (a) Isotropic effective fields for the spin-reorientation of the r, + r, type. (b)Anisotropic effective fieldsfor the spin-reorientation of the r, --, r2 type.

Theory of spin reorientation

in rare-earth orthochromites

491

and orthoferrites

Table 3(a). Symmetry operations and spin configurations for the rotational spin-reorientation of the f., + T, type. c,, represents the screw axis parallel to the ith axis, i the inversion symmetry and R the time reversal Symmetry operations Tz c T s Ts, T, =Z T s Tt T ,,r2 s T s T I

1.1(E, RCti, Rc,,,

c,,:

i. iR&,,

RC,, ; i, 1’~(E. CZ,, RCzY, RC?,; i,

I‘.&.

ic,,,

Spin configurations iR&,,,

iR$,

G,,

ic,,)

iR&,,, iRC:,,)

Table 3(b). Symmetry operations and spin configurations for the spin-reorientation Symmetry operations

spins of the configuration A,’ - (S,,,)I#L Then, by using equations (40), (19) and (24), the normalized interaction energy of the M3+ spin system due to these effective

fields is given, to the order of l *, by

A,,

F,

:

FzR

G,. F,, A,, C,, F,, G,: FrR, CUR,FzR G, : F,*, CUR F z1 C,,

1

of the T4 + f, type Spin configurations

direction for the M3+ down-spins. Then, the antiferromagnetic spins rotate by 90 deg to be parallel to this effective field, keeping the relative spin configuration unchanged. Next, due to the term of equation (15),

Fz’ = (~z’)/(S,v)*~N

= 2r (S+P)$(D,+D,‘)][email protected] =-2[(J+J’)*/(J,)+J~)](s”+scos~) x stos a.

2s(fi, - b,‘) Fz’CrR’ sin 8, (41)

Here, the energy is already minimized with respect to I/J,r#~and 4’ and is expressed as functions of the angles @ and 8. The temperature-dependence of this interaction energy is schematically shown in Fig. 4(a). The point where s = 0 corresponds to the I first Neel temperature T,,,I. This figure shows that the interaction energy favors the M3+ spin configuration with @ = 0 which, following equation (32) or symmetry arguments[7], corresponds to that with 0 = 0. Thus, the interaction with the isotropic parts of the effective fields stabilizes the r,(G,, A,, F,) configuration determined by the M3+ single-ion anisotropy energy. The anisotropic parts of the effective fields are induced along the crystal axes and shown in Fig. 3(b) by open arrows. Due to the terms of equation (.w, 2s(d, - 8,‘) G,‘CuR’ sin 8, new effective fields gpBH,,z=-gpnH2.,Z=

an effective field gpsH

1234s

[email protected] Wd

are produced along the c-axis, that is, in the direction perpendicular to the M3+ up-spins of the configuration G,’ - (S,,,) and in the opposite

[email protected]

(42b)

is produced along the a-axis. This effective field acts in the direction perpendicular to the original spontaneous moment F, - (S<,,)I,!I.Then, the spontaneous moment rotates to be parallel to this effective field, inducing the SR. This effective field, therefore, plays the same role as the external field along the easy axis of M3+ spins does in YMO, [34]. Note that the moments on which the effective fields H, and H, act are the sublattice moment and the weak-ferromagnetic moment, respectively. Then, it is clear that, as long as the two exchange constants (b,-B,‘) and (d,-b,‘) are of magnitudes of the same order, the effective field H, contributes to the SR more than H, does. Furthermore, due to the remaining terms of the interaction Hamiltonian with R3+ spins in equation (15), effective fields gpsH12”=-gpBH3,Y=

2(S,)(b,-DB,‘)

= 2(SR)(d,-fi),‘)

2(S,)[(j-j’) [email protected] - (B, + d,‘) cos Q sin 01 (42~)

are produced along the b-axis and interact with M3+ spins of a new configuration C,’ - (S,,,)$. Using equations (42) (21) and (25), the normalized interaction energy of the M3+ spin system with the anisotropic parts of the effective fields is given, up

492

T.

YAMAGUCHI

F” is easily obtained as

to the order of 8, as (6,-B,,‘) sin 0]sesin @ =-2[(.l--J’)2/(.lR-JC)](~O’[email protected]) X s.sin @. (4j)

Fz = 2c (I--J’)r$‘+

F”‘=

F,+E-[4((J,,+JB) - (J,,+J,j+lc+J~)l x (D,+B,)2/8(J,~+J,,32

-[4(5,-Jc)

- (J,.,--JR.--Jc+J,,)l

x (D,-C,)z/8(J,j--5c)2+{(D-E) +[4(5,,--Jr) - (J,,--,,--Jr+J”)l

As shown in Fig. 4(b), the interaction energy is the x (D,-Cc,)‘/8(J,~--5~)‘-[4(5,,--3~) lowest, for any value of S, for the r?(F*, C,, G,: FrR, CUR) configuration with 0 = 7rI2 and @ f 0. -(J.~+J~--J~-~~)I(~,--C,)*/~(JO-JC)~~ X sin2 0, (44) The interaction energy of the rz4 configuration, the configuration in the SR region where sin 0 = s.sin @ (see equation (32)), is lower than that of the r., which is temperature-independent. Taking account of the contributions of the canting energies of M3+ configuration with @ = 0. The free-energy of the M3+ spin system alone spins, calculations of the free-energies similar to

F;

S

0

o
r/2

,&I-E)+ ,h I

8=0

(a)

FZ

s

4=0

0

(b)

(d)

Fig. 4. Temperature-dependenceof the free-energyvs s = (S,)/(S,,). The valuesof s are positive. The critical points of s = 0, sC1and (s,,~+ 1/A2)*‘2correspond to the temperature T,,, T, and T,, respectively. (a) Interaction energyof W+ spinsdue to the isotropiceffective fields. (b) Interaction energy of Ma+ spins due to the anisotropic effective fields. (c) Anisotropy energy of M”+ ions. (d) Total freeenergyof the M3+ spinsystem.

493

Theory of spin reorientation in rare-earthorthochromites and orthofenites equation (13) show that the coefficient of the sir? 0 term of equation (44) is positive for the spin configuration of the f,, type. The free-energy F”’ is shown in Fig. 4(c). The temperature-dependence of the total freeenergy of the M3+ spin system

(25), the effective fields are obtained as g/d’sws=’ = 2[ (J+F)JI-

(d,+&‘)] = -2[ (J+J’)*/(JD+JB)] x (s,+s~cosaJ),

gpBH56u=-gpBH78u= F=

F”+Fz’+F

%

2[ (.hJ’)r#~‘+

(d,-6,‘)

X sin 01

(45)

is illustrated in Fig. 4(d) for three configurations of the fd, TZdand fZ types. Fig. 4(d) shows that the P,, (0 = 0), the r2., (0 c 13c ~r/2) and the r2 (0 = 7r/2) configurations are stable for 0 6 s s So,, s,, s s < (s,,?+ l/A”)i’2 and (s,,‘+ l/A’)“* s s, respectively. There are two critical points of s, s,, and (s,,~+ l/A”)“‘, where the corresponding temperature is Ts and T,, respectively. Thus, as the sublattice moment of R3+ spins increases due to the decrease of the temperature, the absolute value of the interaction energy Fz of M3+ spins due to the anisotropic parts of the effective fields increases. When the absolute value of this interaction energy exceeds that of the MJ+ anisotropy energy F”‘, a new effective field H, which induces the SR is produced. Since the effective field is always along the crystal axis, the SR ceases when the rotation angle reaches 90 deg. Next, we confine ourselves to the system of R3+ spins and regard the interaction ?Y’lmHas the interaction of R3+ spins with the effective fields acting on the R3+ spin system. Since we have neglected the R:‘” Hamiltonian, %“(, completely in this paper, the polarized R3+ spins are assumed to point exactly along the effective fields which are essentially determined by the configuration of MS+ spins. From equations (15), (19), (21), (24) and

Wd

=-2[ (J-Jy*/(JB-Jc)] x (s,‘sin fItssin @),

(46b) where terms smaller than the order of 8 are omitted. As is shown in Fig. 5, the effective field H,’ is in the direction of the c’-axis in the rotating coordinate, while the effective field H, acts in the direction of the b-axis for (S,, &) spins and in the direction opposite to the above for (S,, &) spins. In the high-temperature phase s is smaller than s,,. In this case where 0 = @ = 0, equation (46) shows that H,’ # 0 and H, = 0. Then, the R3+ spins are directed along the c-axis. This is consistent with the IY, configuration of M3+ spins. However, when s exceeds s,,, 6’ and 41 take non-zero values. Then, H,’ # 0 and also H, # 0. The polarized R3+ spins, therefore, rotate and split into two pairs, resulting in the FzR’ and CUR’ configurations. Physically, since the antisymmetric exchange interaction between M3+ and R3+ spins 2s (B, - b>,‘) Gs’CuR’ sin 0

prefers the R3+ spins to be perpendicular to M3+ spins G,’ sin 0 which are along the c-axis, a new configuration CUR’of R3+ spins is induced along the b-axis. The net R3+ moment FzR’ which is mainly c’

Fig. 5. Effective fieldsacting on the R3+ spin systemfor the spin-reorientationof the pr + p2type.

T.

494

YAMAGUCHI

3(c) Abrupt

spire reorientation

(I‘, -+ I‘,)

Next, we investigate in Sections 3(c) and 3(e), respectively, the mechanisms of the abrupt SR of the types II and V, where the easy/axis jumps abruptly from one crystal axis to another at some temperature T,. We also investigate in Section 3(d) why the SR does not occur in RMO:, of the type IV although they have magnetic R:‘+ ions. In this subsection, we consider the case where the stable spin configuration of a high-temperature phase is I,(G,, A,, F, : FIR) and the easy axis jumps from the u-axis to the b-axis in the o-b plane, resulting in the I‘,(A., G,, CL:CLR) configuration. These two spin configurations are identified by magnetic [ lo], optical[ 111 and specific heat measurements [ 121. Contrary to the predictions of the existing theoretical treatments [20, 2 I], no thermal hysteresis is observed. Suppose, as in the SR of the I‘., + I‘, type, that each spin of M3+ and R3+ ions rotates continuously and coherently in the u-b plane from the I‘, to the T, configuration. The normalized free-energy of the rotating system is derived in the same way as equation ( 15): F=

XCOS~+(B,-C,)G,‘C,‘sinO

(48d) (484

(480

where $I, 4 and 4’ are the overt and two types of hidden canting angles of M”+ spins, respectively, and @ takes a value from 0 to 7~.Inserting equation (48) into equation (47), minimizing equation (47) with respect to I/J,4, $‘, Cpand 0, we obtain two sets of expressions for the stable angles and the freeenergy as a function of temperature: (i) s S sc2. where the definition of a new critical value s,.?is given below.

e =o,

(4%)

$ = [(D,+-B,) -2s(J5+~)1/2(J,)+J,I), f=, (Dz-C,)/2(J,>-JJ,.),

(49b) (49c) (49d) We)

a =o: F(r.,) = F”+E-[4(J,,+J,,) -((J,.,+.I,,+Jc+J,,)l x (Dv+B,)Y/8(Jn+J,,)2 -[4(51,--J,-) - (J,.,-JJH-Jc+J,,)l

x (D,-C,)“/8(J,,--5,.)*-2[

(J+J’)2/

(J,I+J,,)ls(s+s”). This corresponds to the high-temperature (T, 6 T Q T,,). (ii) s > SC?.

Q, = 77, w-d = Fe---[4((J,,-Jc)-

(49f) phase

(504 (50b) (5Oc) (504 W-3

(J,+J,~-JC-JD)l x (B,-C,)‘/8(J,,-Jc)‘-[4(5,~-J~) -(J., -.I,{--Jc+Jn)] (D;-C,)‘/8(J,,-Jc)s

+E~G,r’zcos20+2s[ + (J-.?)C,‘C,“’

-2[J-j’)?/(J,-J~)]s(s-s”‘). (47)

where terms of MS+ ions smaller than the order of E?are omitted. 0 is the rotation angle of the easy axis in the a-b plane. (G,‘, Au’, F,‘, C,‘) and (FzR’, CIR’) are symmetrically-allowed basis vectors of the W+ and RR+ spins, respectively, in the rotating coordinate where the al-axis is along the easy axis of My+ ions. These vectors are expressed in terms of various angles as follows; G,'=cos~1cosq5'cos+-I, A,’ = cos t,hcos I$’ sin $I - $, F,’ = sin J,cos 4’ - *7

F ‘t’= ([email protected])/2, &‘=([email protected])/2,

e = 1~12, $!I=o, 4 =- (D,--c,)P(J,>--Jc), ~‘=-[(B,-cC,)+2s(J-J’)]/2(5,,-Jr),

(J.,---R+~c-J,))G,‘?/2 + (J;,--,,--~+J,~)A,,‘?/2 + (J,,+JJR+JC+J”)F2’“/2 + (J.,+J,t-JC-J,))C;‘2/2 + (D,-CC,)G,‘A,,‘(D,+B,)G,‘F,’ (J+J’)F,‘FZW’ + (d,Dr’)G,‘C,H’ x sin 0 - (D, + d,,‘) G,‘FzR’ cos 0 +(d,+d,‘)A~,‘F,R’cosO+(~,,-~,,~) x A,,‘C,” sin 01,

- d’,

C,’ = cos $sin 4’

induced by the isotropic exchange interaction with the M3+ weak-ferromagnetic moment should be parallel to the M3+ weak-ferromagnetic moment, rotating in the n-c plane.

Wa)

Wb) (48~)

(5Of)

This is the case of the low-temperature phase (T.,?, c T 6 T,). The temperature-dependences of the free-energies, equations (49f) and (50f), are schematically shown in Fig. 6. In the high-temperature phase, the energy of the W+ single-ion anisotropy favors configuration. As the the r,(G,, A,, F,: F,“) temperature is lowered, s increases and then the free-energy of this configuration crosses that of the f, (A,, G,, C,: CZR) configuration at the critical point sr2, where the corresponding temperature is defined as T,. At T, the SR occurs abruptly. Note that there exists no thermal hysteresis of the transition. Although analytical calculations offree-energy

Theory of spin reorientation

in rare-earth orthochromites

1

%2

1

L

/’

//’

1-

,* f !IEl+ L

Fig. 6. Temperature-dependences of the free-energies of the r., and I’, configurations. The abrupt spin-reorientation of the I‘, -+ I‘, type takes place at sr2, where the corresponding temperature is T,.

similar to that in Section 3(a) shows that the 1‘,, configuration in the SR region is unstable, the freeenergy F( r,,) is, for comparison, also shown in this figure by a chain line. F(f,,) is higher than F( f,) and F(r,) in the ranges where s is smaller and larger than s,~, respectively. Then, the f,, spin configuration is never realized. Following equation (49~) and (~OC),the hidden canting angles of the A, configuration after the SR are the same as that of the A,, configuration before the SR. Next, based on the effective field model, we discuss the mechanism of the abrupt SR of the lY,-+ f, type. Suppose that the MS+ spin system is rotated by 0 in the a-b plane. The magnetic interactions with R3+ spins whose interaction Hamiltonian is given in equation (47) produce two sets of anisotropic effective fields on the iW+ spins along the crystal axes. One set of effective fields has, as shown in Fig. 7(a), the same symmetry as the initial f., spin configuration and is given by ghfb3=(r4)

= -hfwui)

G/43H14Y(r4)= -gPoH23”(r4) = 2(S,)

act on M3+ spins of the A,,’ sin 0, G,’ sin tI and C,’ configurations, respectively, and favor the f, configuration. Now, in the high-temperature phase the M3+ anisotropy-energy favors the 1; configuration. However, as the temperature is lowered, the effective field increases due to the increase of the R3+ moment and if the absolute value of the interaction energy with the effective fields of the I‘,

(4

b

= -2cw x (B,+

g/-Mmdr,)

495

antiferromagnetic G,’ cos 0, the weak-antiferromagnetic A,’ cos 0 and the weak-ferromagnetic F*’ configurations, respectively. Then, they obviously favor the I‘, configuration. The other set of effective fields has the symmetry of the final f, configuration as is shown in Fig. 7(b). These effective fields

I/(

.___.

and orthoferrites

b,‘)

FzR,

(5 la)

‘a

= TSR)

x (&+&‘)FzR, (I+?) F,“.

(51b) (51c)

These effective fields act on the M3+ system of the

lb) Fig. 7. Effective fields for the spin-reorientation of the I‘, 4 I‘, type. (a) Anisotropic effective fields of the r, type. (b) Anisotropic effective fields of the I’, type.

496

T.

YAMAGUCHI

type exceeds that of the M3+ anisotropy-energy added by the interaction energy with the effective fields of the Tr type, the SR takes place abruptly. Here, the r,, configuration in the SR region is energetically unstable and is not realized (see Fig. 6). The first-order nature of the SR is also examined from the view point of the symmetry in the SR region. Again, let us suppose, as in the’ SR of the T, + r2 type, that each spin of M3+ and R3+ ions rotates continuously from Tr to ri and that Landau’s second-order phase-transition takes place at the beginning and the ending of the SR. The magnetic symmetry group in this SR region should be the index-two subgroup of those in the high- and the low-temperature phases. As is given in Table 3(b), the spin configuration in the SR region is rd,(G,, A,, A,,, G,, F,, C,: FIR, CZR). By using the basis vectors G,’ and A,’ in the rotating coordinate where the (I’-axis is along the easy axis of M3+ ions, the M3+ spin configurations in the SR region are written as G, = G,’ A,=A,‘sin&

cos

0; G,u = G.,.’ sin 8, A.=A,‘costL

(534 Wb)

Then, the r,( G,, A,, F,) configuration of M3+ spins changes continuously into the I‘, (A,, G,, C,) configuration. Here, the hidden canting C, appears at the beginning of the SR and the overt canting F, vanishes at the ending of the SR. The ferromagnetic FzR configuration of R3+ spins should also change into the antiferromagnetic CZR configuration. However, the spin configuration in the SR region means that R3+ spins are confined along the c-axis even in the SR region. This Ising-like property suggests that the SR of the rr + r, type is of the first-order. SOfar, two critical values of s, s,, in Section 3(a) and s,? in this subsection, have been introduced. If s,, < sc2,the rotational SR of the ra + r2 type will occur. On the other hand, if scr 5 sc2, the abrupt SR of the rd --, I’, type will take place. Experiments show that many of RM03 belong to the former category while ErCr03 and DyFeO, belong to the latter. Furthermore, ifs never reaches s,] and/or sc2 in the whole temperature range between TN, and Tr2, no SR takes place. This is the case of GdFeO, [5]. 3(d) Absence of spin reorientation r2 + rd Next, we investigate whether or not SR of the r2 + r4 type takes place, where the easy axis reorients from the c-axis to the a-axis in the 0-c

plane. The free-energy of the hypothetical spin configuration rotating in the u-c plane is also given in the same way as equation (15). If we replace 0 by 1r/2 -0 and change the sign of all the antisymmetric exchange constants D and D and hence also the sign of s,, and s,,‘, all the equations derived in Section 3(a) for the SR of the r4 + r2 type are shown to hold also for the SR of the lY2+ r, type. Then, for the case (II) of Section 3(a) 0 is found to be zero and other stable angles and the free-energy are given by equation (37) and (38~). For the case. (I. l), one has 0 = rr/2. If the stability conditions of equations (29’) and (30’) are satisfied and the freeenergy F(T,) of equation (38a) is lower than that F(r2) of equation (38c), there exists a possibility for the SR . However, in order that the I-?configuration is realized just below Tsl, equation (14~) determines the first term of equation (30’) to be negative. The second term is positive, but it represents the contribution of the hidden canting A,, of M3+ spins in the final I‘, configuration, for which we have no available experimental information. Particularly, in the rough approximation of the two-sublattice model where S, = SR, S, = S,, S, = S0 and S7 = S,, equation (7) shows that C = A = G” = A” = 0, and the second term of equation (30’) is neglected. Thus, the ra configuration where e = 1~12seems to be unstable and not realized. Finally, for the case (1.2), one obtains cos 0 = A(~*-s,.~~)“~ (see equation (36b)), which shows that 0 = n/2 for s = SC, and 0 decreases as s increases. This is inconsistent with the facts that in the high-temperature range s is small and the r2 configuration with 0 = 0 is realized, so the case (1.2) is not realized. Thus, we have only one stable configuration with 0 = 0; the absence of the SR of the r2 + r, type is proved. The effective fields acting on M3+ spins are obtained as those similar to equations (40) and (42), where the sign of the M3+-R3+ antisymmetric exchange constants is changed and 0 is replaced by ~/2-e. As shown in Fig. B(a), the isotropic parts of effective fields acting along the rotating coordinate axes interact with M3+ spins of the F,‘, A,,’ and G,’ configurations. On the other hand, the anisotropic parts of effective fields are, as shown in Fig. B(b), induced along the crystal a-, b- and c-axes. These effective fields interact with M3+ spins of the F,’ cos e, C,’ and G,’ cos 0 configurations, respectively. Contrary to the SR of the r, + r2 type where effective fields act in the direction perpendicular to the M3+ moment (see Fig. 3(b)), in this case the effective fields are produced just along the moments of the original spin configura-

Theory of spin reorientation

in rare-earth orthochromites

and orthofetites

497

tween two sublattices of R3+ spins. Minimization of the free-energy with respect to these angles results in two possible configurations as follows; (i) For 0 = 0, the same equations as equations (37a, c, d and e), (38~) and (29”‘) hold, where the sign of all the antisymmetric exchange constants and hence also the sign of s0 and s,,’ are changed. Here, the stability condition with respect to 0 tPF/dP

= - 2( D + E) + (D,,+ B,,)y2 (JIj+JR) -2s,[ (J+5;)3/(5,,+JB)]s.cos cp > 0 (54a)

is satisfied at high temperature, since D + E < 0 for the I‘, configuration to be stable (see eq. (14~)) and J,,+J,, > 0 from equation (26). This is the case of the high-temperature phase (T,’ s T C T,Y,). (ii) For 13= @ = a/2, with similar change of sign of constants, equations (50b, c, d and f) also hold. This corresponds to the low-temperature phase (T,,,* c T c T,‘). The stability conditions are written as d2F/dW=

’ s,,% (b) Fig. 8. Effective fields in the rotating system from r2 to I’, configuration. (a) Isotropic effective fields. (b) Anisotropic effective fields of the I‘, type. tion. Then, these effective fields favor retention of

the rZ configuration. Now, just below T,Y,the MY+ single-ion anisotropy favors the I’.’ configuration. As the temperature is lowered, the anisotropic effective field increases due to the increase of R3+ moment and stabilizes the r2 configuration. The SR of the rZ + r4 type, therefore, does not occur. 3(e) Abrupt

spin reorientation

(rz + r,)

Finally, we consider the SR of the rZ + r, type where the easy axis jumps from the c-axis to the b-axis in the b-c plane. Symmetry considerations and the calculation of the free-energy similar to those described above lead to the expression for the various angles and thr: free-energy of the stable spin configuration. Here, (F,‘, A,‘, C,,‘, G,‘) and (FIR’, CuR’), respectively, are the basis vectors of M3+ and R3+ spins, compatible with the spin configuration in the SR region, in the rotating coordinate. We define I,!I,4 and 4’ as the canting angles of M3+ spins which give the F,‘, A,’ and C,’ configurations, respectively, and [email protected] as that beJPCS-Vol.

35, No. 4-C

2[ (J-.f’)*/(JB-Jc)]s(s+s,,‘)

[email protected] = 2(D+E)

+ (D,-CJ2/2(JD-Jc)

> 0, Wb) > 0.

(54c) Equation (48b) is satisfied for any S, ifs,’ is positive and also satisfied for s larger than lsO’I,even if s,,’ is negative. In equation (54c), the first term is negative, while the second positive. The second term represents the contribution of the hidden canting A, of IV”+ spins in the final I’, configuration, which vanishes in the approximation of the twosublattice model. Then, the configuration with 6,= 7rTr/2 is unstable and not realized; the SR does not occur and the spin configuration remains r2. Thus, within the framework of the magnetic interactions between iU3+ and R3+ spins alone, we can not explain the mechanism of the SR of the r2 + r, type. In order that equation (54~) is satisfied, we have to take into account other terms such as the anisotropy energy of R3+ ion. The effective fields produced in the case of the SR from rZ to r, consist of the isotropic and anisotropic ones. These effective fields are shown in Fig. 9(a) and 9(b), respectively. Similar to the SR of the rZ + r, type, the anisotropic effective fields obviously stabilize the r2 configuration, which the M3+ single-ion anisotropy favors. Then, also the SR of the r, + r, type does not occur. We assume that the stability conditions are satistied and that the free-energy of the r, configuration

T. YAMAGUCHI

498

(b) Fig. 9. Effective fields for the spin-reorientation of the I-, + r, type. (a) Isotropic effective fields. (b) Anisotropic effective fields of the I’, type.

is lower, below some temperature T,‘, than that of the r2 configuration. Then, the abrupt SR from I‘, to r, takes place at T,‘, the corresponding value of s being defined as s,~, Consider the case where the r2 configuration is realized in a high-temperature phase. If s does not reach srjr the SR of the r9 + I’, type does not take place and the spin configuration remains r,, since the SR in the a-c plane never occurs. This is known to be the case of type IV in Section 1. On the other hand, ifs reaches scarabrupt SR to the r, configuration takes place at the corresponding temperature T,‘. This can be applied to the type V, NdCrO,. 4. DISCUSSIONS AND CONCLUSIONS

In the previous section we have stressed the role of the anisotropic exchange interactions between M3+ and R3+ spins on the SR of various types. The antisymmetric and the anisotropic-symmetric exchange interaction energies, respectively, between the S-state ions such as Gd3+ and Cr3+ ions in GdCrO, is one and two orders of magnitudes smaller than the corresponding isotropic

exchange interaction energy [6]. Then, the antisymmetric exchange interaction is crucial for the SR in GdCrO,. On the other hand, in garnets the exchange interaction between Fe3+ and R3+ ions is known to be strongly anisotropic and the anisotropit exchange interaction is just as large as the isotropic one[35,36]. Furthermore, the anisetropic-symmetric and the antisymmetric exchange interactions between M3+ and R3+ spins arise from the same integrals and are of the same order of magnitude[37]. Thus, in RMO, containing orbitally degenerate R3+ ions, the anisotropicsymmetric exchange interaction between M3+ and R3+ spins is comparable to the antisymmetric one: both are responsible for the SR. We have not been able to explain the abrupt SR of the r, --, l-, type. As has been suggested in Section 3(e), there is a possibility that the singleion anisotropy energy of R3+ ion which is completely neglected in the present work plays an important role in this type of the SR. Table 2 shows that the anisotropy energy of Gd3+ ion is small enough to be neglected. On the other hand, the large anisotropy energy of Dyy+ ion leads to the model of Ising-like spin, whose treatment is rather simple. However, the anisotropy energies of the other R3+ ions such as a Nd3+ ion are of an order of magnitude between these two limiting cases, that is, comparable to the exchange interaction energies between M3+ and R3+ spins. In addition to this, we know that the SR in NdCrO:, takes place at relatively low temperature[ 161. These facts seem to suggest the importance of the R3+ anisotropy in NdCrOZl. One may not reject another possibility that the abrupt SR is related to the structural phase-transition of the rhombohedral + cubic type. The structure of RA 10s where R = La, Pr and Nd is known to be rhombohedral in a hightemperature phase and to change to simple cubic, while the structure of those having other R”+ ions remains orthorhombically-distorted perovskite [38]. In connection with the latter possibility, it is interesting to note that the abrupt SR of the I’, * l’* type occurs even in (Lao.!,Bi0.,)Cr03 which has non-magnetic La3+ and Bi3+ions [39]. In conclusion, we have developed a theory for the temperature-induced SR in RM03. We have shown that among the magnetic interactions between M3+ and R3+ spins, both the antisymmetric and the anisotropic-symmetric exchange interactions are generally responsible for the rotational SR as well as the abrupt SR.. The temperatureinduced SR of various types can be considered as a kind of the magnetization process by the temperature-dependent effective field due to these two

Theory of spin reorientation

in rare-earth orthochromites

magnetic interactions. (1) For the rotational SR of the I‘., -+ I’:! type, the antisymmetric and the anisotropic-symmetric exchange interactions, (D,- B,‘) and (c,,, - cgz’), produce an effective field for the M:‘+ up-spins in the direction perpendicular to that of these spins and an effective field for the M”+ down-spins in the direction opposite to the above. As the temperature is lowered, these effective fields increase due to the increase of R3+ moment. When the absolute value of the interaction energy

of M:‘+ spins with these effective

fields

exceeds that of the anisotropy energy of MS+ ion, these effective fields rotate the MS+ spins, keeping their original antiferromagnetic configuration. (2) The SR starts when the ratio of the mean values of the R:‘+ and M”+ spins, s = (S,,)/(S,,,), reaches the critical value s,, where the temperature is T?. The SR is completed for s = (s,, + I/AZ) ‘IL, where the temperature is T,. (3) At T, arid Tz the secondorder phase-transition takes place. The magneticsymmetry group in the SR region is the index-two subgroup of those in the high- and low-temperature phases, where the magnetic unit cell is assumed to be the same as the paramagnetic one. (4) Particularly, in GdCrO:, where both Gd:l+ and C?+ ions are in orbitally non-degenerate states, the antisymmetric exchange interaction is crucial for the SR to take place. In RMO:, having orbitally degenerate Rn+ ions, the anisotropic-symmetric exchange interaction as well as the antisymmetric one are responsible for the SR. (5) We have clarified the first-order nature of the SR of the I‘, + I’, type. The antisymmetric and the anisotropicsymmetric exchange interactions, (d,+ d,‘) and (ri,,+ii,,.‘), favor the r., configuration, while (6,-B,,.‘) and (ir,,,-fi,,,‘) the r, configuration. As the temperature is lowered, the effective fields due to these interactions increase. When the absolute value of the latter interaction energy exceeds that of the M”+ single-ion anisotropyenergy and the former interaction energy, the SR occurs abruptly. (6) We have proved that the SR from I‘, to I‘, never occurs, since the IV”+-R3+ magnetic interactions do not produce effective fields which rotate MI’+ spins. (7) The abrupt SR of the I‘, + I‘, type has not been explained in terms of the hP+-Ra+ interaction alone. It has been suggested that other terms such as the Ry+ singleion anisotropy should be taken into account in order to explain this type of the SR or that the SR is related to the structural phase-transition. AcknolvledRernozrs-This work has been done in the collaboration with Prof. S. Sugano, Dr. K. Tsushima and Mr. S. Washimiya. It is a great pleasure to thank them and

and orthoferrites

499

Prof. H. Kamimura for many valuablediscussionsand for the revision

of the manuscript.

The author sincerely

thanks Prof. R. L. White of Stanford University for valuable comments and discussions. He would also like to thank Profs. S. Miyahara, S. Iida, T. Moriya and Y. Uemura and Mr. S. Watarai for their penetrating questions and enlightning discussions. He is particularly grateful to Dr. R. M. Homreich of Weizmann Institute for the discussion of his experimental data prior to publication. Discussions with Prof. W. M. Yen, Drs. K. Aoyagi, T. Teranishi and M. Naito and Mrs. T. Tamaki

and K. Sate have also been most helpful. The author is grateful to all of the members of the research group under Prof. S. Sugano for stimulative discussions. REFERENCES K., Aoyagi K. and Sugano S., J. appl. Phys. 41, 1238 (1970). Blazey K. W. and Burns G., Proc. Plr.vs. Sot. 91,640 (1967). Satoda Y., Ph.D. thesis, University of Stanford, 1966 (unpublished). Schuchert H., Hiifner S. and Faulhaber R., Z. Phys. 222, I05 (1969). White R. L., J. crppl. Phys. 40. 106 I (1969). Moriya T., In: Mtr~ll&,s~~l I (Edited by G. T. Rado and H. Suhl), p. 85. Academic Press, New York ( 1963). Bertaut E. F., In: Magr~erisn~ 111 (Edited by G. T. Rado and H. Suhl), p. 149. Academic Press, New York (1963). Homreich R. M., private communications. Tsushima K. and Tamaki T.. to be published in Pruc.

I. Tsushima 1^. 3. 4. 5. 6. 7. 8. 9.

I11ter11.Couf. on Mtrgrretisr~~.Moscow, 1973. In IO. I I. 12. 13.

GdCrO:,, the SR takes place from I..I at 8°K (= T,) to I‘, at 6°K (= T,). Gorodetsky G., Sharon B. and Shtrikman S., J. oppl. Phys.39, 1317(1968). Meltzer R. S., Phys. Rev. B2, 2398 (1970). Eibschutz M., Holmes L., Maita J. P. and Van Uitert L. G., So/id Srule Conunun. 8, 18 I5 (1970). Bertaut E. F.. Bassi G.. Buisson G.. Burlet P.. Charppert J., Delapalme, A. Mareschal j., Roult G.;

Aleonard R., Pouthenet R. and Rebouillat J. P.,

J. uppl. Phys. 37, 1038 (1966). 14. Homreich R. M., Wanklyn B. M. and Yaeger I., 1111. J. Magnerisnl 2, 77 (1972). 15. Shtrikman S., Wanklyn B. M. and Yaeger I., I,zr. J. Moguerism 1,327 (1971). 16. Homreich R. M:, Komet Y. and Wanklyn B. M., Solid Store Commrn. 11,769 (1972). 17. Belov K. P., Belyanchikova M. A., Kadomtseva A. M., Krynetskii I. B., Ledneva T. M., Ovchinnikova T. L. and Timofeeva V. A., Souief Phys. solid St. 14, 199 (1972). 18. Tsushima K. and Tamaki T., to be published in Proc. I~Ic).II. Co/$ 01, Mug/w/km, Moscow, 1973. 19. Homer H. and Varma C. M., Phys. Rev. Let:. 20, 845 (1968). 20. PierceR. D., Wolfe R. and Van Uitert L. G., J. oppl. Phys. 40, 124 I (I 969). 21. Levinson L. M., Luban M. and Shtrikman S., Phys. Rev. 187,715(1969). 22. SivardiireJ., Solid Stare Commun. 7, 1555 (1969). 23. Malozemoff A. P., Ph.D. thesis. Universitv of Stanford, 1970 (unpublished).

500

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24. Moldover

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