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in c -axis films and superlattices of Er (Borchers et al. 1988), but the
explanation in this case is not quite so evident. The dipolar energies are
unchanged in the films, nor is it likely that the anisotropy and exchange
contributions are decisively different. The strain-dependence of the ex-
change energy, expressed in eqn (2.3.3), can however provide a mecha-
nism. Y has a planar lattice-constant a of 3.648 A, which is over two per
cent greater than that of Er, and the Y substrate therefore imposes a
strain on the Er film, which is measured to be 11 = 22
10 3 .Ifthe
atomic volume is assumed to be unchanged in the film, 33
×
6
10 3 .
The difference in exchange energy between the solid and a thin film may
then be found from (2.3.4), and is equivalent to a field of 13 kOe acting
on the c -axis moment of about 8 µ B . The above estimate of 33 is prob-
ably too great, so this calculation may be considered in reasonably good
agreement with the observation that Er films with thicknesses between
860 A and 9500 A require fields varying linearly between 8 kOe and 3 kOe
to establish a ferromagnetic state at 10 K. It is noteworthy that, since
Lu has a significantly smaller basal-plane lattice-constant than Er, the
cone structure should be favoured in a c -axis epitaxial film grown on Lu.
Many of the characteristic features of rare earth superlattices are
demonstrated by the aforementioned [Dy
12
×
Y] systems, which are observed
to form helical structures over the whole temperature range of magnetic
ordering. When the c -axis is normal to the plane of the film, a coherent
magnetic structure may be formed, in which the phase and chirality of
the helix are maintained over many bilayers, provided that the slabs of
non-magnetic Y are not too thick. The coherence length may be esti-
mated from the widths of the neutron-diffraction peaks, and corresponds
to more than 10 bilayers if the Y layers are less than about 10 planes
thick. If the thickness is increased to about 35 planes, however, the
coherence length, which is inversely proportional to the width of the Y
layers, is less than the bilayer thickness, so that the helix in one Dy layer
is uncorrelated with that in the next. In the long-range coherent struc-
tures, the phase change of the helix across the Dy layers corresponds
to a turn angle which varies with temperature and shows a tendency to
lock in to 30 , with associated bunching. The phase change across the
Y layers, on the other hand, is independent of temperature and the turn
angle takes the much larger value of about 50 , which is characteristic of
the periodic structures formed by dilute alloys of magnetic rare earths
in bulk Y. It therefore appears that the magnetic order is propagated
through the Y layers by a spin-density wave, which is incipient in the
unperturbed metal, and is associated with the very large susceptibility
χ ( Q ) of the conduction electrons. The helical ordering in the Dy layers
of the c -axis superlattice is disturbed by edge effects of the type illus-
trated for the Ho film in Fig. 2.11. Consequently, the ordered helical
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