  We present an experimental and theoretical study of the magnetic field
dependence of the mode frequency of thermally excited spin waves in rectangular
shaped nanopillars of lateral sizes 60x100, 75x150, and 105x190 nm2, patterned
from MgO-based magnetic tunnel junctions. The spin wave frequencies were
measured using spectrally resolved electrical noise measurements. In all
spectra, several independent quantized spin wave modes have been observed and
could be identified as eigenexcitations of the free layer and of the synthetic
antiferromagnet of the junction. Using a theoretical approach based on the
diagonalization of the dynamical matrix of a system of three coupled, spatially
confined magnetic layers, we have modeled the spectra for the smallest pillar
and have extracted its material parameters. The magnetization and exchange
stiffness constant of the CoFeB free layer are thereby found to be
substantially reduced compared to the corresponding thin film values. Moreover,
we could infer that the pinning of the magnetization at the lateral boundaries
must be weak. Finally, the interlayer dipolar coupling between the free layer
and the synthetic antiferromagnet causes mode anticrossings with gap openings
up to 2 GHz. At low fields and in the larger pillars, there is clear evidence
for strong non-uniformities of the layer magnetizations. In particular, at zero
field the lowest mode is not the fundamental mode, but a mode most likely
localized near the layer edges.
