P H Y S I C A L R E V I E W V O L U M E 1 5 5 , N U M B E R 2 10 M A R C H 1 9 6 7

Effect of Antisymmetric Exchange Interaction on the Magnetization and Resonance in Antiferromagnets*

G. ClNADERf

Department of Electronics, The Weizmann Institute of Science, Rehovoth, Israel (Received 7 September 1966)

The magnetization curves and the resonance frequencies are calculated for two types of antiferromagnets with an antisymmetric exchange interaction of the form D#-MiXM,- and a uniaxial magnetocrystalline anisotropy term K sin20. In one type of these antiferromagnets, the D vector is perpendicular to the easy axis. This is presumably the case in the orthoferrites. For this case, we have calculated the resonance fre­quency as a function of an external field applied parallel to the easy axis. In the second type of these anti­ferromagnets the D vector lies parallel to the easy axis. This is the situation in hematite below the Morin temperature. For this case, we have calculated the resonance frequency as a function of an external field applied in the plane perpendicular to the easy axis. In both cases we obtained two resonance modes, and the lower mode was found to have a frequency W-—Q for a characteristic critical field, at which the anti-ferromagnetic axis becomes perpendicular to the easy axis.

INTRODUCTION

THE canting effect responsible for weak ferro-magnetism in essentially antiferromagnetic ma­

terials is attributed primarily to two types of mech­anisms, i.e., single-ion anisotropy1 and antisymmetric exchange. The second is an exchange interaction of the form D#'StXSy. This term was originally suggested by Dzyaboshinsky,2 who showed, on purely symmetrical grounds, that such a term may exist. A theoretical derivation of this term was given by Moriya3 who showed that it was due to the effect of the spin-orbit term on the superexchange interaction. In this paper we shall deal only with antiferromagnets of the second kind, having uniaxial magnetocrystalline anisotropy of the form K sin20. The D vector can be perpendicular to the easy axis, as is presumably the case in the ortho­ferrites (RFeOz where R is a rare-earth element), according to Treves,4 or parallel to the easy axis, as occurs, for example, in hematite below its Morin transition temperature. (Hematite in this temperature range is in the antiferromagnetic phase, because the magnetocrystalline anisotropy energy is lower than the antisymmetric exchange energy.)

D PERPENDICULAR TO EASY DIRECTION

The Equilibrium Position

The situation where the external applied field is perpendicular to the easy axis in materials with D perpendicular to the easy axis had been discussed by Herrmann5 as a special case of a Hamiltonian having a

* The research reported in this document has been sponsored in part by the Air Force Materials Laboratory, Research and Technology Division, AFSC through the European Office of Aerospace Research, U. S. Air Force.

f Present address: Israel Atomic Energy Commission, Yavne,

1T." Moriya, J. Phys. Chem. Solids 11, 13 (1959). 2 1 . Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). 3 T. Moriya, Phys. Rev. 120, 91 (1960). 4 D. Treves, Phys. Rev. 125, 1843 (1962). 5 G. F. Herrmann, J. Phys. Chem. Solids 24, 597 (1963).

more general form of the magnetocrystalline anisotropy. We shall consider in this section the case where the external field is parallel to the easy axis, taking the X axis in the easy direction and the F axis parallel to D; the free energy F is given by

F - / M r M 2 - D - M i X M 2 - •(K/2Mo2)(Mu2+M2*2)

-H(Mt*+M2X), (1)

where Afo=|Mi| is the magnitude of the sublattice magnetization, / is the molecular field constant, D is the Dzyaloshinsky-Moriya vector, K is the uniaxial magnetocrystalline anisotropy constant, and H is the applied field in the X direction. Solving the equilibrium and stability equations obtained by considering the first and second derivatives of F, we found that there is a gradual rotation of the antiferromagnetic (AF) axis (see Fig. 1) with increasing field from a position parallel to one perpendicular to the easy direction. Thus there is no spin flop as would occur in the case of an antiferromagnet.

The equilibrium directions of the sublattice mag­netization vectors Mi, M2 may be defined by means of the angles «i and a% respectively, as shown in Fig. 1, or alternatively, by the canting angle # and the rotation angle 6 of the AF axis. These sets of angles are related by ai=<3>— d; a 2 = ^ + ^ . The equilibrium values of #

FIG. 1. Equilibrium sublattice magnetization directions. -*z

x azy\Mz

155 453

4 5 4 G . C I N A D E R 155

a n d 6 are found t o be

^(Hd+H)/2Hex,

sm0£*HaH/(Hna* ^ 1

-H2),

for for

for for

H^H0

3^ Ho,

(2)

(3)

where

Hex=JM«, Hd=DM0,

and

HK=K/Mo, HUC=1(2H^+HK)HK}112,

# o = i [ - Hd+ (H„»+4H| | c2)1 / 2] . (4)

£T||C is the field at which spin flop would occur in the case of an antiferromagnet. The magnetization in the X direction as a function of H is M x=2M<$m$ Xsin0, and is shown in Fig. 2 for the case of SmFeOs at room temperature. Here we have6 £Tex=4.6X106 Oe, £Td=6X104 Oe, and HK=170 Oe. For the pure anti-ferromagnetic case, where Hd=0, we get from (3) the well-known spin flop at H\\c=Ho. As H* increases, we see from (4) that Ho decreases and the transition of the AF axis from parallel to perpendicular to the easy axis becomes more and more gradual. This is to be ex­pected, as there is a corresponding increase in the ferromagnetic moment.

Resonance Frequencies

To obtain the resonance frequencies, we resort to the usual small-signal approximation. This is done by expressing the motion of the magnetization vectors in terms of their deviation from equilibrium. We follow the customary procedure5 of transforming to two separate coordinate systems, (Si,Ti,Yi) and (S2,T2Y2)y

which describe, respectively, the motion of Mi and M2. The Si and 62 axes are chosen so as to coincide with the equilibrium positions of Mi and M2. Also, Fx, F2, and F are taken parallel (see Fig. 3). The transforma-

H(kOe)

FIG. 2. Magnetization in the X (easy) direction Mx, as a function of an external field H in this direction, for the case of SmFeOs at room temperature. (# e x = 4.6X106 Oe, # d = 6X104

Oe, a n d # x = 170Oe.)

tion is formally given by

Xi=Si cosai+ 7*1 sinai, X2= — 5*2 cosa2+ T2 sin«2, Zi=5isinai—Ticosai, (5) Z2=S2 sina:2+ T2 cosa2 .

In terms of the new variable, (1) becomes

F=JM 02 F i F 2 - (JM02 c o s 2 $ + Z W sin2$)

X (S^+T^+iJMo2 sin2$-£Mo2 cos2$) X (S1T2-S2T1)-iK(S1

2 cos2o:i+522 cos2a2

+ 2Y sin2ai+r2 sm2a2+2SiT! sinai cosai- 2S2T2

Xsina!2 COSQJ2)—HMo(Si cosai+Ti sinai— S2 cosa2

+ r2sina2), (6)

where Xi, Yi, Zi, and Si, Yiy 2\- are the components of unit vectors R; along the sublattice magnetization vectors R»=M*/Afo. The dynamic equations may now be written as

OVM)R;=RiXV,F, *=1,2, (7)

where the gyromagnetic ratio 171 = 1.86X 107 Oe-1 sec-1. In the small-signal approximation, the component of Mi along Si is taken as constant, and (7) yields the set of four equations

M% . dF dF —Ti=Yt S% , * = 1 , 2 , | T | dSt dYi

Mo . dF dF ^ Yi=Sl T% , 1=1,2,

M ST* dS(

(8)

T a k i n g the t ime dependence of the components of M» perpendicular to 5 t- in t h e form eiWt, (8) becomes

coT1+g1Y1+HesLY2=0, cor2+g2F2+^exF1=0, (9)

coFi-&iri+cr2=o, coF2-62T2+cTi=0,

where <a=i(W/\y\), and

c=JQTex c o s 2 $ + E d s i n 2 $ , gi=c+HK cos 2 a i+(— ) i + 1 H cosa,-, bi=gi—HKsin?ai.

6 G. Gorodetsky and D. Treves, Phys. Rev. 135, A97 (1964).

FIG. 3. Equilibrium ~~*"z coordinate system.

155 A N T I S Y M M E T R I C E X C H A N G E I N T E R A C T I O N 455

The condition for obtaining a nontrivial solution of this set of equations is that its determinant vanish. This requirement yields the secular equation

(W/yY+A (W/y)2-B=0, where

A^2cHex-g1bl-g2b2 and

B^(b1b2-c2)(Hex

2-g1g2).

Setting Hd—0 gives, as a special case, the well-known antiferromagnetic modes7 [see Fig. 4(a)]. The two modes obtained from the secular equation are

(W±/yy=K-A±U2+4B)U2l. (10)

This general result can be simplified for the following cases:

(1) In the range H<£B<?/H\\e. Here

A^-(2HU2+Hd

2+2H2) and

giving B^-~(HUc*+Hllc

2Hd2),

fW+\2 r /#iio\2n

(TH'-'-2b>* (ID

7 / ^*ld

(2) In the range Hex2>H^H0. Here,

« i = - ( — < M > a2 = - + $ , c^Hex+(Hd-H) sin®,

g=gi=g2~Hex+Hd sin<i>,

and b=bi=b2=c+H sin<f>-HK cos2*,

giving (W+/y)2^Hd

2+HdH, (W^/y)2^H2+HdH-Hl ,c

2. (12)

It is clear from (12) that the lower mode vanishes at H=Ho. The resonance frequencies as a function of H, for different values of Hd, are shown in Fig. 4. [Fig. 4(e) describes the case for SmFe03 at room temperature].

D PARALLEL TO EASY DIRECTION

The Equilibrium Position

The case wherein D is parallel to the easy axis occurs, for example, in hematite a—Fe203, below its Morin transition temperature (rM=263°K). The case where the external field is parallel to the easy axis, in such a material, is well known.8 A zero-frequency field was

7 C . Battel, Phys. Rev. 82, 565 (1951); T. Nagamiya, Progr. Theoret. Phys. (Kyoto) 6, 342 (1951); S. Foner, in Magnetism, edited by G. T. Rado and H. Suhl (Academic Press Inc., New York, 1963), Vol. I.

8 S. Foner, in Proceedings of the International Conference on

FIG. 4. Resonance frequencies W± as a function of H for dif­ferent values of Hd: (a) Hd = 0 Oe, (b)Hd=10* Oe, (c) # d = 104

Oe, (d) ^ = 3 X 1 0 * Oe, (e) # d = 6 X l 0 4 Oe [(the case of SmFe03 at room temperature), and (f) Hd= 106 Oe. All cases are for # e x=4.6X10 6 Oe and HK=170 Oe.

found at £r|,<J=[(2FeX+i9rx)i3r

x-Jff/]1/2, which is the critical field at which spin flop occurs. Note that for this kind of material, the expression of H\ \ c is different from the corresponding critical field in the previous case. Assuming a uniaxial magnetocrystalline anisotropy EK=Ksin26y one finds, in the two-sublattice approxi­mation, with the applied field^in the plane perpendicular to the easy axis, a gradual rotation of the AF axis (see Fig. 5) from parallel to perpendicular to the easy axis.9 This rotation of the AF axis is caused by the torque due to the antisymmetric exchange interaction.

FIG. 5. Sublattice magnetization vectors.

Magnetism, Nottingham, England, 1964 (Institute of Physics and The Physical Society, London, 1965).

9 G. Cinader and S. Shtrikman, Solid State Commun. 4, 459 (1966).

456 G . C I N A D E R 155

FIG. 6. Magnetization M as a function of an applied field H perpen­dicular to the easy axis.

The free energy for this case may be written in the form

F=/M 1 .M 2 -D-MiXM2-(^ /2Mo 2 ) X(M1*+M2z

2)-H(Mly+M2y), (13)

where D= (OftJ)). Taking Z as the crystal's easy axis (i.e., the [1,1,1]

direction in hematite) and Y as the direction of the applied field, the canting angle <£ and the angle 6 be­tween the AF axis and Z are found to be

sm^HKH/Hu2,

sind^HdH/HUc"

for H<H0

i(Hd+H)/2Hex, iovHex»H^Ho, (14) 2, for H<H0 (15)

^ 1 , for H^H0,

H0=HU2/Hd. (16)

The net magnetization M=2Mo sin<£, as a function of H, is shown in Fig. 6.

Resonance Frequencies

As previously, we use the small-signal approximation and transform to more convenient coordinate systems. The transformed coordinate system used for each sub-lattice is shown in Fig. 7. Formally, the transformations are given by

Xt-= (-)*+1 cos0- tj+ (-Y sin$ sinB-ty* + (—) *+1 cos$ sin0 • £*%

Y»= cos^e^^+sin^E^, (17)

Z*= (—)* sin0e/+(—)* sin$ cos0e/ + (—)i+1 cos$ cosfle/, i= 1,2 .

FIG. 7. Equilibrium coordinates, where X is the easy axis, D = (0,0,D), H = (0,#,0). £/||Mi, zj is tangent to the base bound­ary of the cone, so that when 0=0 we have £ / | | (—) m X. ty

i is denned so as to make (£**,£/,£/) a right-hand set. |£&*|=1; k = x, y, z; i=l, 2. 0 is the angle between the AF axis and the easy axis Z. <j> is the canting angle between Mi and the ZX plane.

By straightforward calculations, we obtain

(W±/y)2= (B^A)(C±Hex)- (G=F708, (18)

where

A = HeX cos2<£+#d sin2<£ sin0, B=A+H sm$+HK cos2<i> cos20, C=A+H sm<f?+HK(cos2$ cos20-sin20), G=Hd cos$cosd, and F=%HK sin<£ sin20.

For # = 0 , (18) may be written as

(W+/y)2^HUc2+H2, (19a)

(W-/y)*9*Hu*. (19b)

This is the result that is obtained for the case of the antif erromagnet.10 This equality is to be expected in this range as E d = - D - M i X M 2 ^ 0 for H^O. In the range

yHnc

h

-

, l .. J.

^ /w« .

~ ^~-̂ \w-

i' i V i

yS '

~

J 30 60

H(kOe)

H0 90 120

FIG. 8. Resonance frequencies W± as a function of H for the case of a—Fe203 below its Morin transition temperature with # e x =10 7 Oe, # d = 2X104 Oe, and ^ = 1 0 0 Oe.

Hex^>H^ Ho, we obtain

e> H2+HdH, y

fWJ l — )^Hd(H-H0),

(20a)

(20b)

which corresponds to Fink's results11 above the Morin transition point, as, it should. From (20b), it is clear that the lower mode PK_ is equal to zero at H=H0. The resonance frequencies TF±, as a function of H, are shown in Fig. 8 for # e x=107 Oe, Hd=2X10* Oe, and HK=100 Oe (hematite at about 240°K). A numerical check of the first derivative of W+ with respect to H shows a discontinuity at H=H0.

12

10 F. Keffer and C. Kittel, Phys. Rev. 85, 329 (1952). 1 1H. J. Fink, Phys. Rev. 133, A1322 (1964). 121 am indebted to R. Hornreich for calling my attention to

the existence of this discontinuity.

155 A N T I S Y M M E T R I C E X C H A N G E I N T E R A C T I O N 457

CONCLUSIONS The critical field H0 is quite large at most tempera­

tures. The simplest method of obtaining such high fields is by utilizing pulsed-field techniques, but, with this method, it is difficult to measure the weak ferromagnetic moment directly, in the absence of spin flop. A measure­ment of Ho using a magnetic-resonance method could avoid this difficulty. Using such magnetic-resonance studies, the theoretical rotating AF-axis picture could be investigated. Also, more information about the canting mechanism, especially at low temperatures, could be obtained, and, from a knowledge of HQ(T), it would be possible to find the temperature dependence of the uniaxial magnetocrystalline anisotropy constant.

I. MODEL

A THEORETICAL discussion of the cooperative orientational ordering of ortho-H2 molecules on

both face-centered-cubic (fee) and hexagonal-close-packed (hep) lattices has recently been given by Raich and James.1,2 For the case of a rigid fee lattice it was shown that the internal-field approximation (or molecu­lar-field approximation) leads to a first-order phase transition between an orientationally ordered phase, stable at low temperatures, and an orientationally dis­ordered phase, stable at high temperatures.

The purpose of this paper is to indicate how the technique of temperature-dependent Green's functions can be applied to a model of the ordering of ortho-H2

molecules on a rigid fee lattice, valid within the frame­work of the internal-field approximation. It is shown

1 J. C. Raich and H. M. James, Phys. Rev. Letters 16, 173 (1966). This paper gives a list of references to previous experi­mental and theoretical work on this problem.

2 H. M. James and J. C. Raich (to be published).

To calculate the resonance modes in the case where the applied field deviates slightly from the special directions considered here is somewhat more difficult as the sym­metry of the equilibrium configuration is reduced. All the above calculations were performed at essentially 0°K using the molecular-field approximation.

ACKNOWLEDGMENTS

I am indebted to Professor S. Shtrikman for sug­gesting the problem and for continuous guidance throughout its solution. I would also like to thank R. Hornreich and Y. Tennenbaum for helpful advice and discussions.

that this technique leads to the same results as the more conventional one, based on the minimization of the free energy.

Leaving off the rotational energy of the molecules, the Hamiltonian for the model to be considered is

ff=iEr*(Q<,0/), (1) *»/

where Q*= (Q%,<l>i) specifies the orientation of molecule i. If we assume that the orientational coupling of the molecules arises from quadrupole-quadrupole coupling, the potential energy of interaction of molecules i and j , Vij can be written as3

F^=(207r/9)(707r)1/2I\i £ C(224; MN) MN

XY2M(Qi)Y2N(&J)YitM+N(aiJ)*. (2)

3 H. P. Gush and J. Van Kranendonk, Can. J. Phys. 40, 1461 (1962).

P H Y S I C A L R E V I E W V O L U M E 1 5 5 , N U M B E R 2 10 M A R C H 1 9 6 7

Orientational Order in fee Solid Ortho-H2. Greene-Function Treatment of the Internal-Field Approximation

J. C. RAICH

Department of Physics, Colorado State University, Fort Collins, Colorado

AND

R. D. ETTERS

Douglas Advanced Research Laboratory, Huntington Beach, California (Received 17 October 1966)

The technique of temperature-dependent Green's functions is applied to a molecular-field model of the orientational order-disorder transition of ortho-H2 molecules on a face-centered-cubic (fee) lattice. The chain of equations for the various Green's functions is decoupled by an approximation similar to the one used by Tahir-Kheli and ter Haar, which enables one to solve the three remaining Green's-function equations of motion. I t is shown that the Green's-function approach leads to the same results as the more conventional methods.