Spin-Wave Interference in Microscopic Rings
J. Podbielski,
*
F. Giesen, and D. Grundler
†
Institut fu ¨r Angewandte Physik und Mikrostrukturforschungszentrum, Universita ¨t Hamburg,
Jungiusstrasse 11, 20355 Hamburg, Germany
(Received 4 November 2005; published 27 April 2006)
We have studied the spin excitations of ferromagnetic rings and observed a distinct series of quantized
modes in the vortex state. We attribute them to spin waves that circulate around the ring and interfere
constructively. They form azimuthal eigenmodes of a magnetic ring resonator which we resolve up to the
fourth order. The eigenfrequencies are calculated semianalytically and classified as a function of magnetic
field by a quantization rule which takes into account a periodic boundary condition. Strikingly each mode
exists only below a characteristic field.
DOI: 10.1103/PhysRevLett.96.167207 PACS numbers: 75.40.Gb, 75.75.+a, 76.50.+g
When the lateral dimensions of a microstructured ferro-
magnet are on the order of the spin-wave wavelength , the
geometrical boundaries impose a quantization condition
for excitations. This has been found experimentally for
straight micromagnets such as wires or rectangular prisms
[1–7]. Curved ferromagnetic devices have recently at-
tracted considerable interest due to flux-closure states
with vanishingly small stray field and the circular symme-
try of magnetization [8,9]. Quantization phenomena for
disks in the vortex state [10] are, however, complex: due
to the presence of a vortex core radial and azimuthal nodes
do not, in general, represent good quantum numbers. A
clear separation in radial and azimuthal spin waves [11–14]
is not always possible [10] and the calculation of the
eigenmodes and eigenfrequencies is involved [15–17].
We have investigated nanostructured permalloy
(Ni
80
Fe
20
) rings which are in the vortex state [8] and where
the core is removed. We observe quantized azimuthal spin
waves up to the fourth order. They appear in a stepwise
manner with overall negative magnetic field dispersion
when we increase the applied magnetic field H. Our semi-
analytical calculations show that the steps originate from
backward volume magnetostatic waves [18] which inter-
fere around the ring. For each eigenmode an upper critical
field exists. With increasing H in the vortex state, a further
mode appears with positive magnetic field dispersion
showing no discrete steps. This behavior is due to a local-
ization phenomenon. Our observation that for propagating
spin waves a ring acts as a ring resonator with quantized
azimuthal modes is stimulating for both further fundamen-
tal spin dynamics research [19] and magneto-logical ap-
plications [20].
The fabrication of rings on the signal line of coplanar
waveguides (CPWs) using electron beam lithography and
lift-off processing has been described elsewhere [21,22].
We investigated three arrays which consisted of permalloy
rings with similar width w 600 nm and outer diameter
2R 2000 nm but had (i) different thickness t, (ii) dif-
ferent ring-to-ring separations and (iii) a different number
of rings. They all displayed the same characteristics in the
vortex state on which we report. Only the absolute values
of the measured spin-wave eigenfrequencies varied (as
expected, e.g., from the different thickness). We focus
here on the data of one sample. It consisted of 750 nomi-
nally identical rings with 2R 195030 nm, w
60030 nm, and t 306 nm. The geometric pa-
rameters were determined by atomic force microscopy
(AFM). The ring-to-ring separation of 2 m excluded
dipolar interaction [23]. Transmission spectra are mea-
sured at room temperature by means of a vector network
analyzer connected to the CPW. The sinusoidal output
signal of power 1 mW causes a high-frequency magnetic
field H
rf
surrounding the central conductor of the CPW and
acting mainly in the plane of the rings. A static external
magnetic field
0
H is applied parallel to the CPW and
orthogonal to H
rf
. Following Refs. [10,14] H
rf
leads to a
spatially inhomogeneous torque ~ m
~
H
rf
in the vortex
state [cf. Figure 3(a)]; i.e., spin-wave excitation is
inhomogeneous.
In Fig. 1 we summarize a series of absorption spectra
taken at different in-plane fields
0
H ranging from 60 to
60 mT. At each field the spectrum was recorded after
applying a saturation field of 90 mT. The rings exhibit
the typical reversal behavior with two irreversible switch-
ing processes [23,24]: for
0
H
0
H
sw
1
2 mT the
rings are in the onion state. Lowering H below H
sw
1
makes
the rings switch to the vortex state. The absorption char-
acteristics change significantly. The vortex configuration is
stable down to
0
H
sw
2
18 mT. For H H
sw
2
rings
form the reversed onion state. The three separate regimes
of dynamic response have already been reported for
250 nm wide rings [21]. Modes A and B, which are
observed at high field in the onion state (see labels in
Fig. 1), have already been explained by localized spin-
wave excitations [21–23]. Mode A resides in ring seg-
ments where the external field is oriented tangentially to
the ring, and mode B in domain walls. These localized
modes at high field will not be discussed.
PRL 96, 167207 (2006)
PHYSICAL REVIEW LETTERS
week ending
28 APRIL 2006
0031-9007= 06=96(16)=167207(4)$23.00 167207-1 © 2006 The American Physical Society