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Nonlinear s-f exchange interaction effect and magnetic properties of rare earth metals
K. Kaino, T. Kasuya
To cite this version:
K. Kaino, T. Kasuya. Nonlinear s-f exchange interaction effect and magnetic properties of rare earth metals. Journal de Physique Colloques, 1979, 40 (C5), pp.C5-24-C5-25. �10.1051/jphyscol:1979508�.
�jpa-00218880�
JOURNAL DE PHYSIQUE
Colloque C5, suppKment au no 5 , Tome 40, Mai 1979, page C5-24
Nonlinear s-f exchange interaction effect and magnetic properties of rare earth metals
K. Kaino and T. Kasuya
Department of Physics, Tohoku University, Sendai, Japan
R6sumB.
-On a CtudiC I'effet non lintaire de I'interaction s-f d'tchange avec la structure de bande simplifite.
Les diagrammes de I'ordre magnktique sont tires pour des parametres divers. L'onde de spin ferromagnktique est aussi tirte.
Abstract.
-The nonlinear effect of s-f exchange interaction is investigated with a simplified band structure.
The magnetic ordering diagrams are drawn for various parameters. The ferromagnetic spin wave is also deter- mined.
1. Introduction.
-Rare earth metals are known as the most typical materials in which the s-f exchange model is applicable, and a lot of data has been ana- lysed on this model, but so far to the second order of the s-f exchange inter'action [I]. However, as was shown by Kasuya [2], the effective Fermi energy is rather comparable with the s-f exchange energy and thus the higher orderyeffect should be important.
To study the higher order effect, we treat the s-f exchange energy in a nonperturbed way and examine the phase diagram and the spin wave dispersion.
2. Model and formulation. -The most fundamental character of the conduction electrons in rare earth metals is the existence of the flat surface perpendicular to c-axis [3], which produces the screw type structure.
To make the calculation simpler, we use here the following simplified band model with a flat surface perpendicular to k,-axis,
where k, = 1 k, I + 1 k, 1 and 8(x) is the step function.
As k, increases, the flat part increases. In actual band, the flat surface has ripples which prevent the loga- rithmic divergences. Here, for the similar purpose, we introduce the life time term i T for the conduction electron as a parameter. The Hamiltonian has three terms, the band energy term, the s-f exchange term and the uniaxial crystal field term;
Now, each f spin is rewritten as
where ( ) means the expectation value. The unperturbed Hamiltonian X, is derived from eqs. (3) and (4) by replacing S, by ( S, ) and then the energy spectrum of conduction electrons is solved rigorously.
Here the magnetic ordering is assumed to be the cone structure, that is,
< S, ) = < S ) (sin 8 cos cp,, sin 8 sin cp,, cos 8) , (5)
where cp,
=Q. R, and Q is chosen in the k, direction.
This includes the screw, 0 = 7112, and the ferroma- gnetic, 8
=0 or 8 = 7112 and Q
=0, structures.
In section 3, the phase diagrams for ( X, ) are shown.
In section 4, the ferromagnetic spin waves are shown.
3. Ordering phase diagrams at T
=0.
-At first, let us consider the case, V
=0. Then the phase diagram is shown in figure 1 in two parameter space,
d"
EI ( S)IE, and za = kav21Ef,
Fig. 1. - Phase diagram for V
=0. Solid lines represent the second order transition and the dotted lines the first order transition.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979508
NONLINEAR s-f EXCHANGE INTERACTION EFFECT C5-25
for several values for r TIE,, where Ef is the Fermi energy for n o s-f exchange. The life time effect, which is expected to be similar to the ripple effect on the flat Fermi surface, is seen mostly on the ferro-cone transition part. Note that the cone structure is stabi- lized easily by the nonlinear s-f exchange effect in the case of a rather smaller flat part. The ferro-cone second order transition line is determined by the second derivative of the total energy,
for d" < 1. The ferro-screw second order transition
line is determined by the equation,
(d2~ldG! 2)0=n12, a=
0 cc( 1 - 3 4"' - 2 J3)13 + + Ea(l
-2,) + E"? = 0 , (7)
for d" > 1. In figure 2, the values for 0 and
0 = Qv,l2 Ef are shown as functions of d" for two fixed values of ia. When d" increases, in general, Q decreases for the screw structure but increases for the cone structure because of the decreasing screw compo- nent. This tendency is consistent with the experimental results [4]. In the next, the crystal field effect is taken into account. In figure 3a, an example for T"
=0 and V < 0 is shown. In this case the screw structure including the z axis is most stable. The boundary for the z-plane screw structure is also shown for comparison. In the vanishing limit of d", the crystal field always stabilizes the cone structure. Note that the ferro-screw transition becomes of the first order.
An example for T" = 0 and V > 0 is shown in figure 3b.
The ferromagnetic component is perpendicular to the z-axis, that is, 0
=7112 and Q
=0 type, so that the fan structure instead of the cone structure is stabilized in a large region. The life time effect for the ferro-fan transition is larger than that for the ferro-cone transition. Note that, even in this case, the cone structure appears in some region against the crystal field. This results may correspond to the case of Holmium. For the case of small T", the higher harmo- nics, 2 Q etc., coexist. The effect on the phase diagram is, however, small.
Fig. 2.
- 13 andQ for various d: a) = 1.2 and 6) %
= 1.3.See the arrows in figure 1.
Ferro
I /
0
b)10 , 2.0
ka
Fig. 3.
-Phase diagrams for
a)
V
(S
)21Ef = -0.001 ( ~ : 1 3
nul vz N )and
b) V (
S
)21Ef =0.008 (~:13
x u , v: N ).
Note that the curve a is the boundary for the s-plane screw structure.
4. Ferromagnetic spin waves.
-Here, only the spin waves in the ferromagnetic ordering are shown.
In figure 4, an overall behaviour of the spin wave spectrum, a,, in k , and k , directions is shown. The spectrum in k, direction is softened at Q = 2 E,/v,.
The stability condition for the ferromagnetic state coQ 3 0 is equivalent to eq. (6). When the reduced wave numbers, k", = k , v,lEf and k", = k, v2!Ef, are used, the dispersion along k , is similar to that along k , except around the peak a t Q.
Fig. 4.
-Spin wavedispersion,
w,,f o r k
= 213,d"
= 113,T"
=0.01 and along the k,-axis for solid line and along the k,-axis for dotted line.
5. Conclusions. - By using the simplified band model, the nonlinear effect of the s-f exchange inter- action was calculated. An effective anisotropy is induced which makes the phase diagram complicated.
Note that, when ( S ) is taken as the expected value at a finite temperature, the phase diagram shown here is applicable at a finite temperature. The tendency of the screw pitch is, for example, consistent with the experiment. Detailed calculation including finite tem- perature and comparison with experiment will be shown in separate papers.
References
[I] LINDGKRD, P. A., Phys. Rev. B 16 (1977) 2168.
[2] KASUYA, T., Magnetism IIB (Academic Press, New York and London) 1967.
[3] KEETON, S. C. and L o u c ~ s , T. L., Phys. Rev. 67 (1968) 672.
[4]
