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Submitted on 1 Jan 1971
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PHASE DIAGRAM AND SOFT MODES OF THE UNIAXIAL ANTIFERROMAGNET
K. Blazey, M. Ondris, H. Rohrer, H. Thomas
To cite this version:
K. Blazey, M. Ondris, H. Rohrer, H. Thomas. PHASE DIAGRAM AND SOFT MODES OF THE UNIAXIAL ANTIFERROMAGNET. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1020-C1- 1021. �10.1051/jphyscol:19711364�. �jpa-00214401�
JOURNAL DE PHYSIQUE Colloque C I, suppliment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 1020
PHASE DIAGRAM AND SOFT MODES OF THE UNIAXIAL ANTIFERROMAGNET
K. W. BLAZEY, M. ONDRIS (*), H. ROHRER and H. THOMAS (**) IBM Zurich Research Laboratory, 8803, Riischlikon, Switzerland
RCum6. - Nous avons 6tudi6 par r6sonance antiferromagnktique la dynamique de spin la limite de la transition de premier ordre entre la phase antiferromagnktique et la phase de spin-flop dans la GdAlOs. Les rksultats sont en accord qualitatif avec les calculs numMques basks sur une thkorie qui tient compte de faqon phknomknologique des prockdks de relaxation des spins. La structure dans le spectre d'absorption, causk par la forte dkpendance, en fonction du champ, des fr6quences de modes infkrieures de rksonance dans chaque phase, est fortement anisotrope par rapport a la polari- sation du .champ haute frkquence.
Abstract. - We have investigated the spin dynamics at the first-order phase boundary between the antiferromagne- tic phase and the spin-flop phase in GdAlO 3 by antiferromagnetic resonance. The results are in qualitative agreement with numerical calculations based on a theory, which takes spin relaxation processes phenomenologically into account. The structure in the absorption spectrum, which is caused by the strong field-dependence of the frequencies of low-lying reso- nance modes in either phase, is found to be strongly anisotropic with respect to the polarization of the microwave field.
We have studied the phase transitions occurring in a uniaxial antiferromagnet and the associated dynamical behavior as function of applied field and temperature both experimentally and theoretically. We discuss here our results on the spin dynamics a t the first-order phase boundary between the antiferromagnetic (AF) phase and the spin-flop (FL) phase in GdA10,. A full description of our results will be given elsewhere.
The theoretical procedures have been described in detail in reference [I]. We therefore indicate here only briefly the main steps. The free energy F(< SA >, < SB >) is calculated in molecular field approximation as a function of the sublattice magneti- zations < SA > and < S, >, the equilibrium values of which are obtained by minimizing F with respect to < SA > and < SB >. The thermodynamic phase boundary between two phases is determined by the condition that the free energies of the two phases are equal. The stability limits of a phase are determined by tne condition that one eigen-vector of the reciprocal static susceptibility matrix
I a Z ~ a 2~ 1
becomes zero, all others being positive. This particular eigen-vector represents the spin deviation with respect t o which the instability occurs.
In the study of the dynamic behavior, we have taken spin relaxation processes qualitatively into account by adding to the RPA equations of motion longitudinal (7,-type) and transverse (z,-type) relaxation terms. By linearizing the equations of motion about the equili- brium state, we find a dynamic susceptibility matrix which is given by
X-l(") - x-l =
(*) Now at the Technische Hogeschool, Delft, The Nether- lands.
(**) Now at the University of Frankfurt, Germany.
Here, ~ ( ~ ' ( o ) and f f ) are the dynamic and static susceptibility tensors of a free spin in an external field equal to the molecular field, with spin relaxation described by z, and 7,. The normal mode frequencies are the zeros of the eigenvalues of ~ - ' ( o ) in the complex o-plane, and the corresponding eigenvectors describe the spin motion of the normal modes. At the stability limit of any phase, one of the normal modes has zero frequency and an eigenvector corresponding to the spin deviation of the static instability (soft mode).
When a uniform rf-field
is applied, energy is absorbed by the spin system at a rate
Q = - ioGH* . ( ~ ( o ) - xt(o)). 6H
where 6H is the six-dimensional vector 6H = (6H, 6HJ.
We have solved the equilibrium conditions and determined the phase boundaries and stability limits as well as the normal mode frequencies and eigenvectors numerically on an IBM 360150 computer for GdAIO, using the parameters determined in reference [2] with zl = 5 x s, z2 = 2.5 x lo-" s. The computed absorption spectrum for given o as a function of H was displayed on an IBM 2250, such that theoretical and experimental results could be compared directly.
The phase boundary between the A F phase and the FL-phase in the (H,, Hz)-plane shows the feature reported previously for T = 0 [3, 41 : It extends only to a maximum angle with respect to the easy axis, where it ends in critical points (Fig. 1). The critical angle is found to go to zero as T approaches the temperature of the triple point, where the AF, FL, and paramagnetic phases co-exist.
We now consider a fidd applied a t an angle
II/ < prit with respect to the easy axis (Results for the case $ = 0 have been reported in reference [ 5 ] . ) The dynamic behavior of GdAIO, in the frequency range of 1 to 5 GHz is determined mainly by the low-lying reso- nance mode in either phase. Figure 2 shows the field dependence of the frequency of these modes for an angle $ close to the critical angle. In the FL-phase, this mode becomes the Goldstone mode $ = 0, which restores the broken uniaxial symmetry. Both modes are strongly field-dependent, although neither of them is
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711364
PHASE DIAGRAM AND SOFT MODES OF THE UNIAXIAL ANTIFERROMAGNET C 1 - 1021
FL- phase /
\
'\
--.
--\-a Phaseboundary "
-
2N / @ /
I C)*
- H O )
AF - phase I I
FIG. 1. - Phase boundary (-) and stability limits (- - -)
at the spin-flop transition in GdAIO3 at T = 1.35 OK. The demagnetizing field has been taken into account, assuming a cylindrical sample with the axis parallel to the easy (z) axis.
The straight lines indicate fields applied at an angle y = 6O with respect to the easy axis.
1
2 . 2 4 1 , , , - J , , , 8
11.6 11.8 12.0 12.2 IbC 11.6 11.8 12.0 1 2 2 121
FIELD (IN k & )
FIG. 2. - Real and imaginary parts of the low-lying resonance modes in the AF and FL phases as function of the applied
field (t+v = 6" ; T = 1.35 OK), (1 kOe = 2.8 GHz).
the true soft mode associated with the stability limits.
The soft modes are found t o be slow relaxation modes, which have no dynamic significance in this frequency range.
The absorption is found to be strongly anisotropic with respect to the polarization of the microwave field 6H. For 6H perpendicular to the easy axis in the plane containing the sublattice magnetizations, we find a strong resonance line in the AF-phase at a field where the real part of the mode frequency crosses the micro- wave frequency, but only a weak structure in the FL-phase (Fig. 3a). This is in good qualitative agree-
11.0 i s l i b 1i.r 10 11
FIELD (IN k@)
FIG. 3. - Absorption spectrum at 3.95 GHz as function of the applied field (t+v = 6O, T = 1.35 OK) for the microwave field perpendicular to the easy axis in the plane containing the
sublattice magnetizations.
a) Calculated Q (H).
6) dQ/dH observed in GdAlOs. The samll peak at H = 10.9 kOe is caused by the discontinuity of Q (H) at the phase boundary.
ment with the observed absorption spectrum (Fig. 3b).
For 6H parallel t o the average magnetization in the FL-phase, on the other hand, we predict a weak reso- nance line in the AF-phase, and a strong line in the FL-phase.
References
[I] THOMAS (H.), Lectures presented a t the Conference [4] CHEPURNYKH (G. K . ) , Soviet Physics - Solid State, on Magnetism, Chania, Crete, June 1969. 1968, 10, 1517 (Fiziba Tverdogo Tela, 1968, 10, [2] BLAZEY (K. W.) and ROHRER (H.), Phys. Rev., 1968, 1917).
173, 574. [5] BLAZEY ( K . W.), MUELLER ( K . A.), ONDRIS (M.) and [3] ROHRER (H.) and THOMAS ( H . ) , J. Appl. Phys., 1969, ROHRER (H.), Phys. Rev. Letters, 1970, 24, 105.
40, 1025.
