Solid State Communications, Vol. 44, No. 6, pp. 801-804, 1982. Printed in Great Britain.

0038-1098/82/420801-04503.00/0 Pergamon Press Ltd.

SPIN-WAVES IN TWO-DIMENSIONAL ANTIFERROMAGNETS OF THE K2FeF4 TYPE

Yu.B. Gaididei and V.M. Loktev

Institute for Theoretical Physics, Academy of Sciences, Ukr. SSR, Kiev-130, U.S.S.R.

(Received 19May 1982 by E.A. Kaner)

The spectrum of spin-waves in two-dimensional antiferromagnet of the K2FeF4 type is calculated taking into account the total single-ion anisotropy and lattice distortions. The formulas for the AFMR frequencies are obtained and it is shown that the observed values cannot agree with theoretical ones without deformations being taken into account. The tem- perature dependence of strain tensor is found in the region of the spin- wave approximation validity.

1. MUCH INTEREST has recently been generated in studying 2-D antiferromagnets whose general chemical formula reads asA2XB4 (A = Rb, K;X = Mn, Fe, Co, Ni; B = F, C1, Br). Two-dimensionality of magnetic pro- perties of these crystals was put forward by Birgeneau et al. [1] based on the fact that there is no dispersion of the spin-waves in one of the crystallographic direc- tions. The elastic properties of such crystals seem not to have this character.

All the above crystals have the axis C4 orthogonal to which the planes of magnetic ordering are situated. In most cases they refer to an "easy axis" symmetry type; the exception is the crystals on the basis Fe 2+ (K2FeF4, etc.), with the magnetic order symmetry of the "easy plane" type. The above case is due to the crystal field effects and the spin-orbital interaction in rather single- ion anisotropy putting spins into basis plane. Until recent years there has already been a great number of experimental and theoretical papers dealing with the description of the behaviour of the spin-wave spectrum of these crystals [2-4] . The peculiarity of this spectrum is a comparatively large gap in the lowest branch of magnons. Its nature has not been practically discussed in literature. In theoretical papers the appearance of this gap is caused by adding to the Hamilton operator the effective (staggered) magnetic field which changes its direction relative to the magnetic sublattice. In [5] it was shown that the cause for the gap in the magnon spectrum of 2-D crystals of "easy plane" type may be the magnetoelastic interaction leading to the orthor- hombic distortion of a square plane. As a consequence, two domains with mutually easy axes of magnetization in a plane may appear. The existence of such domains was mentioned in [6].

The calculations of the magnon energies in such sys- tems using Holstein-Primakoff transformation lead to the certain disagreement with the Goldstone theorem:

the gap in the spectrum does not vanish if the in-plane anisotropy disappears [2]. Many efforts have been made to obtain the correct result, though in [7] it was shown that quantum mechanical self-consistent calculation considers completely the single ion anisotropy and deter- mines the correct limit behaviour.

In the present paper we perform the quantum mechanical calculation of the spin-wave spectrum in these crystals assuming anisotropy to be small in com- parison with the exchange interaction and show that just the lattice deformations are important in stabilizing the 2-D ordering. In the region of the spin-wave approxi- mation we also estimate an additional temperature con- tribution to the energy of magnons which arises due to the temperature dependence of the tensor of homo- geneous deformations.

2. The crystal field in which the Fe 2+ ion occurs consists of cubic and tetragonal parts and can be represented as [81

= B404 + (1) B2 02 + B4 04, ~ ¢ ~ . f . 0 0 0 0 4 4

where Bn m are the crystal field parameters, and On m are the irreducible spin operators (the Stevence operators) proportional to the corresponding spherical harmonics. In order to obtain the operator form of the single-ion anisotropy, it is necessary to find the effective spin Hamiltonian of the Fe 2+ in the tetragonal field using the operator (1) and the operator )~LS of the spin- orbital interaction ()~ is the constant, L and S are the orbital and spin momenta operators). Such a calculation with an accuracy to the fourth order in the spin--orbital interaction was performed in [9] where it was shown that the spin Hamiltonian sought for has the form

= Y. = + h.c.]} n, ln

801

802 TWO-DIMENSIONAL ANTIFERROMAGNETS OF THE K2FeF4 TYPE Vol. 44, No. 6

+ ½J . ~ p Snc~Sme+P (2)

(n and m are the lattice point vectors, a is the magnetic sublattice index, 6"4 It OZ, p connects the nearest neigh- bours) and the following constants were estimated: D ~ 10a ~_ 4.2 cm -1, e ~ 0.018 cm -1. The value of exchange interaction of ions can be determined for example from the AFMR data. It is not difficult to see that an identical operator can be written down for the symmetry reasons, in this case the in-plane anisotropy is provided by the operator proportional to the constant e > 0 which leads to the spin orientation along the axes OX and OY. In the paper [9] an attempt was made to calculate the magnon spectra using the Holstein- Primakoff transformation. The variant of decoupling proposed in this paper was incorrect and the fourth- order operator was decoupled inaccurately: (S÷) 4 + (S-) 4 -+ ((S+) 2 + (S-) 2 ) [(S+) 2 + (S-)2], where ( . . . ) is the thermodynamieal average with the operator (2). Thus, it follows that the operator of the D4h symmetry resulted in the orthorhombic distortion with D2h sym- metry, and the constant before this operator should exceed e nearly by an order. It was necessary to associ- ate the spin orthorhombic distortion with asymmetry of the electric field gradient on the nucleus Fe which was determined from the Mtissbauer effect [10]. The depen- dence of the value of the electric field on a nucleus on the crystal direction proves to be induced only by the elastic deformation tensor distinct from zero whose appearance, as is shown in [5 ], is associated with the magnetoelastic interaction:

afme = ~ Bu x 2 :~ 2 • . [ ( s m e ) - ( s . o t ) l - ½Jnu Y~ n ~ IICCj

X X x ( S , ~ Sme+l~ - ~ Y " - - ( 3 ) Snc~Sna+p), U =Uxx Uyy,

where B indicates the dependence of crystal field on the value of homogeneous ion displacements and r/describes the value of magnetoelastic interaction due to the mag- netic dipole-dipole interaction• The sign of the B and 7/ are determined by the relation of elastic moduli of a crystal. According to [5] ,1 ~_ 0.01 cm -1, as to the value of B, for microscopic reasons it is difficult to estimate this parameter, so we shall consider it as a phenomeno- logical constant•

3. We calculate the spin waves spectrum of the operator afs = a£ef~ + afm.~, according to [7]. We separate the single-ion part in the operator afs having divided it into two parts

a{'s = E ~-.".' + ½ ~ Vme mO (4) noz~

where afme describes the spectrum of the ion with the spin S in the self-consistent field of the environment, besides, we represent

~Cme = ~C. ~°~ + Vine; (5)

~f(o) = --JzsS~a; (6) n ~

A + 3Bu A --Bu vine - - - ( s ~ y - Jzr~usS~ot

2 2

x [(S2ot) 2 -- (S~)=I - e { [(S~ot) 2 - (S~ot)=] 2

a

+ [S~,~S~ + h . c . ] =} + ~- {S(S + 1) - - ( S ~ ) :

-- [ (S~) = -- (SnYa)=]} =, (7)

where z is the number of nearest neighbours, s is the spin projection value in the ground state of the operator (5) onto the quantization axis which, for convenience, is chosen in a plane. The eigenfunction of equation (5) have the form

~ - M s ) ( S , Ms) = ~, CMsM~IMs)n~, M~

where [Ms)me are the eigenfunction of the operator SZ~. For the arbitrary spin S we failed to find the exact func- tion '#~-Ms)(s , Ms). However, we can do it considering Vine as perturbation. In the first approximation it is suffi- cient to restrict ourselves by the two nearest states qs(n°)(S; S) and +,'(.~(S, S -- 1) so that in the same approximation s = S; then from equations (6) and (7) we have

~I'(n°a)(S;S) = IS)me + ( 2 S - 1)

A --Bu -- 3e(S -- 1)(2S -- 3) + a(3S -- 2) X

4JzS

x IS -- 2)me; (8)

xI ( 1 ) ( K ' ' S - - 1) m I S -- 1 )me.

We substitute S~ot = xbno t + yb~a , where

x = ( , I ,g>(S;S) I S ~ I ' I ' ~ ( S ; S - 1) );

y = < , I , ~ > ( s ; s - I ) I S~l*~>(s;s)>

and the operators bn,~ and b ~ are the Hubburt tran- sition operators, but at T ~ T N they satisfy the Bose commutation relations. Then it is easy to write the operator (4) in the form

Jfs = E k e b ~ b n ~ + J Z not n t p

x x + y y . x {(1 --r?u)SnlSnl+ P SnxSnt+P},

Ae = JzS(1 +'qu) + ( 2 S - - l)~[A + 3Bu + 5e (S - - 1)

Vol. 44, No. 6

x (2S -- 3) + a(3S - 2)].

Using the k-representation we can obtain the AFMR frequencies from equation (9) in a usual way:

~2~(k) = / A e + [ ( 2 S - - 1 )

A --Bu -- 3e(2S -- 3)(S - 1) + a(3S -- 2) X ""

2JzS

flu JzSTO¢) --J2z2S2720¢ ) 1 - ' (10) 2

1 eikp ) = -: Z Z P

In the range ka "~ 1 we obtain from equation (113)

a~(k ) ~ ~2~(0) + ~J2z2 S~(ka)2, (11)

where

~_(0) = 2 (2S- - 1) I[Bu + 2e(S-- 1)(2S-- 3)1

x Jz S + B u + 2 e ( S -- 1 ) ( 2 S - - 3)1 / ; 2 S - - 1

~2+(0) = (2S - - 1) { [A + Bu + e(S - - 1)(2S - - 1)

+a(3S--2)][Jz 2__~S + A + B u + e ( S - - 1 ) [ 2S-- 1

l ] 1 / 2

× ( 2 S - - 1 ) + a ( 3 S - - 2 ) ~ ] , (12)

and we have omitted the terms proportional to ~7 because of their small value. The expressions (11) and (12) differ from those obtained in [9] by a more complete consider- ation of anisotropy and are exact in the approximation of the self-consistent field at T ~ T N. They also express the correct behaviour of the spectrum at B, e, a -~ 0. I f we make use of the magnon spectrum data given in [9, 10]: ~_(0) = 18.7 cm -1, ~2+(0) = 49cm -1, Ae = ~2± ([ k z B [ = 1r/a) = 98 cm- 1 and choose the parameters a and e according to [9], then we can find from equation (12) that the best agreement with experiment data is achieved whenJ = 11.48 cm -1 = 16.5 K, A = 3 cm -1 = 4.32K, Bu = 0.28 cm -~ = 0.4K. It is seen that the con- sideration of only the constants of the fourth-order anisotropy e is insufficient to fit the values obtained by the formulas (12) with experiment and the lattice distor- tion makes a major contribution to the ~2(0) stabilizing 2-D ordering.

4. The spectrum (12) can be defined more exactly, for example, in the RPA or by using the matching matrix

TWO-DIMENSIONAL ANTIFERROMAGNETS OF THE K2 FeF4 TYPE 803

(9) method elements [2]. In so doing we can find either the correction to the spectrum at T = 0 or study its tempera- ture dependence. We would like to note that already at low temperatures (in region of spin-wave approxi- mation) a certain dependence on T will appear in the expressions (12) due to the temperature behaviour of the deformations. In order to estimate this dependence it is necessary to find the free energy F of a crystal according to the expression

F = kBTlnSpexp ( - ~ s + ½1aVu2 + BT , (13)

where kB is the Boltzmann constant, Eo is the crystal ground state energy, V is the crystal volume, ~ is the elastic constant. Then from the equation 6F/6u = 0 it is easy to find that

u ~ B ~ ( x 2 y 2 [(S.~) -- (Sn~) l).

The expression (13) can be directly calculated in spin- wave approximation which result in

4 ( 2 S - 1)BkBT u(73 = u(0) +

rr#vJzS

X In ({1- exp exp (--kBT]J 1 V

u(0) = ~ [ (2S-- 1)SB +JzS2rl], V -A t . (14)

From equation (14) it is seen that in the region of small T,~ ~_(0) the dependence on T is negligibly small but in the region intermediate T [~ f2_(0)] such a depen- dence will contribute to the temperature behaviour of the AFMR frequencies. The function u (T) undoubtedly determines the temperature behaviour of the gradient of the electric field on the nucleus Fe.

5. Thus, we can conclude that though allowance for the fourth-order anisotropy may, in principle, explain, some experimental data under consideration (for example, the appearance of the spin-wave spectrum gap), the descrip- tion and interpretation of the whole complex of exper- iments, to say nothing of the qualitative agreement, require that the magnetoelastic interaction should be employed. Besides, the above formulas for the AFMR frequencies can be used to obtain more accurate values of crystal parameters.

Acknowledgements - We are much obliged to the authors of [9] who inform us of the results of their paper before its publication.

804 TWO.DIMENSIONAL ANTIFERROMAGNETS OF THE K2FeF4 TYPE Vol. 44, No. 6

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