ELSEVIER

Journal of Magnetism and Magnetic Materials 148 (1995) 30-31 h i Journal of

magnetism and magnetic

, 4 ~ materials

Spin-wave dynamics of two-dimensional isotropic dipolar honeycomb antiferromagnets

C. P i c h * F. S c h w a b l

lnstitut fiir Theoretische Physik, TU Mi~nchen, D-85747 Garching, Germany

Abstrac t The question of long-range order in two-dimensional isotropic dipolar Heisenberg antiferromagnets on a honeycomb

lattice is studied. The excitation spectrum is evaluated, yielding via linear spin-wave theory a non vanishing order parameter at finite temperatures. The N6el temperature is calculated by the Green function method of Callen. For MnPS3, a quasi two-dimensional antiferromagnet, good agreement with the measured transition temperature is obtained. In addition it is shown that the dipolar interaction leads to an intermediate phase (T----- 0) when an external magnetic field is applied.

Recently [1,2] it has been shown for a square lattice that in two-dimensional isotropie HeJsenberg antiferromag- nets long-range order is possible when the dipole-dipole interaction is included. Owing to the anisotropy of the dipole-dipole interaction, the H o h e n b e r g - M e r m i n - Wagner theorem [3,4] does not hold and a non-vanishing order parameter occurs. Evaluating the N6el temperature for some quasi-two-dimensional halides led to good agree- ment with experiments.

The honeycomb lattice is another example of a two-di- mensional lattice which has an unfrustrated ground state. We expect lhat it will behave like the square lattice leading to long-rang~ order.

The classical ground state of the isotropic dipolar anti- ferromagnet is no longer continuously degenerate. The spin orientation is perpendicular to the plane owing to the dipolar energy, if the dipole-dipole interaction is small. The Hamiltonian for the honeycomb lattice with lattice constant a reads:

H= -- E E(Ju'8,~.a +A~tP)SrSf" -gp'nHoESf, l ¢ l ' a13 I

O) with the exchange interaction Jw, the usual dipole-dipole interaction A~/3 and an external field, H 0, pointing along the z-direction. Transformation of the spin operators to Bose operators for the two sublattices via Holstein- Pdmakoff we obtain the spin-wave spectrum to order 1/S. An energy gap appears in the excitation spectrum because

" Corresponding aulhor. E-mail: cpieh@physik.lu-muenehen.de; fax: + 49-89-3209-2296.

it costs energy to rotate the spin configuration homoge- neously ( H o -- 0):

• 1 / 2 Eo=2S[2 (A~Z- -X~Z-A~X+X~x) l J I z ] , (2)

for nearest-neighbor exchange, 1.71>> (gl~a)2/a 3. Here A~ are the Fourier transforms of the dipole tensor on one sublattice and A~ are the Fourier transforms for the inter- action between the two sublatlices [5]. This equation is equivalent to the analog equation for the square lattice [1]. The magnon frequencies are shown in Fig. 1 for several directions in the Brillouin zone (Fig. 2). The dipolar energy lifts the twofold degeneracy. In addition the disper- sion curves no longer have inversion symmetry as in the isotropic case. This results from the broken inversion symmetry of the honeycomb lattice and the non-diagonal component of the dipole tensor, ~,~Y.

Next we consider the influence of an external field, H o ~ 0. For finite external field the N~el ground state becomes unstable at the critical value

g/xBHg = Eo, (3)

as for the square lattice [1]. Above this field the systems changes to a canted ground state. Calculating the ground state energy for finite magnetic fields we find that, owing to the dipolar energy, the system changes from the N6el phase to an intermediate phase. The spin orientation varies continuously to a spin-flop orientation at a second critical field, H~ f. The width of the intermediate phase is propor- tional to the dipole energy:

XX ~-XX 2 J I z + A o --Ao" --A~ z - A ~ z H~ ~ H o <~ Hdf = H~

2 J l z + n ~ z - - . 4 ~ Z - A ~ X - , ~ x '

(4)

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C. Pith, F. Schwabl/Journal of Magnetism and Magnetic Materials I48 (1995) 30-31 31

Fig. 1. Spin-wave frequency for the Heisenberg: antiferromagaet on a honeycomb lattice. The dashed curve is for pure exchange interaction and the solid curve with additional dipole-dipole interaction, (gpn)‘/(a3 I J 1) = 0.1.

and thus”vanishes for pure isotropic exchange. Note that the intermediate phase is absent if the easy-axis anisotropy in the Hamiltonian is of single-ion form [6]. Above the second c$ical field a spin-flop phase occurs as long as the field is balow a third critical field,

~,u~H&~‘= 2S(2 1 J 1 t+Agx --x;” -A$= vi-$=), (5)

above which the paramagnetic phase is stable. The two easy-plane phases (intermediate and spin-flop) with in- plane staggered magnetization will not lead to long-range order at finite temperatures as already stated for the spin- flop phase [71, so the nature of the transition is still not quite clear. We assr.tme a phase similar to a Kosterlitz-

Thouless phase.

QY P K

X

N A I- J-7

J-4 Fig. 2. Brillouin zone of the honeycomb lattice with qX E [--4m/3a, 4rr/3a] and qy E[-%T/fi, 2P/&].

Now we turn to the N6el temperature, which is calcu- lated by the Green function method of Callen [S,9]. lntro- ducing the retarded double-time Green function

G(Rk-RR,, t)= -i@(t)([Sl(t), ebsiS;(0)]}, (6)

and using the Tyablikov decoupiing scheme the phase transition temperature is evaluated in close analogy to the square lattice 121. If the dipolar energy is weak in ccmpari- son with the exchange energy, (g&*/a3 -+c 12 I, the expression for the temperature can be approximated by

IJI TN=

ln(lJI/&) *

MnPS, is a quasi-two-dimensional antiferromagnet with a nearest-neighbor exchange energy, 1 J 1 = 9.1 K and a lattice constant, a = 3.5 A [lO,ll]. The measured transi- tion temperature is TN = 78 K. With the above theory we get a theoretical value of T$ = 73 K, which is in good agreement with the experimental value.

This work has been supported by the German Federal Ministry for Research and Technology (BMiTT) under the contract number 03-SC2TUM.

References

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