Journal of Magnetism and Magnetic Materials 123 (1993) 255-261 North-Holland

Spin wave excitations in canted body-centered orthorhombic antiferromagnets

P. T h a l m e i e r

Physikalisches Institut der Universith't Frankfurt, 6000 Frankfurt, Germany

Received 19 June 1992; in revised form 20 October 1992

A model for the spin Hamiltonian of body-centered orthorhombic (bco) antiferromagnets with strong intra-/interlayer exchange anisotropy and a canting due to antisymmetric exchange is discussed. Closed solutions for the AF resonance modes in a four-sublattice model are derived and discussed. It is found that single-ion anisotropy and especially interlayer exchange strongly influence the soft mode behaviour of the AF resonance modes. This would in principle allow determiation of the interlayer exchange in systems with weakly coupled AF layers. In addition the spin wave dispersion for the bco structure is derived for the general model including all anisotropies and the asymmetric exchange.

1. Introduction

The magnetic properties of the body-centered orthorhombic(bco) antiferromagnetic (AF) com- pounds La2MO 4 (M = Cu, Ni) have attracted a lot of attention [1-5]. Their magnetic behaviour is quite subtle due to an interplay of intra(inter)layer exchange, their anisotropies and the Dzyaloshin- ski-Moriya (DM) asymmetric exchange. The lat- ter leads to a canting of spins out of the MO 4 planes and the possibility of a metamagnetic tran- sition. To investigate these various interactions through spin wave excitations by AF resonance method would be important but attempts have so far been unsuccessful.

It is the aim of this paper to study some aspects of spin waves in the canted bco AF struc- ture as illustrated in fig. 1. The field dependence of k = 0 modes is calculated for B parallel to the easy axis c in a classical four-sublattice model. In ref. [5] this has been done for anisotropic ex- change appropriate for the above compounds. Here we consider the situation of isotropic ex- change which is more interesting because it al-

Correspondence to." Dr. P. Thalmeier, Physikalisches Institut der Universit~it Frankfurt, 6000 Frankfurt, Germany.

lows the occurrence of a soft mode. The influ- ence of interlayer exchange on the soft mode behavior is investigated. This part is a generaliza- tion of an earlier two sublattice calculation [6]. A similar though different four sublattice model has been applied to the copper dichlorides [7].

Furthermore the spin wave dispersion for the AF structure in fig. 1 using the most general model will be calculated within a standard quan- tum mechanical RPA procedure. The results may be useful for bco AF systems with a larger inter- layer exchange than in the LazMO 4 compounds.

2. Four-sublattice model of the canted AF

The magnetic structure relevant for all follow- ing discussions is shown in a projection along the orthorhombic b-axis in fig. 1. The general spin Hamiltonian for this structure is

H=_~.,(KaSX2+KcS~2)_2. , £ i~SiS~iJ ~ ,~ i (ij)a

1 , , S j ) - ~ E I i jS iSi - ~ E D ° ' ( S i × (6) (ij)

-- g•BB E S ,. (1) i

0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

256

• o • o •

[ ] o • b

b , z : I a c - p l a n e

Fig. 1. Projection of the bco AF structure (a c for simplic- ity) along the b-axis. Moments point essentially along [010] with a small canting out of the ac plane. The conventional magnetic unit cell as well as a basis for the four-sublattice model (connected by dashed lines) are indicated. The primi- tive unit cell for the base-centered orthorhombic A, B sublat- tices in the simplified two sublattice model is also shown

(left). Its translation vectors are a,/~, c with [~ = (c + b ) / 2 .

The various terms describe single-ion anisotropy (K); in-plane (ac ) exchange ( I ) anisotropic in spin space; interlayer exchange ( I ' ) ; Dzyaloshin- ski-Moriya interaction (D) and Zeeman term respectively. ( ) denotes nearest neighbors (in- plane) and ( } the next-nearest neighbors (inter- plane). Furthermore, i = (l,r), where l = 1 . . . . . % is the lattice index of any given sublattice ~- ( r = 1 . . . . . n L = 4), i.e. N = n L N ~ is the total number of sites. The k = 0 spin wave modes will be calcu- lated in classical approximation for which the mean field energy per unit cell corresponding to (1) is needed. Introducing the sublattice magneti- zation M, with M~ = (ST"> in units of gP'B, where a = x , y, z or a, c, b, the energy can be written as

E T : - K a E M x 2 - K~EM¢ 2 7" "7

- E J ~ ( M T M g + M ~ ' M 2 ) o,

- J ' ( M , + M 2 ) ( M 3 + / 1 4 4 )

- D ( M t X M 2 + M 3 X M 4 ) - g l z B B ~. ,M~. "r

(2)

The connection to the interaction constants in (1) is given by (a = a, b, c): J , = z I , ; J ' = z ' l ' ; D =

z ' D o = D x where z = 4 and 2 z ' = 8 are the num- ber of nearest (in-plane) and next-nearest neigh-

P. T h a l m e i e r / Sp in waue exc i ta t ions in b c o a n t i f e r r o m a g n e t s

bors (planes above and below), respectively (fig. 1). An additional term due to an orthorhombic real space anisotropy J j 4 : J ] of the interlayer exchange would be - j ' (M 1 - M e) ( M ~ - M 4)

with j ' = ( J 2 - J ' ) / 2 . The term is important be- cause the sign of j ' decides whether a metamag- netic transition for B IIb occurs [5]. For the situa- tion discussed in the following (B 1[ c), however, j ' can be neglected.

3. Equilibrium configuration for B II c

The directions of sublattice moments are de- termined by minimization of the energy given by (2). It is convenient to introduce effective interac- tion fields in units of Tesla:

H .... = 2 K , , , c M , H~_~ = - J . M ,

H d = D M , h c× = - y ' M . (3)

where M is the sublattice saturation magnetiza- tion in units of g/xB. In the following an isotropic exchange Hex will be assumed. The canted mo- ments will then stay in the bc plane and turn continuously until the resultant moment of the sublattices becomes parallel to the field at B,,.

This configuration is shown in fig. 2. The moment directions are characterized by two angles (0, ~b)

1.4 i / ....~ ~ i14

12 [- B,~ - ~ : o / / . . - ~ " 12 - 4

08- / " / ' / / b~ ~o~,, /~ 0 8 . ~

0 0.6 i - - - ~" 0.6 ,~

0.4 ~ ~ " r~, ~ \ ' - , \ 1 0.4

0.2 I- .... \m 0 2 4 6 8 10 12 14 16 18

B [T]

Fig. 2. Field dependence o f the t i l t ing angle 0 (ful l l ine) and canting angle ~ (dashed line) describing the four-sublanice equi l ibr ium conf igurat ion with moments m~. A standard set of effective fields 1t,:, - 10 ~ T, h~.~ - 0.5 x 102 T, f t a - 10 T, H c = 0.1 T, H a - 0 (uniaxial anisotropy) was taken. At the

critical field B ~ = l l . 7 3 T o n e h a s 0 "rr/2.

P. Thalmeier / Spin wave excitations in bco antiferromagnets 257

or a 1 = 4' - 0, a 2 = 4' + 0. The total energy then reads

ET( O, 4' ) /gtXBM

= -2He(cos20 - sin24' cos 20)

- 2Hex cos 24'

- 4hex cos 20 sin24'

- 2 H d sin 24' - 4B sin 4' sin 0. (4)

Because Hd/Hex << 1 the canting angle will al- ways be small and it is sufficient to expand ET(O , 4') up to second order in 4'. Minimization then leads to the equilibrium configuration

B <Be ,

B4' B4' sin 0 = ---

H ¢ - 24'2( H ~ - 2hex) H c

H d + B sin 0 4'=

2Hex + ( n c - 2hex ) cos 20

--4'0 1 + 2Hc(Hex-hex) . (5)

The approximations hold for small fields B << B e; furthermore, 4'0 = Hd/(2Hex - 2h~x + H c) is the zero field canting angle. At Bc, M 1 + M 2 and M 3 + M 4 are parallel to B, for larger fields one has constant 0 and a canting 4' linear in B.

B >Bc ,

Hd+B 0 = aT/Z, 4' = 2(Hex + h e x ) _ H c . (6)

In both cases the magnetization per ion is given by Mi/i.t B = gM4' sin 0. The critical field is ob- tained as

Hex + hex ( 2hexHd

Be n e x _ he x nex + hex ½nd

1 2 1/2} +(~Hd + 2Hc(Hex-h~x)) . (7)

In the case hex = 0 (effectively the two-sublattice model of ref. [6]) this reduces to the known result

B e - H J 2 + (14o2/4 1/2 = + 2HcHCx ) •

In fig. 2 the field dependences of the canting angle 4' and tilting angle 0 are shown for a given parameter set. In the previous considerations the moments stay in the bc plane and turn around the a-axis when B is increased. If there are anisotropic exchange fields and Hax - Hbx is large enough there may be a spin-f lop transition out of the bc plane for a field smaller than B c. This situation was discussed in ref. [5] (note that B c is equivalent to B~ in ref. [5]).

4. Field dependence of spin wave frequencies (BIIc)

The central topic of this paper is the calcula- tion of field-dependent spin wave modes for the canted four-sublattice model introduced above in the configuration of fig. 2 and defined by the energy in eq. (2). This is done for isotropic ex- change Hex where no out-of-plane spin-f lop oc- curs. This calculation is a generalization of the one done for the canted two-sublattice model, adding the effect of interlayer exchange and the AF ordering of resultant moments. The geometry and equilibrium configuration is that of section 3. As in the case of the two-sublattice model it is convenient to transform m~ = M , / M to a local coordinate system m, --, (YT, S,, TT) to set up the equations of motion. For the (m~, m 2) pair one has

max = YI ,

mly = - S 1 cos O~ 1 - - T 1 sin a 1, (8a)

m l z = S l sin al - T 1 cos 0~1,

m 2 x ~ Y2 ,

m2y = S 2 cos a 2 - T 2 sin 0/2, (8b)

m2z = S 2 sin a 1 + T 2 cos a 2.

For the other pair replace (1, 2) ~ (3, 4) and al, 2 --+--a3, 4. The total energy e T = E T / M for the excited configuration can be separated into intra- and interlayer contributions whose very lengthy form can be obtained from (2) and (8), but is not written explicitly.

e v ( V , , S,, T,, 0, 4') = e(1, 2) + e(3, 4)

+ e ' (12, 34). (9)

258 P. Thalmeier / Spin wave excitations in bco antiferromagnets

Again e(3, 4) is ob ta ined by the r ep lacemen t de- scribed above. In the equi l ibr ium state one has Y = T~ = 0, S~ = 1 and O(B), 4,(B) given by (5) or (6). The spin wave modes and f requencies are de te rmined by the classical equat ions of mot ion

(T = g / z B / h ) :

1 aT~ OC T a,6 T

1 0Y_ 0e-!. 0e I, (i())

with ~. = (T 1, T 2, Y!, Y2) and ~" = (T 3, T 4, Y3, Y4) the l inearized equat ions of mot ion (S~ = 1) can be wri t ten as (T,, Y~, ~ exp(iwt) , S'2 = w / y ) :

A ~ + B ~ = 0 , B ~ + A ~ = 0, (11)

0 A =

- b I

c

0

0 B =

- h 1

h o

/ i J0 He× c iJ'2

- b 2 0 i.O )

0 hex he~ /

1) h~x h0~ h o 0 "

h 2 0 0 )

The various matrix e lements contain interact ion constants and equi l ibr ium angles (c~!, a 2) and (0, 05) f rom (5) or (6) and are given by

c H~× c o s 2 4 , + H d s in24 , ,

gl = ¢ + Hc c°s2°z! -1- B cos ol!

/ % ( c o s 2c~ l - cos 20) ,

g2 = c + H c COS20~2 - - B cos 0{ 2

- he , (cos 2(x 2 - cos 20 ) ,

g~2=gl . x - Ha, b!,2= g!.2 - Hc sinZce!.2

h o = h~x cos 20, hi. 2 = hex cos 2o<1. 2.

12)

The four spin wave f requencies are then solutions

of secular equat ions det( A + B) = 0 and det (A - B) = 0. Explicitly, they are given by

= + + 48 , ), = ' , ( - A , _+ + 4 8 , ) , 113)

AI = 2(c + h o ) ( Hex +h~, ) - (b I + h l ) (g i ' + h . , )

- ( b 2 + h 2 ) ( g ~ +h~x).

B! = [ ( b ! + h l ) ( b 2 + h 2 ) - ( c +]10) 2 ]

× ((He× + h,.×) 2 ( g T + h e × ) ( g ~ + h e × ) ] ,

A, = 2 ( c - h o ) ( l l , . , - l M ) - ( b! - h , ) ( g7 - I t< , )

(b2-h~)(g~ - he , ) ,

B 2 = [ ( b ! - h , ) ( b 2 - h : ) - ( c - h o ) 21

× [ ( H ~ - h e , ) 2 - ( g ' ! ' l % ) ( g ; ' - h e x ) ] .

Equat ions (12) and (13) const i tute the comple te analytical solution for the canted four-sublat t ice AF in the field conf igurat ion of fig. 2 and for isotropic exchange. Notice that A~, B i are ob- ta ined by subtract ing large but a lmost equal num- bers. For numerical calculat ions it is therefore advisable to pe r fo rm this subtract ion analytically by expressing the A,, B i through c ' c - H e x = Hex(COS 24, - 1) + H d sin 24, ra ther than by c it- self. The result ing expressions in ( , however, are very lengthy and are not given here.

To obtain a feeling for the possible excitations, numerical calculat ions for various cases with dominat ing in-plane exchange H~, >> h~x, H d, H, , H e have been pe r fo rmed . In fig. 3 the AF reso- nance f requencies for a fixed set of Hex, H,!, H~., H,~ = 0 (uniaxial) and increasing inter layer ex- change he~ are shown. For he, = 0 (a) one obtains the (now twofold degenera te ) modes of the two sublatt ice model; a finite he, (b) couples (ra l, m,) to (m 3, m 4) and leads to a splitting of the degen- era te modes . Especially the soft modes -(21. 2 (h~× = 0) split rapidly with increasing he× such that

,O~ stays soft and S2] becomes finite. The split- ting A . , = ~ 2 2 ( B c ) - I 2 ~ ( B ~) as a function of he~ is shown in fig. 4. Al ready for he×/He, -- 10 3 the

1). Thalmeier / Spin waue excitations in bco antiferromagnets 259

24 a)

20 S2 IT]

16

12

8

4 0 ] ~ I ~ I I I I i

24 Bc b)

ET] 16 ~ ~-

1248 Q ~

0 i I I t I |1 I I 0 2 4 6 8 10 12 14 16 18

B I T ]

Fig. 3. A F spin wave resonance frequencies (in effective field units, O = (0/3, ) in the four-sublatt ice model. For he, = 0 one recovers the result of the two sublattice model with a twofold degenerate soft mode at B c (cf. fig. 2). For he× 4:0 the degeneracy is lifted, especially p ronounced for the soft mode (see also fig. 4). The effective fields are Hex = 103 T, n d = 10

T, H c = 0.1 T, H a = 0 (uniaxial) and hex, as indicated.

splitting ratio As/g2~,2(0) = 0.63 X 10 - t is quite large. Thus a determination of As(B c) should be a sensitive way to determine the interlayer cou- pling in the canted bco AF.

10

- B=Bc 8 -

" ~ 4 4 I

2 2 /~ hcx~ t 5 T

8 10 t2 B IT] 0 I I I t I J I I i

o 20 40 oo 80 loo

hex [T] Fig. 4. Soft mode splitting in the uniaxial case at B c as a function of interlayer exchange hex; o ther parameters as in fig. 3. Inset shows indentification of A s = 122 -- O { soft mode

splitting.

n i T ] . a)

1 6 -

12

8

4

0 I i I I Ii I i

S~[r] Bc b )

12

8

4

0 I I I I I 11 I I

0 2 4 6 8 10 12 14 16 18

B IT] Fig. 5. Inf luence of nonuniaxial anisotropy ( H a =~ 0) on the A F resonance frequencies. At H a = 0.1035 T three soft modles at B c which signify the onset of an out-of (bc)-plane spin-f lop . Again He, = 103 T, hex = 50 T, H d = 10 T, Hc = 0.1 T (same

as fig. 3b) and Ha, as indicated.

In fig. 5(a) the influence of nonuniaxial anisotropy where H a 4:0 is shown (cf. fig. 3b) If H a reaches a critical value one has the curious situation of three soft modes (fig. 5b). For even larger H a two of the modes become unstable in a region around B c. This means that the inplane (bc) ground state configuration (fig. 2) is no longer stable, i.e. one expects an out-of-plane spin-f lop below B e. This would also be the case if one assumes a sufficiently anisotropic exchange in- stead of a single ion anisotropy [5].

For zero field, compact formulas for the spin wave frequencies in the uniaxial case ( H a = 0) may be given:

02 + =

s~2-=

g2~ + =

0 2 - =

For hex

H~x + h ex 2 Hex_ hex Hd + 2Hc( Hex + hex)'

2Hc(Hex + hex ) , (14)

H 2 + 2 H c ( H e x - h e x ) ,

2 n c ( n e x - h e x ).

= 0, 0 2+ and 0 2 - (i = 1, 2) are degener-

260 P. Thalmeier / 5"pin waue excitations in bco ant(ferromagnets

ate and one recovers the result for the two-sub- lattice model [6] with 122+= H 2 + 2H~H~x and ~1~-= 2H~H¢×. As already mentioned, the split- ting of these two modes, especially at B C, would provide a direct measure for the interlayer ex- change h ~ (figs. 3 and 4). One might suppose that a measurement of the spin wave dispersion along b could also be used to determine h~x directly. As it turns out in the next section, this is not possible in the bco structure.

5. Spin wave dispersion for the canted bco anti- ferromagnet

A standard quantum mechanical equation of motion technique in RPA was used to calculate the spin wave dispersion for the model Hamilto- nian (1) in zero field but now with generally anisotropic I~ . In the limit D / I ~ << 1, the cant- ing angle 4' is small and one can treat the prob- lem as a two sublattice (A, B) model, i.e. neglect the doubling of the unit cell along b due to the canting. As a consequence the zone boundary modes in the (A, B) model correspond to two zone center modes (k = 0) in the four-sublattice model. The dynamical degrees of f reedom are described by the deviation variables o2~ = S t - M A, tr m = S m - M B, where l, m run over the A, B sublattice, respectively, which have each approxi- mately base-centered orthorhombic structure in the case that lattice constant a and c are almost equal. The Fourier- t ransformed variables

Ak = U, ' /=Ee~*~'¢, , Bk = N , - I / = E e i * R ' c G l m

(15)

then fulfil the quantum mechanical equations of motion. Using a standard RPA decoupling and transforming to circularly polarized A~ =A~, _+ i A~, etc., whose time dependence is given by Aft - -exp(_+i to t ) , etc., one obtains the coupled system of equations (12 = w/T) :

- a~ ( 17 + s~) a~ /7 ' ) zl~ - I'" - k a~ ( / 7 - ~)

[ A; A~

× B{ =0. (16)

BX

with matrix elements given by

I~ ~ = ½(H~, + H~x)y k + hexy ~. + ½&ondYk,

I ~ = 1 t ex~/k ~( H,, + H(~) + H~, + h '

+ ½GHa( l + ~k), I a = - - ' ~&oHdYk, (17)

• 1 t 1

~&oHd( Yk A'k = 7( H,, - Hb) -- l -- ).

Here it has been convenient to use a different form _ ( , x2 , ~2 K.~S i + KbS f ) for the single-ion anisotropy term. The connection to (1) is given by K~I = K,, - Kc, K b = - K~ and for the anisotropy fields H , ~ , b = - - 2 K~',,b; accordingly, H~ = H c - H~, and H~ = H c.

The dispersion enters through the structure factors of the sublattices

~k = [COS w ( k J k ] ) + cos ~r (ky /k~! ) ] /2 ,

7~. = cos ~r(ky/2k~,).)cos ¢ r ( k : / k ° ) , (18)

= c o s (kjZk ) c o s

where kl~ = ~v/a, k~= w/c , k~ = w / ( b / 2 ) . The wavevector k lies within the BZ corresponding to the base-centered orthorhombic unit cell (a - - c ) of one sublattice. The frequencies of the two spin wave branches are then obtained from the secular equation of (16), explicitly:

+ : ( r ; 2 + 2 ) - ( r f +

+_ 2( r ;~7 , - t~a~,) . (19)

Equations (17-19) are the closed solution for the spin waves of the model described by (1). It is interesting to look at certain limiting cases:

(a) Isotropic exchange, uniaxial anisotropy and no DM term, where H a , b = H c , H~x = Hex, H d = 0, which leads to two degenerate spin wave branches:

a 2 = 2H~Hex + H~x(1 - y~)

+ 2he×Hex(Y k y~) 2 c2 c _ + h e x ( , Y k _ ,ya2~ k 1"

(20)

P. Thalmeier / Spin wave excitations in bco antiferromagnets 261

(b) same case with DM term for k = 0, k0:

122_(k=0) =12~ +, a2+(ko) = a ~ - , (21)

+,

where k o = k°z is a zone boundary wavevector. Thus the two zone-center and two zone-boundary modes obtained here are identical to those four k = 0 modes from the four-sublattice model (B = 0) in the clasical method (cf. 14).

(c) Dispersion I[ b, k = kzz ( H a = 0, Ha, b - Hc), in which case k is perpendicular to the AF layers and Yk = 1, y~ = y~ = cos Ir(kz/k °) (a = x, y):

g]2- = 2[ Hc + ( H ~ _ H~)]H~x 1 a b t + 2(H¢-~(H~x-H~x))hexyg,

Of + = 2[ n¢ + ( n : c - nbx)] nCx

1 a Hex)) h exYl¢" (22) + 2 ( H c + $ ( H , x - o ,

The magnitude of the dispersion A J2 z = O2(ko) - giZ(k = 0) is then given by

AO2:~= 4(He T- (H~x-Hbx)/Z)hex. (23)

If there is no single-ion and ab plane exchange anisotropy there is no dispersion for k 11 b even though hex 4= 0. This is due to the fact that in the bco structure one has top to center configurations of moments in adjacent AF layers (fig. 1), i.e. two parallel and two antiparallel next-nearest neigh- bor moments in the adjacent plane with respect to each moment. Therefore in the case without anisotropy the exchange fields for k II b modes cancel and no dispersion results. Thus according to (22) the dispersion along the b-axis does not give direct information about the magnitude of hcx, but is also dependent on the anisotropies. On the other hand, for a simple orthorhombic AF structure with top to top positioning of moments one would expect that AI22---h2x, providing a direct measure for hex.

6. Conclusions

A discussion of a model spin Hamiltonian has been given which is appropriate for canted bco antiferromangets. Closed analytic solutions for the AF resonance modes have been derived within a classical four-sublattice calculation including the effect of interlayer exchange. It was found that for suitable anistropy fields a soft mode occurs at a critical field B c connected with the in-plane turning of sublattice moments similar as in the two-sublattice model. Its twofold degeneracy is lifted by small interlayer exchange. Observation of AF resonance frequencies around Be would thus be an excellent method to measure small interlayer exchange couplings in bco antiferro- magnets.

In addition the general expression for the spin- wave dispersion for the realistic case of small canting angle has been given. It is found that the dispersion along the orthorhombic axis is gener- ally small due to the bco structure where ex- change fields cancel and will not provide a direct measure for the interlayer exchange.

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