DYNAMICS OF THE SPIN FLOP IN ANTIFERROMAGNET

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DYNAMICS OF THE SPIN FLOP IN ANTIFERROMAGNET

T. Nagamiya, T. Nishikubo

To cite this version:

T. Nagamiya, T. Nishikubo. DYNAMICS OF THE SPIN FLOP IN ANTIFERROMAGNET. Journal

de Physique Colloques, 1971, 32 (C1), pp.C1-769-C1-770. �10.1051/jphyscol:19711268�. �jpa-00214099�

JOURNAL DE PHYSIQUE Colloque C I , supplbment au no 2-3, Tome 32, Fbvrier-Mars 1971, page C 1

-

⁷⁶⁹

DYNAMICS OF THE SPIN FLOP IN ANTIFERROMAGNET

T. NAGAMIYA and T. NISHIKUBO

Faculty of Engineering Science, Osaka University, Toyonaka, Japan

R6sumB.

-

La dynamique du retournement de spin dans une substance antiferromagnktique orthorhombique a kt6 ktudike analytiquement et puis, par une mi5thode numkrique, ceci a kt6 appliquC au cas de CuCl2.2H20 (orthorhombique) et MnF2 (uniaxiale).

Abstract. - The dynamics of the spin flop in an orthorhombic antiferromagnet is studied analytically, and then a computational method is used to study the spin flop in orthorhombic CuC12. 2H20 and in uniaxial MnFz.

1 . The phenomenon of spin flop in an antiferromag- net at a critical value of the magnetic field applied along the easy axis of magnetization has been known since N6e17s prediction in 1936 [I]. When the magnetic field and the spin axis are along the easy axis, the energy of the antiferromagnet is -(1/2) H,, and when the magnetic field is along the easy axis but the spin axis is perpendicular, it is -(1/2)

X,

H z

+

^K,

where K is the increase in the anisotropy energy.

Since

X, > ^XI,

^{for T}

^{TN, X,}

being approximately constant and tending to zero for T + 0, the second configuration is favored when H exceeds

A little more careful consideration, taking into account the anisotropy energy in the formula for the perpendi- cular susceptibility, will show that there is a small range of H in which the spin flop takes place. At a certain Hcl the parallel configuration becomes no more locally stable, i. e., one of the two antiferromag- netic resonance frequencies vanishes, and the energy of the parallel configuration is higher than that of the perpendicular configuration. At a slightly lower cri- tical field H,, the latter configuration becomes no longer locally stable and its energy is higher than that of the parallel configuration. Between Hcl and H,, there is the third critical field Hc, where the ener- gies of the two configurations become equal and both configurations are locally stable. Hence one should have a hysteresis effect in the spin flop.

So far, however, the dynamical behavior of the spin flop process has not been known. The present authors, with H. Suzuki, have made a theoretical study of it and published the results in two papers in 1969 121, 131. The present report gives a short account of the study.

2. The basic equation is the torque equation :

the same sublattice, that of the form AM,.M, be- tween the spins of different sublattices, and an aniso- tropic exchange interaction of the form

where @ and @' are diagonal tensors having compo- nents @,, @,,, 0 and @:, @:, 0, respectively. The total energy is the sum of these energies plus the Zeeman energy,

-

H.(M,

+

M,). H j is derived by differen- tiating the total energy with respect to Mj.

3. Approximate solution.

-

When H is close to Hc (discrimination of three critical fields being dis- regarded) or damping is large, we can expect a slow motion of the magnetization axis in the flop process.

We define the magnetization axis to be the direction of M' ⁼ M I

-

M,. In an orthorhombic antiferro- magnet such as CuC12.2 H 2 0 one may expect the spin axis to move nearly in the plane determined by the easy axis and the second easy axis (we shall call it the easy plane). The net magnetization

will be small, and it will oscillate with a frequency that corresponds to the higher antiferromagnetic eigenoscillation, as will do the spin axis at the same time. Neglecting H z compared with (AM')' and assum- ing the anisotropy field to be even smaller, we find in the crudest approximation that the direction of M' is governed by an equation of the same form as that for the motion of a classical pendulum in a viscous fluid :

.

_tp

^. _{- A' 40 +}

k2 sin 2(9 - qe) = 0, A' = ---- ^AM0 , Y where Mo =

I

M I

I

⁼

I

M,

1,

h = H/2 AM,, and

M~ = y(Mj x Hj)

-

AMj x (Mj x Hj)

,

h,2 = [(@,,

-

@;) - (GX

-

@L)]/2 A . where j = 1, 2 or

+, ^-

denotes the two antiferromag-

netic sublattices, M j each magnetization vector, H j ^(PH denotes the angle which H makes with the easy the internal magnetic field on Mj, and the second axis, the angle for the direction of M', and q e the term is the ~ ~ ~ d ~ ~damping - ~ iterm which f ~ ~ hequilibrium value of i ~ ~ cp which is determined from makes Mj tend toward Hj. We confine ourselves to

T = 0. To derive the relation between M;'s and Hz's,

h2 sin 2(q,

-

9,) = hc sin 2 2 cp

..

we assume an isotropic exchange interaition of All these angles are measured within the easy plane.

form (r/2) (MI .MI

+

^M,.M,) among the spins of To study the small oscillational motion superposed

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711268

C 1

-

⁷⁷⁰ T. NAGAMIYA AND T. NISHIKUBO on the large/ pendulum motion, we return to the basic

equation and linearize it with respect to the oscillatio- nal amplitude. As we have four degrees of freedom corresponding to the four angles specifying the direc- tions of MI and M,, we obtain apparently four eigen- oscillations which, however, consist of two modes each having

+

^{o and}

-

w . The lower frequency mode is nothing but a small change in the pendulum motion caused by a small change in the initial conditions.

Hence the higher frequency mode is left, which super- poses on the pendulum motion. This mode may be observed by antiferromagnetic resonance during the flop process (Nagamiya [4]). The differential equation for the oscillation is inhomogeneous, possessing terms of driving forces, and solving it, we can determine the amplitude as a function of time or as a function of cp.

We find that the two angles specifying the direction of M' have large amplitudes, whereas the two angles specifying the direction of M, relative to M' as well as that of M2 (two angles, since M = MI

+

^{M, is}

perpendicular to M' = M I

-

M,) have small anpli- tudes. The resonance occurs, however, due to the oscillation of M.

4. Numerical solutions. - We also worked out numerical solutions of the basic torque equation with the use of an electronic computer, assuming various values for anisotropy parameters and damping para- meter and various initial conditions. Two examples will be given here (for other examples, see reference [3]). One of the two is for CuC1,.2 H,O shown in figure 1 and the other for MnF, shown in figure 2.

The locus of the plus sublattice magnetization vector M f (one of MI and M, which points initially near the direction of H and near the easy axis) is shown in the cp8 plane, where q ( q f in this case) is the angle measured from the easy axis in the easy plane and 8+

the angle of deviation from the easy plane in the case of CuC1,.2 H,O, whereas they denote in the case of MnF, the angle between the unique easy axis and the projection of M+ onto a plane that contains the easy axis and the angle of deviation from it. Time markers are given on each locus with attached nume.

rals that represent the elapse of time in units of Ax shown in the figure, where x =

-

(2 yAMo)t (y < 0).

It is found that in the case of an orthorhombic anti- ferromagnet the superposed oscillation has a frequency agreeing very well with the frequency obtained in the approximate analytical way. In figure l a there is no damping and the motion is purely periodic, in (b) there is a small damping and M + performs many turns before it tends to the equilibrium direc- tion, and in (c) there is a large damping so that M+

tends smoothly to the equilibrium direction. In figure 2 a small damping is assumed, and in this case the vector performs many rotations around the axis of maximum potential energy before reaching the equilibrium posi- tion.

[l] N ~ E L (L.), Ann. Physique, 1936, 5 , 232.

[2] NISHIBUKO (T.), SUZUKI (H.), and NAGAMIYA (T.), J. Phys. SOC. Japan, 1969, 27, 333.

&

-0.2 1 1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

++

^{( radians)}

( a 1

+'

^(rodions1

( c )

FIG. 1.

-

CuC12.2 H z 0 in a magnetic field of H. = Hcs and

+ +

P)H = -lo, with initial angles pi = Bi = 0 for M+ and anti-

+

+

parallel M-. p, , 19, are the angles of the equilibrium direction of M'.

1-

^1.2

FIG. 2.

-

MnFz in a magnetic field of H = 1.2H03 and pa =

+ +

- 5 O , OH = 0, with. initial angles pi = Bi = 0 for M+ and

+ +

antiparallel M-. a, , Be are the angles of the equilibrium direction of M+.

rences

[3] NISHIBUKO (T.), J. Phys. Soc. Japan, 1969, 27, 343.

[4] NAGAMIYA (T.), Progr. Theoret. Phys. (Kyoto), 1954, 11, 309.