The Formation of the Concertina Pattern: Experiments, Analysis, and Numerical Simulations [Elektronische Ressource] / Jutta Steiner. Mathematisch-Naturwissenschaftliche Fakultät

Jutta Steiner

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• Physik

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• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2011
• Nombre de lectures: 13
• Langue: English
• Poids de l'ouvrage: 4 Mo

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TheFormationoftheConcertinaPattern:
Experiments,Analysis,andNumericalSimulations
Dissertation
zur Erlangung des Doktorgrades (Dr. rer. nat.)
der Mathematisch-Naturwissenschaftlichen Fakultat¨ der
Rheinischen Friedrich-Wilhelms-Universitat¨ Bonn
vorgelegt von
JuttaSteiner
aus
Munchen¨
Bonn, Dezember 2010Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultat¨
der Rheinischen Friedrich-Wilhelms-Universitat¨ Bonn am Institut fur¨ Angewandte
Mathematik
Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn unter
http://hss.ulb.uni-bonn.de/diss online elektronisch publiziert
1. Gutachter Prof. Dr. Felix Otto
2. Prof. Dr. Stefan Muller¨
Tag der Promotion: 12. Mai 2011
Erscheinungsjahr: 2011Abstract
The concertina pattern is a ubiquitous pattern observed in ferromagnetic thin-film
elements. It occurs during the switching process due to the reversal of an applied
homogeneous magnetic field. The pattern-forming quantity is the magnetization,
which we think of as a unit-length vector field. The pattern consists of stripe-like
quadrangular and triangular regions – called domains – with a uniform, in-plane
magnetization that is, in particular, constant in the direction of the film thickness.
The domains are separated by sharp transition layers in which the magnetization
quickly turns – called walls.
Figure 0.1.: Concertina in a very elongated (length 2 mm) sample of width 50μm and thick-
ness 50 nm (left) and in a sample of width 35μm, thickness 40 nm and length
110μm (right). The left image shows only the center of the stripe which is less
than 10 percent of the whole sample. As indicated by the blue arrows, the gray-
scales encode the transversal component of the magnetization in the domains.
By courtesy of R. Schafer¨ .
The term concertina was introduced by van den Berg in [vdBV82] for this bellow-like
structure which is shown in Figure 0.1. In that reference, he discusses its formation
in thin rectangular-shaped ferromagnetic elements. He provides an explanation of
the domain pattern in a fairly thick (350 nm), not too elongated Permalloy sample
(width 15μm and length 50μm). He argues that the stripe-like pattern grows into
the sample from the tips due to boundary effects as the strength of an external
homogeneous magnetic saturation field – parallel to the long edge – is reduced.
We claim that in very elongated (length 2mm) thin (thickness 10 to 150 nm) fer-
romagnetic samples (width 10 to 100μm) the concertina does not grow from the
tips into the sample. For these extreme aspect ratios experiments rather suggest
that a bifurcation is at the origin of the concertina pattern, see Figure 0.2: As the
strength of an applied homogeneous magnetic field is reduced and finally reversed,
the uniform magnetization becomes unstable and buckles. As the strength of the
iiiAbstract
destabilizing field increases, the oscillatory buckling of the magnetization grows
into the concertina pattern – simultaneously all over the sample. Cantero and Otto
performed a linear stability analysis on the basis of the micromagnetic energy func-
´ ´tional, see[CAO06a]and[CAO06b]. Theyidentifiedathin-filmregimeinwhichthe
most unstable perturbation, the so called unstable mode, has the form of an oscil-
latory buckling. They find that the period of that instability is determined by the
width and the thickness of the sample together with the exchange length, a material
´parameter. In [CAOS07] a reduced energy functional was deduced as the scaling
limit of the micromagnetic energy in the oscillatory buckling regime. Numerical
simulations of the reduced energy functional showed that the unstable mode grows
intoaconcertinapattern. Thebifurcationisslightlysubcriticalbutexhibitsaturning
point. This means that the bifurcating branch of stationary points is unstable but be-
comes stable after the turning point (both under perturbations of the same period).
Acomparisonoftheperiodoftheunstablemodewiththeexperimentallymeasured
period yields a good agreement over a wide range of widths and thicknesses. How-
ever, there is a clear tendency that the experimental period is always larger by a
factor up to approximately two. In the experiments, one additionally observes that
theconcertinapatternexhibitsseveralcoarseningeventsasthestrengthofthedesta-
bilizing external field increases: Folds collapse, increasing the average period of the
pattern until it finally disappears. In order to understand these observations, it is
necessarytostudythestabilityw.r.t.perturbationswhoseperiodisamultipleofthe
period of the unstable mode or of the concertina, respectively.
The genesis of the concertina pattern is a prototypic example of a hysteretic process.
Theaimofthisworkisanextensiveunderstandingoftheexperimentalobservations
intheformationprocessoftheconcertinapatternonthebasisofthereducedenergy
functional. In particular, we explain the deviation of the period from
the period of the unstable mode and investigate the coarsening of the concertina.
This is achieved by an application of a mixture of rigorous analysis, numerical sim-
ulations and heuristic arguments.
• The application of a heuristic sharp interface model, namely domain theory,
showsthattheoptimalperiodoftheconcertinaisanincreasingfunctionofthe
(destabilizing)externalfield. Thisisrigorouslyconfirmedonthelevelofthere-
ducedenergyfunctionalbasedontheconstructionofappropriateAnsatzfunc-
tions and new nonlinear interpolation estimates providing Ansatz-free lower
bounds. Domain theory is (partially) justified by a compactness result for min-
imizers of the reduced energy functional.
• Domain theory suggests that the concertina becomes unstable under long
wave-length modulations as the destabilizing external field increases. The in-
stability is analyzed and confirmed by a Bloch wave analysis of the Hessian
of the reduced energy functional in combination with numerical simulations
of the r energy functional. Simulations show that the instability finally
leads to the coarsening of the concertina pattern.
ivAbstract
• A (generalized) bifurcation analysis shows that the deviation of the period of
the unstable mode from the experimental observations is due to a non-linear
modulation instability. This instability is in turn related to the so called Eck-
haus instability.
• Domain theory and numerical simulations are applied to investigate the ef-
fect of a uniaxial transversal and longitudinal anisotropy, respectively. This
confirms the experimental observation that a transversal anisotropy has a sta-
bilizing effect while in case of a longitudinal anisotropy the concertina cannot
be observed at all.
• Based on a linearization of the reduced energy functional, the ripple-like struc-
ture, which occurs in polycrystalline material, is investigated. In the exper-
iments, one observes that the ripple continuously grows into the concertina
pattern. The analysis shows that both the ripple and the concertina are driven
by the same physical mechanisms. Numerical simulations confirm this result
and reproduce the transition from the ripple to the concertina.
In Chapter 1, we review the previously known results and extensively present and
physically interpret our new insights. For proofs, explanations of the methods ap-
plied, and further investigations, we refer to the subsequent chapters.
The experiments that we discuss and present were carried out at the IfW Dresden
by J. McCord, R. Schafer¨ , and H. Wieczoreck.
Danksagung
An erster Stelle mochte¨ ich mich herzlich bei Herrn Prof. Dr. Felix Otto fur¨ die
Betreuung dieser Arbeit und seine intensive For¨ derung und fortwahr¨ ende Unter-
stutzung¨ bedanken.
¨Herrn Prof. Dr. Stefan Muller¨ danke ich fur¨ die Ubernahme des Zweitgutachtens.
Frau Prof. Dr. Ursula Hamenstadt, Herrn Prof. Dr. Herbert Koch und Herrn Prof.¨
Dr. Karl Maier danke ich fur¨ ihr Mitwirken in der Promotionskommission.
Herrn Dr. Jeffrey McCord, Herrn Dr. Rudolf Schafer¨ und Herrn Dr. Holm Wiec-
zoreck vom IfW in Dresden danke ich fur¨ die fruchtbare Zusammenarbeit. Den
Mitgliedern der dortigen Arbeitsgruppe danke ich fur¨ ihre Gastfreundschaft.
Herrn Prof. Dr. Alexander Mielke gilt mein Dank fur¨ seine Hinweise zur Eckhaus-
Instabilitat.¨ Herrn Martin Zimmermann danke ich fur¨ die technische Hilfe.
Mit Sicherheit war¨ e ohne meine Kollegen und Kommilitonen die Entstehung dieser
Arbeitnichtdenkbargewesen: Ihnendankeichherzlichfur¨ dieintensivenDiskussio-
nen, die Hilfe beim Beseitigen von IT-Problemen jeder Art, die aufbauenden Worte
und die gemeinsamen Erholungspausen.
Besonderer Dank gilt meiner Familie und meinen Freunden fur ihre Unterstutzung¨ ¨
und vor allem Artur fur¨ seine liebevollen Aufmunterungen und seine Geduld.
DieseArbeitwurdeunterstutzt¨ durchdenSFB 611unddieBonnInternationalGrad-
uate School in Mathematics.
vFigure 0.2.: Formationoftheconcertinapatternintheexperiment: Thepicturesshowasectionnearthecenteroffourdifferentelongated
thin film elements for different values of the external field. The two upper series show samples of 30 nm thickness of
low anisotropy. The two lower series show samples of 30 nm thickness of higher transversal anisotropy. The width is
30μm and 50μ