Read Back Signals in Magnetic Recording - Research Group Fidler

DIPLOMARBEIT
Read Back Signals in
Magnetic Recording
ausgeführt am
Institut für Festkörperphysik
der Technischen Universität Wien
unter der Anleitung von
Univ.Doz. Dipl.Ing. Dr.techn. Thomas Schrefl
durch
Otmar Ertl
Reinprechtsdorferstraße 20/16
1050 Wien
Mat. Nr. 0026810
Wien, am 15. Juli 2005 ________________

Kurzfassung
Diese Arbeit widmet sich der Berechnung des Signals von Leseköpfen, wie sie in
herkömmlichen Festplatten Verwendung finden. Dabei wird zuerst auf analytische Methoden
zurückgegriffen, die es erlauben, den Verlauf des Signals für ein vereinfachtes 2-
dimensionales Modell zu ermitteln. Insbesondere wird das Reziprozitätstheorem verwendet.
Der Großteil dieser Arbeit beschäftigt sich dann mit der Simulation des Lesevorgangs mittels
der Methode der finiten Elemente. Dabei wird die Landau-Lifshitz-Gilbert Gleichung gelöst,
die das dynamische mikromagnetische Verhalten eines magnetischen Modells beschreibt. Um
die Auswirkungen des Sensorstroms auf den Lesekopf zu berücksichtigen, muss die
Stromverteilung für inhomogene Leiter berechnet werden. Da es sich bei einem modernen
Lesekopf um einen magnetoresistiven Sensor handelt, hängt die Leitfähigkeit von der
Magnetisierung des Lesekopfes ab. Die berechnete Stromverteilung erzeugt wiederum ein
Magnetfeld, das auf die Magnetisierung des Lesekopfs einwirkt. Die programmtechnische
Umsetzung und die Realisierung solcher Simulationen von Leseköpfen waren die primären
Ziele dieser Diplomarbeit.
Im letzten Teil dieser Arbeit wird anhand eines 3-dimensionalen Modells eines Lesekopfes
die Funktionalität des Computerprogramms erprobt. Es werden der Gleichgewichtszustand,
die Sensorkennlinie und das Relaxationsverhalten berechnet. Weiters wird der Einfluss der
magnetischen Schilde des Lesekopfes untersucht. Zum Schluss wird das Lesesignal von
perfekten magnetischen Übergängen beziehungsweise von periodischen Bitmustern bestimmt.
Mit dem im Zuge der Diplomarbeit erstellten Computerprogramm ist ein Werkzeug
geschaffen worden, das die komplette Simulation des Lesevorganges, und auch die
Entwicklung von Leseköpfen am Computer erlaubt. Dabei können die dynamischen und
statischen Eigenschaften von Leseköpfen bestimmt werden.
2

Abstract
This thesis concentrates on calculating the read back signals of read heads, as used in
conventional hard disks. First the read back signal of a simplified 2-dimensional model is
determined by analytical means. Here the reciprocity principle is used.
The main part of this work deals with the simulation of the read back process with the Finite
Element Method. The Landau-Lifshitz-Gilbert equation is solved, which describes the
dynamic micromagnetic behavior of a magnetic model. To take effects of the sense current of
the read head into account, the current distribution for inhomogeneous conductors has to be
evaluated. Modern read heads use magnetoresistive effects. Thus the conductivities depend on
the magnetization of the read head. The calculated current distribution generates a magnetic
field, acting on the magnetization of the read head itself. The programming and the
implementation to make such simulations possible were the primary targets of this work.
In the last part of this thesis the functionality of the program is tested with a simple 3dimensional
model of a read head. The equilibrium state, the sensor curve, and the relaxation
behavior are calculated. Moreover the influence of the magnetic shields is examined. At last
the read back signal is calculated for perfect transitions and for periodic bit patterns.
This work resulted in a tool for simulating the whole read back process, which also allows the
computer-based development of read heads. Furthermore the dynamic and static
micromagnetic properties of read heads can be calculated.
3

Contents
Contents
KURZFASSUNG................................................................................................................................................... 2
ABSTRACT........................................................................................................................................................... 3
CONTENTS........................................................................................................................................................... 4
1 INTRODUCTION ....................................................................................................................................... 7
2 BASICS....................................................................................................................................................... 11
2.1 OHM’S LAW........................................................................................................................................ 11
2.2 MAXWELL’S EQUATIONS .................................................................................................................... 11
2.3 BASIC EQUATIONS OF MAGNETOSTATICS ........................................................................................... 12
2.4 INHOMOGENEOUS CONDUCTORS......................................................................................................... 14
2.5 GIBBS’ FREE ENERGY ......................................................................................................................... 15
2.5.1 Magnetostatic Energy.................................................................................................................... 16
2.5.2 Stray Field Energy......................................................................................................................... 16
2.5.3 Exchange Energy........................................................................................................................... 16
2.5.4 Magnetocrystalline Anisotropy Energy ......................................................................................... 17
2.6 EFFECTIVE FIELD ................................................................................................................................ 18
2.7 LANDAU-LIFSHITZ-GILBERT EQUATION ............................................................................................. 18
2.8 MAGNETORESISTANCE........................................................................................................................ 20
2.8.1 Magnetoresistance of Ferromagnets............................................................................................. 20
2.8.2 Anisotropic Magnetoresistance Effect........................................................................................... 21
2.8.3 Giant Magnetoresistance Effect .................................................................................................... 22
2.8.4 Tunneling Magnetoresistive Effect ................................................................................................ 26
2.9 RECORDING SCHEMES......................................................................................................................... 26
2.9.1 Longitudinal Recording................................................................................................................. 27
2.9.2 Perpendicular Recording .............................................................................................................. 27
3 ANALYTICAL CALCULATIONS ......................................................................................................... 29
3.1 RECIPROCITY PRINCIPLE..................................................................................................................... 29
3.2 LONGITUDINAL RECORDING ............................................................................................................... 29
3.2.1 Head Field..................................................................................................................................... 30
3.2.2 Signal Field ................................................................................................................................... 34
3.2.3 Signal of a Perfect Transition........................................................................................................ 34
4

Contents
3.2.4 Signal of Written Bits..................................................................................................................... 35
3.3 PERPENDICULAR RECORDING ............................................................................................................. 36
3.4 SIGNAL FIELD IN GAP ......................................................................................................................... 38
4 NUMERICAL METHODS....................................................................................................................... 41
4.1 FINITE ELEMENT METHOD.................................................................................................................. 42
4.1.1 Dirichlet Boundary Conditions ..................................................................................................... 44
4.1.2 Neumann Boundary Conditions..................................................................................................... 44
4.1.3 Linear equation solver................................................................................................................... 45
4.2 STRAY FIELD CALCULATION............................................................................................................... 46
4.3 ADAPTIVE CROSS-APPROXIMATION TECHNIQUE ................................................................................ 48
4.4 CONDUCTOR MODEL .......................................................................................................................... 49
4.4.1 Conductivity .................................................................................................................................. 49
4.4.2 Electric Potential........................................................................................................................... 50
4.4.3 Current Distribution...................................................................................................................... 50
4.4.4 Magnetic Field of Current............................................................................................................. 51
4.4.5 Hybrid FEM/BEM for Current Field............................................................................................. 52
4.4.6 Example: Long Wire...................................................................................................................... 54
4.4.7 Example: Coil................................................................................................................................ 55
4.5 TIME INTEGRATION ............................................................................................................................. 57
5 READ HEAD DESIGN ............................................................................................................................. 59
5.1 BIAS SCHEMES.................................................................................................................................... 59
5.1.1 Demagnetizing Factor................................................................................................................... 59
5.1.2 Hard Bias ...................................................................................................................................... 60
5.1.3 Exchange Bias ............................................................................................................................... 60
5.1.4 Synthetic Antiferromagnet............................................................................................................. 61
5.1.5 Sense Current ................................................................................................................................ 62
5.1.6 Crystalline Anisotropy................................................................................................................... 64
5.2 SHIELDING .......................................................................................................................................... 64
5.3 THERMAL STABILITY .......................................................................................................................... 64
6 FEM SIMULATIONS............................................................................................................................... 65
6.1 MODEL................................................................................................................................................ 65
6.1.1 Magnetic Model............................................................................................................................. 65
6.1.2 Conductor Model........................................................................................................................... 67
6.2 RESULTS ............................................................................................................................................. 69
6.2.1 Equilibrium State........................................................................................................................... 69
6.2.2 Transfer Curve .............................................................................................................................. 70
6.2.3 Influence of Shields ....................................................................................................................... 72
5

Contents
6.2.4 Relaxation...................................................................................................................................... 73
6.2.5 Signal of a Perfect Transition........................................................................................................ 74
6.2.6 Read Back Signal........................................................................................................................... 75
7 OUTLOOK................................................................................................................................................. 77
APPENDIX A ...................................................................................................................................................... 78
APPENDIX B ...................................................................................................................................................... 80
LIST OF FIGURES ............................................................................................................................................ 83
BIBLIOGRAPHY ............................................................................................................................................... 86
DANKSAGUNG.................................................................................................................................................. 90
LEBENSLAUF .................................................................................................................................................... 91
6

Introduction
1 IntroductionEquation Section 1
The idea to use magnetism for recording originates from Oberlin Smith (1840-1926), an
inventor, industrialist, and mechanical engineer. His idea based on a visit to Thomas Edison’s
laboratory in 1878. Lacking time to pursue further development - it is not known whether he
built a working device - he published a description of magnetic recording in Electrical World,
Sep. 8, 1888 [1]. Figure 1.1 shows the original drawing of Oberlin Smith. It demonstrates the
basic principle of magnetic audio recording used up to now. The audio waves are transformed
by the microphone (A) into an electrical signal and are recorded in the form of magnetization
patterns on a string covered with iron filings (C), passing through an electromagnet (B). To
reduce noise a battery (F) supplies bias current. The string is moved from the supply reel (D)
to the take up reel (E).
Probably inspired by Oberlin Smith’s article Valdemar Poulson built the first working
magnetic recording device (see Figure 1.1). He used an inductive head moving over a wire
wrapped drum. In the following decades a lot of improvements were invented, but as
information carrier mainly wire wrapped drums were used. Later tapes coated with magnetic
powder replaced them.
Figure 1.1: The original drawing of Oberlin Smith (1888) and Valdemar Poulson’s
telegraphone [2].
7

Introduction
In the late forties and the early fifties the first computing machines where developed. To store
program and data information punched paper cards or punched paper tapes where used. Later
magnetic tape recording was used, which had a big disadvantage. The random data request of
tapes was very time consuming, because the non-sequential access required long rewind and
fast-forward times. That is why there was the demand for a fast random access storage
system.
In the late 1940s a new computer design program started at the University at Berkeley to
develop an intermediate sized computer based on a magnetic drum memory. To get rapid data
access, high rotational speeds were used, which generated interest in non-contact recording
heads. Among the tested head designs, there was also a head with a wide gap and a drum with
a soft magnetic layer beneath the usual magnetic layer, the so called Soft-Under-Layer (SUL).
The result was that the magnetite dipoles were oriented perpendicular to the surface and not
along the surface circumferentially [3] (1949).
In 1956 IBM announced the first commercial magnetic disk drive, the Random Access
Method of Accounting and Control (RAMAC) 350. This drive had a storage capacity of 5 MB
and a data density of 2 kbit/in 2 . 50 24-in aluminum disks coated with iron oxide paint acted as
media. The whole stack of disks had a weight of 250 kg. Air pressure held the distance of
25 µm between head and disk. The complex air pressure mechanism was very expensive. That
is why only one actuator-arm assembly with a head pair - one for the upper surface and one
for the lower surface of a disk - was used (see Figure 1.2). Nevertheless the access time was
less than a second.
Figure 1.2: The first computer disk drive RAMAC. The pneumatically driven actuator arm
and some of the 50 disks are visible [4].
8

Introduction
In 1955, before the RAMAC was announced, IBM decided to start the development of the
next generation hard disks. The targets of the so called Advanced Disk File (ADF) project
were a 10 times larger capacity and a 1/10 the access time compared to RAMAC. The faster
access time could only be reached by using one head for each disk surface. It was clear that
the uneconomic air pressure system had to be replaced by flying heads (air bearing sliders).
Flying heads allow a smaller head-disk spacing, which finally allows higher linear data
densities. The experiences of the RAMAC lead to the decision to use a harder media. They
chose soft magnetic steel disks. Oxidizing in a steam atmosphere leads to a hard thin layer on
their surface. In addition the manufacturing costs were less than that of RAMAC’s aluminum
disks.
The project was very risky, because of lacking knowledge about air bearing heads and about
perpendicular recording. In 1960, due to several failures and major problems (e.g. the surface
quality of the steel disks was unacceptable), IBM turned away from perpendicular recording
and towards longitudinal recording. Due to major achievements in longitudinal recording
perpendicular recording became uninteresting for many decades.
All magnetic storage devices were using coils to read out magnetic information. In 1971
however, a new type of transducer using the anisotropy magneto-resistance effect (AMR) was
reported by R. B. Hunt [5]. AMR heads generate larger read back signals than inductive heads
for common relative velocities between disk and head. Another advantage of AMR heads
over inductive heads is their lower resistance and inductivity resulting in lower Johnson noise.
In 1991 the first commercial hard disks using AMR read heads were available and broke the
100 Mbit/in 2 recording density barrier.
In 1988 Baibich et al. [6] discovered a new magneto-resistance effect. Due to its large change
in resistance compared to AMR this effect was called Giant Magneto-Resistance (GMR).
GMR read heads were first introduced in commercial hard disks by IBM in 1997. The
invention of AMR and GMR read heads has accelerated the growth in data density (see Figure
1.3).
9

Introduction
Figure 1.3: The development of bit sizes and storage densities over the last years [7].
Before the introduction of magnetoresistive (MR) sensors in read heads the density doubled
every three years. When AMR read heads reached the market the doubling time was reduced
to two years. And when the GMR head was introduced, the density has been doubled each
year from 1998 to 2001. In the last years the growth rate decreased to 25-40%.
It is clear that this exponential growth cannot last forever. The major limitation, which
worries disk-drive builders most, is the so called superparamagnetic limit.
Superparamagnetism occurs for very small bits. Then the energy barrier is very small, so that
the magnetization can switch due to thermal fluctuations. The limit actually depends on the
media.
There are several different attempts to increase these limit with special media. One possibility
is to use antiferromagnetically coupled media [8] [9]. So the media exists of a second
ferromagnetic layer, which is antiferromagnetically coupled with the actual data layer. Both
layers are always magnetized in opposite directions, leading to a better thermal stability.
Other approaches are the reintroduction of perpendicular recording, the usage of patterned
media [10], where the bit cells are predefined as isolated magnetic domains in a nonmagnetic
matrix, or thermally or optically assisted magnetic recording [11]. At last to reach a maximum
data density there are experiments adapting an atomic force microscope to store information
at the scale of individual atoms [12].
10

2 BasicsEquation
Section (Next)
2.1 Ohm’s Law
Basics
Georg Simon Ohm (1789-1854) found that the current I through most materials is
proportional to the applied voltage V.
V = R⋅ I
(2.1)
The proportional coefficient is called resistance R. The more general form of this equation is
shown below.
σ⋅ E = j (2.2)
E [V/m] denotes the electric field, and j [A/m 2 ] the current density. σ [A/Vm] is a property
of matter named conductivity. Generally the conductivity is a tensor, but here all conductors
will be assumed to be isotropic.
2.2 Maxwell’s Equations
James Clerk Maxwell (1831-1879) was the first who joined the theories of electricity and
magnetism. He proposed following four equations to describe both theories (in SI).
� ∫ DdA= ∫ρdV
(2.3)
∂V
V
� ∫ BdA= 0
(2.4)
∂V
∂
� ∫Eds=− d
∂t
∫B
A
(2.5)
∂A
A
∂
� ∫ Hds= ∫jdA+ d
∂t
∫D
A
(2.6)
∂A
A A
These four integral equations can also be written as equivalent differential equations.
11

Basics
div D =ρ
(2.7)
div B = 0
(2.8)
∂B
curlE
=− (2.9)
∂t
∂D
curl H = j + (2.10)
∂t
D [C/m 2 ] is the electric displacement, B [T] is the magnetic induction, E [V/m] is the electric
field, H [A/m] is the magnetic field and j [A/m 2 ] denotes the current density. The third
equation (2.9) is the Law of Induction found by Michael Faraday (1791-1867), who first
recognized that an alternating magnetic field produces a current. The last equation (2.10) is
the generalization of Ampere’s Law:
� ∫ Hds= ∫jdA.
(2.11)
∂A
A
The magnetic induction B and the magnetic field H are related by
( )
B=μ H+ M =μ H+ J, (2.12)
0 0
and the displacement D and the electric field E by
Here
D=ε 0E+
P. (2.13)
μ = π× is the permeability of vacuum, and
−7
0 4 10 Vs/Am
ε = × is
−12
0 8.8542 10 As/Vm
the absolute dielectric constant of vacuum. M [A/m] denotes the total magnetization, J [T] the
magnetic polarization, and P the electric polarization [C/m 2 ].
Taking the divergence of both sides in (2.10) and using (2.7) lead to the continuum equation
∂ρ
div j + = 0 . (2.14)
∂t
2.3 Basic Equations of Magnetostatics
In case of vanishing field E = 0 , non-dielectric materials ( P = 0 ), and vanishing current
density j = 0 , (2.10) simplifies to
12

Basics
curl H = 0 . (2.15)
So H is a irrotational field and can therefore be expressed as gradient of a scalar field Ψ ,
called magnetic potential.
H =−grad Ψ
(2.16)
The magnetic potential of a volume V with a given magnetic polarization J can be calculated
by solving the Poisson equation
ρm
ΔΨ = − μ
with the jump condition
0
out in
∂Ψ ∂Ψ σ
− =−
∂n ∂n μ
∂V
m
0
(2.17)
(2.18)
at the boundary ∂ V . Here ρ m and σ m are magnetic charge densities in the volume V and at
its boundary ∂ V . These charges are only virtual, because there are no magnetic monopoles in
reality. These virtual charges are useful to calculate the magnetic potential of a polarization
distribution. They can be obtained by taking the divergence and the face divergence of the
magnetic polarization J.
ρ =−div J, σ =−Div
J (2.19)
m m
The face divergence is defined as difference of the normal components of outer and inner
field.
( )
out in out in
Div J = J −J ⋅ n = J −J
(2.20)
n n
Assuming that the magnetic potential is regular at infinity,
1
Ψ→ for r →∞, (2.21)
r
the solution of (2.17) and (2.18) is
1 ⎛ ρ σ ⎞
Ψ= dV + dA
4
∫ ∫� . (2.22)
m m
⎜ ⎟
πμ0 ⎝ r r
V ∂V
⎠
13

Basics
2.4 Inhomogeneous Conductors
To calculate the read back signal of magnetoresistive heads, it is necessary to calculate the
current distribution, because the conductivity does not only depend on locus, but also on the
magnetization:
σ=σrMr (, (,)) t . (2.23)
Consequently a general treatment is necessary to calculate the electric potential and the
current density. The electrical field can be splitted into two parts, an irrotational and a
solenoidal part:
with
and
E= Efree + E eddy
(2.24)
curlE 0
(2.25)
free =
divE 0 . (2.26)
eddy =
It can be shown, that the irrotational part of the electrical field, denoted with E free , is caused
by external applied voltage or current. The solenoidal part denoted by E eddy is due to magnetic
induction described by Faraday’s law (2.9) and originates from eddy currents. For the moment
we can neglect eddy currents. Later they can be partially taken into account by choosing
higher damping constants in the LLG equation (see Section 2.7) [41] [42]. This implies, that
the total electrical field is irrotational
curlE = 0
(2.27)
and can be written as gradient of a scalar potential function Φ , also called electric potential
E =−grad Φ.
(2.28)
Furthermore the whole problem is considered to be quasi-static. In other words the
displacement current term in (2.10) can be neglected
curl H = j. (2.29)
14

Basics
This is justified, because the sense current through MR sensors is normally held constant and
the current density is quite large for the small dimensions of a MR reading head. (The current
density is only limited due to the heat dissipation. Typical values for the critical current
density are 10 7 -10 8 A/cm 2 .) So local charge concentrations are immediately compensated.
Applying the divergence operator on both sides of equation (2.29) gives
div j = 0 . (2.30)
Thus the source term of the continuum equation (2.14) vanishes. Combining (2.2), (2.28), and
(2.30) leads to a generalized form of Laplace equation
div( σgrad Φ ) = 0.
(2.31)
If the electric potential is known, it is possible to evaluate the current density using
j =−σgrad Φ.
(2.32)
2.5 Gibbs’ Free Energy
The equilibrium state of an isothermal and isobaric thermodynamic system is characterized by
the minimum of its Gibbs’ free energy E.
∫
E = F − J⋅H dV
(2.33)
ext
F = U −T⋅ S+ p⋅ V is the free energy, where U is the inner energy, T the absolute
temperature, S the entropy, and V the volume of the magnetic system. For constant
temperatures and neglected magnetostrictive effects, which means keeping the volume and
the temperature constant, the free energy is reduced to U. The inner energy U can be
calculated as the sum of the stray field energy E S , the exchange energy E exch , and the
magnetocrystalline anisotropy energy E ani , so the total Gibbs’ free energy can be written as
E = ES+ Eexch + Eani+ EH.
(2.34)
Here E H is the magnetostatic energy and arises from the external field H ext . The equilibrium
state of such a magnetic system is determined by
15

Basics
δ E
= 0 . (2.35)
δJ
2.5.1 Magnetostatic Energy
The energy associated with a magnetized body in the presence of an external field H ext is
called magnetic field energy and can be calculated via
∫
E =− J⋅H dV . (2.36)
H ext
V
This energy will cause the magnetization to align parallel to the external field in order to
minimize the energy.
2.5.2 Stray Field Energy
The stray field energy term writes similar to the magnetic field energy.
1
ES =− ⋅ S dV
2 ∫ J H (2.37)
V
The additional factor ½ takes into account, that the stray field H S , which is also called
demagnetizing field, is caused by the magnetization itself. This is in contrast to the magnetic
energy where an external field is given, which is not influenced by the magnetization. The
stray field can be determined by calculating the magnetic scalar potential Ψ S as described in
Section 2.3. Then the stray field can be calculated using (2.16):
H =−grad Ψ . (2.38)
S S
2.5.3 Exchange Energy
In the approach of micromagnetics the quantum mechanical spin operators are substituted by
classical magnetization vectors. Thus, the exchange energy in the Heisenberg model [13] can
be expressed as
N N
exch =−∑∑ i j
ij ⋅S ⋅S
. (2.39)
i= 1 j≠i E J
16

Basics
Here J ij represents the exchange integral. In case of ferromagnetism the exchange integral is
positive and leads to the observed strong coupling and parallel alignment of spins.
Considering only nearest neighbor (n.n.) angular momentums and assuming a constant
exchange integral Jij � J lead to
N n. n.
i j
Eexch =−J ∑∑S ⋅S
. (2.40)
i= 1 j≠i The assumption of a constant exchange integral is justified, because experiments show only
small differences when measuring along different crystallographic directions.
As angular moments are described by a continuous function J = J() r in micromagnetics, the
sum can be approached by a volume integral,
A
E = ( ∇ J ) + ( ∇ J ) + ( ∇J
) dV
2 ∫ . (2.41)
2 2 2
exch
J s V
x y z
Here J s is the spontaneous polarization, and the exchange constant A [J/m] is proportional to
the exchange integral J. The value of A has been found experimentally for a lot of magnetic
materials.
2.5.4 Magnetocrystalline Anisotropy Energy
In general a ferromagnetic material does not show isotropic magnetization behavior.
Experimentally certain directions can be found, which favor magnetization (easy axes) and
others which do not (hard axes). This leads to a new energy term.
ani ani
V
( )
E = ∫ f J dV
(2.42)
Here f ani denotes a direction dependent function. In case of hexagonal systems (e.g. Co) a
uniaxial anisotropy can be assumed. Therefore f ani can be expressed as series expansion
f ( θ ) = K + K sin θ+ K sin θ+ ... . (2.43)
ani
2 4
0 1 2
θ denotes the angle between J and the c-axis of the hexagonal system and 0 K , 1 K , 2
are the so called anisotropy constants. In practice all higher terms are neglected.
K , etc.
In case of cubic systems, their cubic symmetry can be taken into account, if f ani is written as
17

Basics
2 2 2 2 2 2 2 2 2
( ) ( )
f ( α , α , α ) = K + K α α +α α +α α + K α α α + ... . (2.44)
ani
1 2 3 0 1 1 2 2 3 3 1 2 1 2 3
α 1 , 2 α , and α 3 denote the direction cosines of the cubic lattice vectors to the direction of the
magnetic polarization J.
In both cases the first term K 0 can be neglected, because it does not depend on angle and is
therefore a constant term.
2.6 Effective Field
The effective magnetic field H eff which acts on the magnetization can be calculated by the
variation of Gibbs’ Free Energy.
δE
2A δfani
( J)
H =− = ΔJ− + H + H + H
eff 2
S ext curr
δJ J s δJ
(2.45)
Here the first term corresponds to the exchange field H exch , and the second one to the
anisotropy field H ani . The additional contribution H curr is the field due to currents. In fact this
field is a contribution to the external field H ext , but here it is written separately, because from
now on ext H is used to describe homogeneous fields, whereas H curr are fields due to currents,
and can be calculated using the Biot-Savart Law
1 r−r′ Hcurr () r = ×
4π
∫ j
r−r′ V
3
dV ′ , (2.46)
which was empirically found by Jean Baptiste Biot and Félix Savart in 1820.
2.7 Landau-Lifshitz-Gilbert Equation
To find out the equilibrium state of a magnetic system equation (2.35) can be used. To get
information about the magnetization dynamics, i.e. about the dynamic behavior of the
magnetic polarization J, it is necessary to apply a full 3-dimensional treatment of the time
evolution of the magnetic polarization, which is described by the Gilbert equation of motion
18

s
Basics
∂J α ∂J
=−γ J× Heff + J × . (2.47)
∂t J ∂t
This equation is mathematically equivalent to the Landau-Lifshitz-Gilbert equation (LLG)
∂Jγ γ α
=− J× Heff − J× ( J× H eff ) . (2.48)
∂ t +α J +α
2 2
( 1 ) s ( 1 )
Here J s is the spontaneous polarization and γ is the gyromagnetic ratio
5
γ= 2.210175× 10 m/As . The first term in this equation describes the precessional movement
of the magnetic polarization vector J around the effective field H eff . The second
phenomenological term stands for the energy dissipation during the movement and causes a
movement towards the direction of the effective field. The motion of the magnetization is
visualized in Figure 2.1a. The strength of the damping is characterized by the dimensionless
Gilbert damping constant α . Large α means strong damping. There are many processes
which contribute to the damping in magnetic solids, for example magnon-magnon, magnon-
phonon interactions, or eddy currents. So the damping constant is an intrinsic property of
matter and can also be varied in practice, for instance by doping [14]. In general, except for
thin layers (see Section 6.2.4), high damping constants lead to coherent motion of the
magnetization and therefore to faster reversal processes. Figure 2.1b shows the field of a
simulated write head as function of time for different Gilbert damping constants. It can be
seen that for damping constants of about 0.5 the fastest switching time is achieved [43].
∂ J
In equilibrium, thus if = 0 , equation (2.47) simplifies to
∂t
J× H eff = 0,
(2.49)
which means that in equilibrium the magnetic polarization J is aligned either parallel or
antiparallel to the effective field H eff . The unrealistic case of antiparallel alignment is
irrelevant, because this solution is not stable, and in this case small deviations lead to a
movement in direction of the effective field due to the damping term in (2.48) (see also Figure
2.1a).
19

a)
Heff
J
−J x Jx Heff
−J x Heff
b)
Basics
Figure 2.1: a) Illustration of the damped precession of the magnetic polarization J around
the effective field Heff, taken from [15]. b) Influence of the Gilbert damping constant on the
field rise time in magnetic read heads. Head field strength as a function of time for different
Gilbert damping constants within the head [43]. The dashed line depicts the current profile.
The solid lines give the head field for α = 1, 0.5, 0.1, and 0.02. The fastest field rise time is
achieved with intermediate damping (α = 0.5).
2.8 Magnetoresistance
2.8.1 Magnetoresistance of Ferromagnets
A change in resistance of a material under an applied field is known as magnetoresistance.
This effect was first discovered in 1856 by Lord Kelvin who was examining the resistance of
an iron sample. He found a 0.2% increase of resistance when he applied a field in longitudinal
direction, and a 0.4% decrease for the transversal direction.
Thus, a positive and a negative change of resistance is possible, although intuitively one
expects only positive magnetoresistance, because electrons are forced to take longer paths due
to the magnetic field, which leads to more scattering. Nevertheless negative
magnetoresistance occurs, which has to be explained in a different way. Mott [16] [17] could
give a first explanation for this behavior. Figure 2.2 shows the typical band configuration of a
transition metal. The bands are spontaneously split without external field. This phenomenon is
also called band ferromagnetism. Scattering of electrons from the s- to the d-bands occurs
only, if there are any unoccupied states available. Figure 2.2 shows the electron configuration
at low temperatures. Here the spin up d-band is fully occupied. Thus scattering can only occur
20

Basics
for spin down electrons. An applied magnetic field may increase the spin polarization and
allows fewer s to d transitions, resulting in lower resistance.
Figure 2.2: The spin-split bands of a ferromagnet [18].
2.8.2 Anisotropic Magnetoresistance Effect
The resistance of several ferromagnetic materials shows a dependence on the angle between
magnetization and current. This effect is known as anisotropic magnetoresistance (AMR). Its
origin is connected with the spin-orbit interaction and its influence on s-d scattering.
This phenomenon can be described by assuming different resistivities for currents parallel and
perpendicular to the magnetic field and leads to following relation:
0
2
( 1 RAMR
cos )
ρ=ρ + ⋅ θ . (2.50)
Here θ is the angle between the current and the magnetization, and R AMR is the
magnetoresistance ratio. Typical values of R AMR are about 2%.
The concept of using the AMR effect for reading heads was first proposed by Hunt in 1971
[5]. AMR read heads were used from 1992 to 1998 in commercial hard disks. Their advantage
was the much higher signal amplitude for smaller bits (higher data densities) compared to
inductive read heads. Additionally magnetoresistive heads have reduced noise and inductance,
and the output voltage is not dependent on the relative velocity of the head and the data layer.
21

2.8.3 Giant Magnetoresistance Effect
Basics
Giant magnetoresistance was first reported by Baibich et al. in 1988 [6]. They discovered that
the resistance of Fe/Cr multilayers decreased 50% when they applied a magnetic field (see
Figure 2.3).
Figure 2.3: Giant magnetoresistance in Fe/Cr multilayers [18].
Without applying an external field the magnetic layers are antiparallel coupled. If a strong
enough field is applied to align the magnetization of adjacent magnetic layers, the resistivity
of the multilayers drops significantly. Thus resistivity is lower in case of parallel alignment
and higher for the antiparallel configuration. Multilayers require very high saturation fields to
overcome the strong antiferromagnetic coupling. Therefore the field sensitivity of such
multilayers is actually lower compared to sensors using the AMR effect. These considerations
motivated the invention of an alternative GMR structure for sensors shown in Figure 2.4. It
consists of four main layers. There are two ferromagnetic layers which are separated by a
nonmagnetic layer, called spacer layer (mostly Cu). All these layers are only few nanometers
thick. The magnetization of one ferromagnetic layer is fixed via the exchange anisotropy of an
antiferromagnetic layer (see Section 5.1.2). The magnetization of the other ferromagnetic
layer can rotate freely and is therefore called free layer. In contrast to multilayer structures
this structure does not show any hysteresis.
22

Basics
Figure 2.4: Schematic model of a spin-valve GMR head (CIP mode) with exchange-pinned
layer and longitudinal hard bias [19].
The change in resistance of the GMR effect can be mathematically approximated by
following formula:
( θ θ )
⎛ 1−cos 1− 2 ⎞
ρ = ρ0⎜1+ RGMR
⋅
⎟
⎝ 2 ⎠
(2.51)
Figure 2.5 shows the dependence of the resistivity as function of the angle between the
magnetization orientations of the two ferromagnetic layers. To achieve highest sensitivity, the
working point of the GMR spin valve should be chosen at maximum slope. If there is no
external field, the magnetization of the free layer should be ideally perpendicular to the
pinned layer magnetization. To adjust the magnetization of the free layer, hard bias magnets
are used as shown in Figure 2.4.
Figure 2.5: Change in resistance of a spin valve compared to AMR (NiFe) vs.
magnetization orientation [19].
23

Basics
There are two different modes a spin valve can be operated, the current-in-plane (CIP) mode,
and the current-perpendicular-to-plane (CPP) mode. The difference between these two modes
is the current direction. In the first case the sense current is parallel to the layers of the GMR
element, while for the second case the current is applied perpendicular to the layers. The two
different modes are illustrated in Figure 2.6.
Figure 2.6: The two different modes of a spin valve. The current is applied either
longitudinal (CIP) or perpendicular (CPP) to the layer structure [7].
Up to now only the CIP configuration is used in commercial hard disks. For smaller read head
designs the CPP mode becomes more and more interesting, because the output voltage for
CPP heads is roughly inversely proportional to the square root of the sensor area. In addition
the change in resistance is higher, because the sense current is not shunted by the conductive
spacer layer, which does not contribute to magneto-resistance. Besides the CPP structure
allows smaller read gaps, thus smaller distances between the two shields, because the shields
can be used as leads. This design allows read gaps down to 20 nm, compared to 60 nm for
CIP heads. CPP heads are thermally more stable, because the layer structure is in contact to
the relative large shields leading to a better heat dissipation.
The origin of the GMR effect is the spin dependent scattering of FM/NM/FM multilayers.
The spin dependent scattering at interfaces is the main contribution. Figure 2.7 shows a
schematic picture of the potential landscape of a GMR layer structure for the two spin
24

Basics
channels in the parallel and in the antiparallel state, indicating the different origins of
scattering. Only deviations from the periodic atomic potentials are shown, as no scattering
takes place in a regular periodic lattice. The size of the potential step and of the scattering
potentials at imperfections of the interfaces depends on the degree of matching of the
electronic structures of the ferromagnetic layers and the nonmagnetic spacing layer. In order
to obtain a large GMR ratio, good electronic structure matching is required for one spin
direction and bad matching for the opposite spin direction. It is clear that giant
magnetoresistance only occurs in layers with thicknesses much smaller than the electron
mean-free path.
Figure 2.7: Potential landscape inside the FM/NM/FM active part of a spin-valve, for the
two spin channels in the parallel state (a,b) and in the antiparallel state (c), indicating the
different origins of scattering [20].
The GMR effect can be qualitatively described by a very simplified two current model. The
current densities are assumed to be homogenous for both spin directions. If the resistances for
the ‘+’ and ‘-‘ channels in the parallel state are denoted with R + and R − , then the total
resistance for the parallel state is
25

R
↑↑
R+ R−
=
R + R
+ −
Basics
. (2.52)
R+ + R−
The resistance of both channels for the antiparallel configuration is set to resulting in
2
a total resistance of
R + R
= . (2.53)
4
+ −
R ↑↓
For the GMR ratio follows
( ) 2
R − R
Δ R R − R ↑↓ ↑↑ + −
= = . (2.54)
R R 4R
R
↑↑
+ −
This value is always positive, which means that the resistance in the anti-parallel
configuration is higher than in the parallel configuration, which is consistent with (2.51).
2.8.4 Tunneling Magnetoresistive Effect
Another magnetoresistive effect is the tunneling magnetoresistive (TMR) effect. The design
of a TMR element is quite similar to a GMR element in CPP mode. The main difference is the
insulating layer between pinned and free layer. The origin is also different, because TMR is
due to spin dependent tunneling and not due to spin dependent scattering. With TMR
elements changes in resistance up to 25% can be achieved at room temperature. The
application in read heads is restricted because of the high resistance, which limits the
operating frequency and causes Johnson and Shot noise.
2.9 Recording Schemes
There are three possibilities to align the written bits in the data layer. Longitudinal to track,
perpendicular to track, or transversal to the data layer called longitudinal, perpendicular and
transversal recording respectively. Only first both recording schemes are of technical interest,
because transversal written bits are hard to detect, because of their small stray fields.
Although there were also attempts to use the perpendicular scheme in the early years, the
longitudinal mode is well established and is used in all commercial hard disk drives.
26

Basics
Nowadays there are again efforts to use perpendicular recording because of the theoretical
higher data density limit.
2.9.1 Longitudinal Recording
The principle of longitudinal recording is illustrated in Figure 2.8. For the writing process a
ring head is used, which generates a stray field at the gap above the recording layer, so that it
is magnetized in longitudinal direction. To retrieve the written information, the read head
detects the stray field of the data layer, which has its maxima at the transitions of two opposite
magnetized bits. In longitudinal recording the read back signal is roughly proportional to the
first derivation of the magnetization.
Figure 2.8: A comparison of longitudinal recording and perpendicular recording [7].
2.9.2 Perpendicular Recording
As shown in Figure 2.8 an additional soft magnetic underlayer is necessary for perpendicular
recording. This soft under layer (SUL) closes the magnetic circuit. The yoke of the write head
has a small write pole, which bundles the flux to achieve high writing fields, and a wide return
27

Basics
pole, so that the flux density is small and can not influence the magnetization of the recording
layer. Here the read back signal is directly proportional to the magnetization of the data layer.
28

Analytical Calculations
3 Analytical CalculationsEquation Section (Next)
3.1 Reciprocity Principle
For any two volumina V 1 and V 2 with given magnetization the following equation is valid.
∫ ∫
Mr ( ) ⋅ H( r) dV = Mr ( ) ⋅H(
r ) dV
(3.1)
1 2 1 1 2 1 2 2
V V
1 2
Here 1 H and 2
H denote the magnetic fields caused by the magnetizations in volumes 1 and 2
respectively. This fact is called reciprocity principle, and its proof is illustrated below.
Partial integration of (3.1) and taking equations (2.16) and (2.19) into consideration lead to
�∫
∫
σ ( r) ⋅Ψ ( r) dA − ρ ( r) ⋅Ψ ( r)
dV =
m 1 2 1 1 m 1 2 1 1
∂V1
V1
�∫
∫
σ ( r ) ⋅Ψ ( r ) dA − ρ ( r ) ⋅Ψ ( r ) dV .
m 2 1 2 2 m 2 1 2 2
∂V2
V2
The final substitution of both magnetic potentials using (2.22) shows equality.
3.2 Longitudinal Recording
Longitudinal recording means that the magnetization of written bits shows in track direction.
To read information it is necessary to detect the stray field of the stored bits. For longitudinal
recording the stray field reaches its maximum at the transitions, where opposite oriented
magnetizations come together.
(3.2)
Figure 3.1 shows a schematic model of a read head. In the middle of two shields there is the
sensing layer. In case of a GMR sensor this layer is equal to the free layer. The shields are
assumed to have infinite permeability and infinite extension. The plane, which contains the
bottom sides of the two shields and the sensing layer, is called air bearing surface (ABS).
29

Analytical Calculations
With the help of the reciprocity principle we are now able to calculate the effective signal
field
eff
H sig caused by the data layer at the sensing layer. Here only the z-component of the field
will be considered. The field components in the other direction can be neglected since the
sensor only registers the z-component anyway. The effective field
field H = H ( r ) , which acts on the magnetization of the sensing layer (SL).
sig sig
eff
H sig is the average signal
eff
1
Hsig = Hsig= Hsig() dV
V ∫ r (3.3)
SL VSL
We assume that the magnetization of the sensing layer is uniform. With this condition the
reciprocity principle is applied on the volume of the sensing layer V SL and the volume of the
data layer.
eff
V M H = Mr () ⋅H()
r dV
(3.4)
SL SL sig H
V
∫
DL
Here H H is the so called head field. It is the field which is generated by the magnetization of
the sensing layer.
3.2.1 Head Field
z
x
Shield
Shield
t
µ → ∞ µ → ∞
g g
Figure 3.1: A schematic 2-dimensional model of a read head for longitudinal recording.
The head field of the reading head due to the magnetization of the sense layer can be
calculated using the Karlquist approximation [21]. This approximation assumes a linear
MR
d
Data Layer
30

Analytical Calculations
developing of the magnetic potential Ψ between the shields and the sensing layer at the ABS.
The magnetic potential is assumed to vanish at the ABS of the shields due to their high
permeability.
⎧ 1 x < t/2
⎪
⎪ g+ t/2+ x
Ψ ( x,0) = C⋅MSL ⋅⎨ t/2 ≤ x < g+ t/2
⎪ g
⎪
⎩
0 g+ t/2≤ x
Figure 3.2: The magnetic potential at the ABS under Karlquist approximation.
This function is shown in Figure 3.2. The amplitude of this function is proportional to the
magnetization of the sensing layer. The magnetization of the sensing layer can be replaced by
(3.5)
top
face charges m 0 SL M σ =μ at the bottom and bottom
m 0 SL M
σ =−μ at the top side under the terms of
(2.19). The top face charge can be neglected, because the sensing layer height is usually much
larger than the distance between ABS and data layer surface. Common flying heights are
about 10 nm. The magnetic charge at the bottom side leads to a jump of the magnetic field
H − H =− M . Due to the symmetry of the surface charge in the near region
in out
z z SL
z
H (0,0) = M / 2
(3.6)
H SL
Ψ(x,0)
C⋅MSL
-t/2-g -t/2 t/2 t/2+g x
can be assumed. This is the necessary condition, which allows the determination of the
unknown proportional factor C in (3.5).
31

Analytical Calculations
To calculate the magnetic potential for the whole half space with the boundary condition
given in (3.5) one has to solve the Laplace equation
ΔΨ ( xz , ) = 0.
(3.7)
This can be done analytically by a Fourier transformation in x [19]. The general solution of
this boundary value problem is
∞
z Ψ(
x,0)
Ψ ( x, z) =− dx′
π ∫ . (3.8)
′
−∞
( ) 2 2
x− x + z
Applying (3.8) on the boundary condition (3.5) yields
with
C⋅MSL Ψ ( x, z) = Γ ( x+ g, z) −Γ ( x, z) +Γ( − x+ g, z) −Γ( −x,
z)
g ⋅π
( )
( ( ) ) ( )
2 2 + /2
(3.9)
z ⎛ x t ⎞
Γ ( xz , ): = ln x+ t/2 + z − x+ t/2
arctan⎜
⎟.
(3.10)
2
⎝ z ⎠
Now we are able to evaluate the head field by taking the gradient of the magnetic potential.
⎛γ x( x + gz , ) −γx( xz , ) −γx( − x+ gz , ) +γx( −xz
, ) ⎞
CMSL
⎜ ⎟
H H ( xz , ) = ⋅ 0
g ⎜ ⎟
(3.11)
π ⎜γ z( x + gz , ) −γ z( xz , ) +γz( − x+ gz , ) −γz( −xz
, ) ⎟
⎝ ⎠
Here x γ and γ z are the derivatives of Γ .
∂ ⎛ x+ t/2⎞
γ x ( xz , ): = Γ ( xz , ) =−arctan⎜ ⎟
∂x ⎝ z ⎠ (3.12)
2
2
( ( ) )
∂
γ z ( xz , ): = Γ ( xz , ) = 1+ 1/2⋅ ln z + x+ t/2
∂z
(3.13)
Now we have to determine the proportional constant C. Here we assume that 2g � t,
which
yields
H
2CM ⎛ g+ t/2⎞
4CM
(0,0) = ln ⎜ ⎟�
. (3.14)
πg ⎝ t/2 ⎠ πt
z
H
SL SL
Comparing with (3.6) gives
32

Analytical Calculations
t
C
8
π
= . (3.15)
Inserting in (3.11) leads to the final result for the head field
⎛γ x( x + gz , ) −γx( xz , ) −γx( − x+ gz , ) +γx( −xz
, ) ⎞
MSL t ⎜ ⎟
H H ( xz , ) = ⋅ 0
8 g ⎜ ⎟
. (3.16)
⎜γ z( x + gz , ) −γ z( xz , ) +γz( − x+ gz , ) −γz( −xz
, ) ⎟
⎝ ⎠
Now a normalized head field H H can be introduced, which is independent from the sensor
magnetization
H
n
H
⎛γ x( x + gz , ) −γ x( xz , ) −γ x( − x+ gz , ) + γ x(
−xz
, ) ⎞
t ⎜ ⎟
( xz , ) = ⋅ 0
8 g ⎜ ⎟
. (3.17)
⎜
z( x + gz , ) − z( xz , ) + z( − x+ gz , ) − z(
−xz
, ) ⎟
⎝γ γ γ γ ⎠
Figure 3.3 shows the normalized head field at z = − 10 nm and for t = 5 nm and g = 27.5 nm ,
so that the total gap is 60 nm. For longitudinal recording only the x-component contributes,
because the magnetization points mainly in this direction.
H x
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-100 -50 0 50 100
x [nm]
H z
-0.02
-100 -50 0 50 100
x [nm]
Figure 3.3: The x- and the z-component of the normalized head field of a read head with
60 nm gap, and sensing layer thickness 5 nm, 10 nm below the ABS.
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
33

3.2.2 Signal Field
Analytical Calculations
Now we are able to calculate the effective signal field from (3.4)
eff 1
n
Hsig = () ⋅ H () dV
V ∫ Mr H r . (3.18)
SL V
DL
In case of a moving head with velocity v the effective signal field can be written as function
of time
eff 1
n
Hsig () t = ( ) ⋅ H ( − t) dV
V ∫ Mr H r v . (3.19)
SL V
DL
The advantage of using the reciprocity principle is that the head field has to be calculated only
once. Then the signal field can be determined by folding the head field with the magnetization
of the recording layer.
3.2.3 Signal of a Perfect Transition
The maximum signal field is achieved for perfect transitions, which are defined in case of
longitudinal recording by
J ( x, z) =± J ⋅ sgn( x)
. (3.20)
x s
Here J s is the spontaneous polarization, which is in the range of 0.3 T to 0.5 T for common
recording layers. For the head field shown in Figure 3.3 we get a pulse for the effective signal
field, which is shown in Figure 3.4 for a 10 nm thick data layer positioned 10 nm below the
ABS. The maximum signal field is reached whenever the center of the head is exactly over
the transition. The full width at half maximum is about 50 nm and is therefore in the order of
the distance between the two shields (60 nm).
34

µ 0 Hsig [T]
0.030
0.025
0.020
0.015
0.010
0.005
Analytical Calculations
0.000
-200 -100 0 100 200
x [nm]
Figure 3.4: The effective signal field as function of the relative position of the read head in
respect to a perfect transition (Js = 0.5 T).
3.2.4 Signal of Written Bits
The maximum read back signal is achieved for perfect transitions. In practice written bits do
not have perfect transitions. Figure 3.5 shows the results of two recording simulations with a
head velocity of 10 m/s and 20 m/s.
Figure 3.5: These two figures show the x-component of the magnetization (Js = 0.5 T) of
the data layer after a write process with head velocity 10 m/s (left) and 20 m/s (right). Red
color represents magnetization in positive x-direction and blue color magnetization in
negative x-direction. The scale is in µm, thus the bit length is approximately 40 nm. The
data layer thickness is 12 nm.
35

Analytical Calculations
The head velocity influences the bit quality. The higher the velocity, the worse the quality of
the bit transitions is.
For both data layer structures the read back signals were calculated. Again a head with 60 nm
gap, 5 nm free layer, and a flying height of 10 nm was assumed. The spontaneous
magnetization of the data layer was again J s = 0.5 T , but the data layer thickness was 12 nm.
Figure 3.6 show smaller amplitudes for the effective field compared to the amplitude of the
perfect transition. The worse transition quality of the faster written bits leads to a smaller
signal.
µ 0 Hsig [T]
0.020
0.015
0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.020
0 50 100 150
x [nm]
200 250 300
µ 0 Hsig [T]
0 50 100 150 200 250 300
x [nm]
Figure 3.6: The effective fields of the recording layers (shown in Figure 3.5) as function of
the head’s position. The dashed lines indicate the boundaries of the recording layer model.
3.3 Perpendicular Recording
For perpendicular recording we can use the reciprocity principle again. We only have to take
the SUL into account (see Figure 3.7). Analogically to the shields the SUL is also assumed to
have infinite permeability. Then the problem is equivalent to assuming a second read head,
mirrored at the SUL surface as shown in Figure 3.7 [22]. In addition the two shields with their
high permeability act as second mirror. Therefore the head field for perpendicular recording
can be approximated by a series expansion of head fields of the original head and its
reflections at the SUL surface and the ABS. If the distance ABS to SUL is denoted with L, the
series expansion of the head field for the perpendicular case can be written using (3.17)
0.020
0.015
0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.020
36

Analytical Calculations
n n
⎛HH, x( x, z− 2 iL) + HH, x(
x, −z− 2( i+ 1) L)
⎞
∞ ⎜ ⎟
n
H H ( xz , ) = ∑⎜0
⎟.
(3.21)
k = 0 ⎜ n n
HH, z( x, z−2 iL) −HH, z(
x, −z− 2( i+ 1) L)
⎟
⎝ ⎠
Figure 3.7: The simplified read head model (left) for perpendicular recording. In the right
picture the mirror heads are indicated, which replace the SUL with its high permeability
[22].
The headfield for perpendicular recording is shown in Figure 3.8. Here the gap was again
60 nm and the sensing layer thickness 5 nm. The flying height was assumed to be 10 nm and
the distance ABS to SUL 20 nm. The series expansion was aborted after the first 10 terms.
For perpendicular recording, where the data layer is mainly magnetized in z-direction, only
H contributes to the signal field. So for perpendicular recording the signal field is
n
H , z
approximately proportional to the magnetization of the data layer. In contrast to longitudinal
recording the bits are detected and not their transitions.
37

H x
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-100 -50 0 50 100
x [nm]
Analytical Calculations
H z
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
-100 -50 0 50 100
x [nm]
Figure 3.8: The x- and the z-component of the normalized head field for perpendicular
recording.
3.4 Signal Field in Gap
After calculating the signal field H sig of the data layer at the air-bearing surface we are now
able to estimate the run of the magnetic field between the two shields. By averaging H sig over
the MR sensor’s height h we can determine the effective field
eff
h
1
sig sig
h 0
sig
eff
H sig acting on the sensor.
H = H = ∫ H ( z) dz
(3.22)
To get Hsig ( z ) we assume that the field in gap- (x-) direction Hg( z ) is constant between
sensor and shield and only depends on z.
As shown in Figure 3.9 we use Ampere’s Law on the dashed marked path, which gives
and finally
H ( z) ⋅ dz + H ( z + dz) ⋅g−H( z) ⋅ g = 0
(3.23)
sig g g
dH g
Hsig ( z) =− g . (3.24)
dz
38

z
Analytical Calculations
Figure 3.9: a) Application of Ampere’s Law on the dashed path, b) application of Gauss’s
Law on the dashed box.
In addition we use Gauss’ Law for a box with infinitesimal small height dz over the sensing
layer:
x
a) g g
b)
−μ ⋅μ ⋅H ( z) ⋅ t+ 2 ⋅μ ⋅H ( z) ⋅ dz+μ ⋅μ ⋅ H ( z+ dz) ⋅ t = 0.
(3.25)
r 0 sig 0 g r 0 sig
Here μ r is the relative permeability of the sensing layer. The last equation is equivalent to
1 dH sig
Hg( z) =− ⋅μr ⋅t⋅ (3.26)
2 dz
Combining (3.24) and (3.26) gives a differential equation.
2
dHsig
2
1
Hsig ( z) = ⋅μr⋅g⋅t⋅ (3.27)
2 dz
Assuming the boundary conditions
ABS
H (0) = H and H ( h)
= 0
(3.28)
sig sig sig
leads to the solution
⎛h−z⎞ sinh ⎜ ⎟
λ
Hsig ( z) = Hsig⋅
⎝ ⎠
with λ=
⎛h⎞ sinh ⎜ ⎟
⎝λ⎠ ABS r
μ ⋅g⋅t . (3.29)
2
Here λ is the flux propagation length. Now the effective field can be calculated under the
terms of (3.22):
Hsig
t
Hg Hg
Hg
dz
z
x
Hsig
t
dz
g g
39

H = H
eff ABS
sig sig
λ cosh( h / λ) −1
h sinh( h/
λ)
Analytical Calculations
For large flux propagation lengths λ � h , which is normally the case, (3.30) simplifies to
(3.30)
h
eff ABS λ ⎛ − ⎞
λ
Hsig = Hsig⎜1−e ⎟.
(3.31)
h ⎝ ⎠
So the signal field roughly decreases exponentially between the shields from the ABS
upwards.
40

4 Numerical MethodsEquation
Numerical Methods
Section (Next)
Micromagnetic simulations require a discrete representation of the continuous magnetization
distribution on a computer. Due to the strong ferromagnetic coupling the magnetization does
not vary rapidly over short distances. So the discretization size can be much larger than the
atomic lattice constant. Usual lengths of significant magnetization changes (widths of domain
walls) are in the order of some nanometers for hard magnetic materials, for soft magnetic
materials even one or two orders of magnitude larger. To keep numerical errors small, the
discretization size should be smaller than the domain wall widths.
Dynamic micromagnetic calculations require two major tasks. The first part is to calculate the
stray field. For this purpose a partial differential equation (PDE) of the form
Lu [ ( r)] = s(
r )
(4.1)
has to be solved. Here L denotes the differential operator, s is called source term and u is the
unknown function, which fulfills this PDE. For stray field calculations u represents the
magnetic potential. We also have to solve an equation of type (4.1) for the electric potential of
an inhomogeneous conductor (see (2.31)). For this purpose the Finite Element Method is
used.
As second task we have solve an initial-value problem (IVP) of the form:
y� = f(, t y), y( t ) = y . (4.2)
0 0
In case of the LLG equation y represents the magnetization. Therefore y 0 would be the initial
magnetization. Here the time has to be discretized. We use a time integration routine, which
automatically adapts the time steps. So for fast physical processes a smaller time step is
chosen to increase accuracy.
41

4.1 Finite Element Method
Numerical Methods
The Finite Element Method (FEM) is a well established tool for solving partial differential
equations (PDE). The main advantage of FEM is the good applicability on arbitrary shaped
geometries. Usually tetrahedrons are used to fractionalize the model. In addition the
discretization size, i.e. the size of the tetrahedrons, can be easily varied over the whole model.
The general form of a PDE is
Lu [ � ( r)] = f(
r)
. (4.3)
Here L denotes the differential operator, f is called source term and u� is the unknown exact
solution, which fulfills this PDE. The main idea of FEM is to discretize the solution. So a set
of N basis functions ϕi ( r) is assumed. Now all linear combinations of these basis functions
are considered:
N
∑
u() r = u ⋅ϕ () r . (4.4)
i=
1
i i
Then the numerical solution u is that linear combination, which fits best to the true solution
u� . Thus the coefficients u i have to be determined.
To measure the quality of different solutions the residual function R( r) is introduced
R(): r = Lu [()] r − f()
r . (4.5)
The smaller the residual the smaller the error of the solution is. Now we can make different
demands on the residual in order to get a good solution. One popular method is the Galerkin
Method, whose conditions for the residual are
∫ R() r wi() r dr = 0 1≤i≤
N . (4.6)
V
In other words, the solution has to fulfill the PDE only in average. Here wi ( r ) are N weight
functions, which have to satisfy certain conditions to lead actually to a small residual. One
possibility is to set these weight functions equal to the basis functions.
w() r =ϕ() r 1≤i≤N
. (4.7)
i i
The basis functions usually are piecewise linear functions with the property
42

Numerical Methods
ϕ ( x ) =δ . (4.8)
i j ij
Each basis function ϕ i refers to the i-th node of the tetrahedral mesh. Figure 4.1 shows such
functions for a 2-dimensional mesh, which are also called “hat functions” due to their
appearance.
Figure 4.1: Typical basis functions of two neighboring nodes of a 2-dimensional mesh.
For our simulations we need to solve Laplace and Poisson equations of the form
( )
∇ σ() r ∇ u() r = s()
r . (4.9)
Here s( r ) is the source term, which vanishes in case of a Laplace equation. The additional
factor σ( r ) also includes the generalized Laplace equation for inhomogeneous conductors
(see (2.31)). Now we apply the Galerkin Method, which leads to
( ( ) )
∫ ∇ σ() r ∇u() r −s() r ϕ i () r dV = 0,
(4.10)
V
Partial integration yields
∫ ∫ ∫
σ() r ∇u() r ∇ϕ i() r dV = � σ() r ∇u() r ϕi() r dA− s() r ϕi()
r dV . (4.11)
V ∂V
V
We take into account that the solution u is a linear combination of all basis functions ϕ i :
N
∑ ∫ ∫ ∫
u σ() r ∇ϕ () r ∇ϕ () r dV = � σ() r ∇u() r ϕ () r dA− s() r ϕ () r dV . (4.12)
j j i i i
j= 1 V ∂V
V
With the definitions
43

Numerical Methods
Aij = ∫ σ() r ∇ϕ j () r ∇ϕi()
r dV , (4.13)
V
si =−∫s() r ϕi()
r dV , (4.14)
V
∂u()
r
bi = ∫ σ() r ∇u() r ϕ i() r dA= ∫ σ() r ϕi()
r dA
∂n
� � (4.15)
∂V ∂V
(4.12) can be written as
N
∑ Au ij j
j=
1
= bi + si = ri
. (4.16)
Here the system matrix A = ( Aij
) is symmetric and positive definite. s i is the source term,
and b i is due to boundary conditions and only contributes, if the i-th node is situated at the
surface ∂ V ( r i ∈∂V
). In this case we have to determine the integral in (4.15) by taking
boundary conditions into account. Here we have to distinguish between Dirichlet and
Neumann boundary conditions. r i denotes the right hand side and is the sum of the boundary
and the source term.
4.1.1 Dirichlet Boundary Conditions
In case of Dirichlet boundary conditions the value of the i-th node u( r ) = b is known. So the
whole i-th equation can be replaced by
i i
ui = bi.
(4.17)
This would destroy the symmetry of the matrix A ij . To maintain its symmetry, which is
convenient for solving a system of linear equations, the matrix entry A ii is just multiplied by a
6
large number ( ~10 ). Then the right hand side is set to b i multiplied with the new coefficient
A ii .
4.1.2 Neumann Boundary Conditions
In case of Neumann boundary conditions the normal component of σ() r ∇u()
r , that is
n⋅σ() r ∇ u() r =− j()
r , is given at the boundary of the mesh. Here j() r can be interpreted as a
44

Numerical Methods
current flowing out of the volume. Normally the Neumann boundary conditions are given
either for each boundary triangle
j() r = j , ∀r ∈Δ
(4.18)
ijk ijk
or for each boundary node of the mesh
j( r) = j , ∀r ∈∂V
. (4.19)
i i i
For the first case we get for the boundary integral term in (4.15)
1
σ() r ∇u() r ϕ () r dA=− j() r ϕ () r dA=−∑ j ϕ () r dA=− ∑A
j
3
�∫ � ∫ ∫
. (4.20)
i i ijk i ijk ijk
∂V ∂V Δijk Δijk
j< k
Δijk
j< k
For the second case we get for the current j by linear interpolation between the sampling
points (points at the boundary) using the test functions i ϕ
∑
j() r = j ⋅ϕ () r . (4.21)
i
i i
Then the result of the boundary integral is
�∫ �∫
σ() r ∇u() r ϕ () r dA=− j() r ϕ () r dA=
i i
∂V ∂V
1
−∑ ∫ ( jiϕ i() r + jjϕ j() r + jkϕk() r ) ϕ i() r dA=− ∑Aijk
( 2 ji + jj + jk)
.
12
Δijk Δ
Δ
ijk
ijk
j< k j< k
4.1.3 Linear equation solver
The FEM method leads to a system of linear equations (4.16)
(4.22)
Au ij j = ri
⇔ Au ⋅ = r (4.23)
with a sparse symmetric and positive definite matrix A. To solve this system of linear
equations a Cholesky factorization method for sparse matrices seemed to be practicable [23].
The advantage of this method over iterative methods is the insensibility of “bad” meshes. The
convergence fails with iterative methods if the tetrahedrons show much difference from
regular tetrahedrons, so that they have small angles. This is problematic for meshes of thin
layers, as used in GMR elements.
45

4.2 Stray Field Calculation
Numerical Methods
The numerical calculation of the stray field is not very efficient, if it is calculated using (2.22),
because for each point in space one has to calculate the whole volume integral in (2.22). So
the calculation requires computing of
calculate the stray field more efficiently with numerical means.
2
ON ( ) . Fredkin and Koehler [24] proposed a way to
As described in Section 2.3 the field of a given magnetization, in this case the stray field, can
be represented as gradient of a magnetic scalar potential.
H =−grad Ψ . (4.24)
S S
The magnetic potential of a given magnetization in a volume V can be calculated by solving
the Poisson equation
ΔΨ = divM
(4.25)
S
with the jump condition at the boundary ∂ V
out in
∂Ψ S ∂Ψ S − =−Mn ⋅
∂n ∂n
∂V
. (4.26)
Now the idea of Fredkin and Koehler was to split the magnetic scalar potential into two parts
Ψ=Ψ S,1 +Ψ S,2.
(4.27)
Inside the volume V Ψ S ,1 is the solution of a Poisson equation
ΔΨ = M (4.28)
in
S ,1 div
with Neumann boundary conditions
in
∂Ψ S ,1
= ⋅
∂n
Mn
∂V
. (4.29)
Outside of V this part of the magnetic potential is set to zero:
Ψ = . (4.30)
out
S ,1 0
Ψ S ,1 can be solved now as performed in Section 4.1. The boundary term and the source term
are calculated as follows:
46

Numerical Methods
∫ ∫ ∫
si =− div Mϕ i( r) dV = M⋅grad ϕi( r) dV −� Mϕi( r) dA,
(4.31)
∫
V V ∂V
bi = � Mϕi() r dA.
(4.32)
∂V
Both terms contribute to the right hand side, resulting in
N
r = M⋅grad ϕ ( r) dV = M ϕ ( r)grad ϕ ( r ) dV . (4.33)
∫ ∑ ∫
i i j j i
V j=
1 V
Here the integral can be calculated analytically for each tetrahedron of the mesh.
Since the total potential S S,1 S,2
satisfy the Laplace equation
in out
S,2 S,2
Ψ =Ψ +Ψ must fulfill equations (4.25) and (4.26), Ψ ,2 must
ΔΨ = 0, ΔΨ = 0
(4.34)
with boundary conditions
and
out in
∂Ψ S,2 ∂Ψ S,2
− = 0
∂n ∂n
∂V
out in in
S,2 S,2 S,1
∂V
S
(4.35)
Ψ −Ψ =Ψ . (4.36)
Fortunately the last equations also describe the magnetic scalar potential of a dipole layer with
moment Ψ S ,1 at the surface ∂ V . The scalar potential of such a dipole layer is known:
1
r−r′ Ψ () r = � Ψ ( r′ ) dA′
. (4.37)
∫
S,2 4π
∂V
in
S,13
r−r′ i
In practice the magnetic potential Ψ S ,2 can be evaluated at each node S,2 S,2 i
i
with FEM. Generally the potential vector ( S ,2 )
j
with the vector ( Ψ S ,1 ) ,
i j
S,2 ij S,1
Ψ =Ψ ( r ) , if used
Ψ can be calculated by a matrix multiplication
Ψ = M Ψ . (4.38)
47

Here the matrix = ( M ij )
Numerical Methods
M describes a dipole interaction of all boundary nodes of V, with all
nodes of V, and is normally fully populated. This matrix has to be calculated only once,
because it does not depend on the boundary conditions. For large models this matrix requires
a lot of memory. So it is more convenient to calculate only the interaction between boundary
and boundary nodes, and evaluate the magnetic potential Ψ S ,2 by solving the Laplace
equation in (4.34) with Dirichlet boundary conditions. In addition the interaction matrix is
compressed as described in next section.
4.3 Adaptive Cross-Approximation Technique
The discretization of boundary integral equations leads to large dense matrices that have no
explicit structure in general. However, by suitable renumbering and permuting the boundary
nodes, the dense matrix can be written in a form that contains blocks that can be approximated
by low-rank matrices [25]. The renumbering of the nodes is done by geometrical criterions.
Consecutive boundary nodes should be located close to each other. A simple example of the
clustering is given in Figure 4.2. Boundary nodes that are located close to each other are
combined in a cluster. A large distance between cluster results in entries in the renumbered
boundary matrix that are well separated from each other. For example the cluster with the
nodes {6,7,8,9,10} and the cluster {1,2,3,4,5} are admissible cluster pairs. This pair of
clusters leads to a block in the boundary matrix which can be approximated by a low-rank
matrix. The boundary matrix corresponding to the clustering is also shown in Figure 4.2. The
off-diagonal blocks of the matrix represent far-field interactions.
This technique leads to compressed interaction matrices, which save a lot of memory.
Additionally the algorithmic complexity for setting up the boundary matrix and for matrix-by-
vector products is reduced from
2
ON ( ) to ON ( ) .
48

Numerical Methods
Figure 4.2: The left figure shows a set of 10 nodes after renumbering and the clustering.
The right figure shows the associated boundary matrix structure. The large off-diagonal
blocks, corresponding to far field interactions of two clusters, can be approximated by lowrank
matrices [25].
4.4 Conductor Model
The sense current through the MR sensor generates a magnetic field, which has to be taken
into account. The current is even used for biasing in AMR heads (see Section 6.1.2).
Therefore the sense current cannot be neglected.
So we have a conductor model in addition to the magnetic model. As there are materials
which are magnetic and conductive these two models can overlap. The current model consists
of materials with different conductivities and magnetoresistive effects as described in Section
2.8. So the current density must be assumed as function of locus and as function of
magnetization.
σ=σrM (, )
(4.39)
Again we use a tetrahedral mesh and the Finite Element Method to calculate the electric
potential and further the current distribution.
4.4.1 Conductivity
In case of non-magnetoresistive materials we have constant conductivities. In
magnetoresistive materials, however, the conductivity has to be evaluated from the
magnetization (see (2.50) or (2.51)). To specify conductivity and magnetoresistance there are
49

Numerical Methods
several material parameters, listed and described in Table 4.1. The listed parameters are
assigned for each material. The tetrahedrons inherit the properties of the material, which they
are assigned to.
If a magnetoresistive material is specified, the conductivity has to be calculated first, in order
to be able to calculate the current density. In case of a AMR (Type = 1) or GMR element
(Type = 2) the magnetic material or two magnetic materials, respectively, have to be
specified, which influence the conductivity. To calculate the conductivity for a tetrahedron of
a GMR material, the nearest points to the midpoint of the tetrahedron in both materials are
searched first. This is done using the Approximate Nearest Neighbor (ANN) Search
Algorithm [26]. Then the effective conductivity can be obtained using (2.50) or (2.51).
Parameter Description
σ Conductivity in Ω -1 m -1
Type 0 Normal conductor, 1 AMR, 2 GMR
MR-Ratio The maximum change in resistance.
Only relevant if Type is not equal to 0.
Magnetic Material 1 Only relevant for AMR or GMR, defines number of (1 st )
magnetic material, which influences the conductivity.
Magnetic Material 2 Only relevant for GMR, defines number of 2 nd material,
which influences conductivity.
Table 4.1: Material parameters for the conductor model.
4.4.2 Electric Potential
To get the electric potential Φ Equation (2.31)
div( σgrad Φ ) = 0
(4.40)
has to be solved. In addition we have to specify boundary conditions. There are two
possibilities: Either the potential or the out-/inflowing current has to be declared. Finally this
problem can be solved by applying the FEM approach as described in Section 4.1.
N
Φ () r = Φ ⋅ϕ () r (4.41)
∑
i= 1
i i
The result gives the values of the electric potential at the nodal points.
4.4.3 Current Distribution
Then the current distribution can be evaluated using (2.32)
50

Numerical Methods
j =−σgrad Φ.
(4.42)
With the calculated electric potential (4.41) we are able to determine the current distribution
for each tetrahedron ijkl. Inserting in (4.42) gives
( grad grad grad grad )
j =−σ Φ ⋅ ϕ +Φ ⋅ ϕ +Φ ⋅ ϕ +Φ ⋅ ϕ . (4.43)
ijkl ijkl i i j j k k l l
4.4.4 Magnetic Field of Current
When the current distribution is known for each element, the next task is to determine the
additional magnetic field due to the current. The current is uniform for each tetrahedron. So
the Biot-Savart Law (2.46) can be written as
or
1
Hijkl () r = jijkl
×
4π
∫
Vijkl
r−r′ r−r′ 3
dV ′
(4.44)
1 1
Hijkl () r = j ′ ijkl × ∇ dV ′
4 ∫
. (4.45)
π r−r′ Vijkl
Integration delivers (see Appendix A)
1 1
Hijkl () r = jijkl × d ′
4 �∫
A . (4.46)
π r−r′ ∂Vijkl
So the field of a tetrahedron with uniform current density can be calculated by integrating
over its surface triangles:
1 ⎛ 1 1 1 1 ⎞
Hijkl () r = jijkl × ⎜ d ′ + d ′ + d ′ + d ′ ⎟
4 ⎜ ∫ A
− ′ ∫ A
− ′ ∫ A
− ′ ∫ A . (4.47)
π
− ′ ⎟
⎝
r r r r r r r r
Δijk Δjlk Δilk Δijl
⎠
Fortunately the integrals in (4.47) can be solved analytically for triangle surfaces, resulting in
quite complicated formulas [27] [28]. The total field due to the current can be obtained by
summation over all tetrahedral elements ijkl. Now we are able to determine the magnetic field
at all nodes r i of the magnetic model. Generally the effective field can be written as
multiplication of the interaction matrix and the current density vector.
∑
Hr ( ) = A ⋅j
. (4.48)
i ij j
jl
51

Numerical Methods
Here A ij are 3× 3-submatrices
describing the interaction of the j-th tetrahedron of the
conductor model with the i-th node of the magnetic model. Thus the matrix A= ( A ) is a
( 3n) ( 3m)
× -Matrix, where n denotes the number of nodes of the magnetic model and m the
number of tetrahedrons in the conductor model. Usually the matrix A is fully populated and
requires a lot of memory. The multiplication with such a matrix is very time consuming. To
speed up the calculation and save memory an adaptive cross-approximation technique can be
used (see chapter 4.3). Unfortunately this technique needs a lot of memory when calculating
the approximation of the interaction matrix A for large magnetic and conductor models.
4.4.5 Hybrid FEM/BEM for Current Field
To overcome the storage problem we proposed an alternative way to calculate the magnetic
field in the magnetic model. Here we use a common mesh for the magnet and the conductor
model. Similar to the stray field calculation (see Section 2.5.2) we can reduce the interaction
matrix to a boundary interaction matrix.
Starting from Equation (2.29)
curl H = j, (4.49)
applying the curl-operator on both sides, and considering that div H = 0 lead to a Poisson
equation for each component of H.
Δ H =−curl
j (4.50)
with boundary condition (see Appendix B)
out in
∂H ∂H
− = n× j
∂n ∂n
∂V
. (4.51)
Now the magnetic field can be calculated quite similar to the stray field, described in Section
4.2. We split the magnetic field into two parts
H = H1+ H 2,
(4.52)
where H 1 is the solution of
Δ H =− j (4.53)
in
1 curl
ij
52

Numerical Methods
within the volume V. Outside the magnetic field H 1 is set to zero
out
H 1 = 0 . (4.54)
Then the boundary condition writes as
∂H
∂n
in
1
=− n× j
∂V
. (4.55)
Now H 1 can be calculated within the volume V as described in Section 4.1. The source term
is determined by
∫
s = curl jϕ ( r ) dV . (4.56)
i i
V
Here the source term is a vector. Each component refers to one of the three Poisson equations
in (4.53). Partial integration gives
si = � ∫ n× jϕi() r dA − ∫j×
grad ϕi()
r dV . (4.57)
∂V
V
The boundary integral becomes
in
∂H1
bi = ∫ ϕ i() r dA =− × ϕi()
dA
∂ ∫ n j r
n
� � . (4.58)
∂V ∂V
So the first integral in the source term and the boundary term cancel each other. The right
hand side is therefore
∫
r =− j× grad ϕ(
r ) dV . (4.59)
i i
V
The gradient of the test function and the current density are constant for each tetrahedron, so
the integral (4.59) can be calculated as sum over all tetrahedrons, which are enclosed to the
i-th node.
The second part H 2 has to fulfill
and further
Δ H = 0, Δ H = 0
(4.60)
in out
2 2
53

out in in
2 2 1
∂V
Numerical Methods
H − H = H (4.61)
to guarantee (4.50) and (4.53). Again we have an equivalent problem as in Section 4.2, but for
three dimensions. Analogously the field H 2 can be evaluated by
( − ′ )
1 r r n
H () r = ( ′ ) dA
π � ∫ H r
. (4.62)
′
2
4 ∂V
in
1
r−r 3
To save memory we only create an interaction matrix from boundary to boundary, and solve
the Laplace equation (4.60) again to calculate H 2 .
4.4.6 Example: Long Wire
To verify the program and the calculated fields, some relative simple models are simulated,
which can also be calculated analytically. The first example is a long straight Cu-wire with
length l = 200 mm and wire radius r = 2 mm . The conductivity of Cu is
7 -1 -1
σ= 5.961⋅10 Ω m . At least we have to specify the boundary conditions of the wire. At one
side the electric potential was set to V 1 = 0 . The boundary condition of the opposite end was
chosen in a way, that the total inflowing current has a value of I = 1 mA . With these
parameters the total resistance of the Cu-wire can be calculated: R = 267 µ Ω . Using Ohm’s
Law (2.1) we get for the electric potential V 2 = 0.267 µV . This analytically calculated value
matches well with that of the FEM calculation 2
0.278 µV
FEM
FEM
V = . The error ( 2
V is about
4.2% larger than V 2 ) is traced back to the mesh. The mesh has to approximate a cylindrical
surface with triangles. So the cross sectional area of the mesh model is smaller than that of the
realistic model. This leads to a larger resistance and further to a larger voltage.
I
2
The current density of the realistic model is j = = 79.58 A/m . The magnetic field of an
2
r π
infinite long straight wire with constant current can be easily calculated due to the symmetry
in respect of rotations around the wire’s axis. The only non-zero field component is that in
azimuthal direction and can be determined using Ampere’s Law (2.6).
� ∫ H⋅ ds= ∫j×
dA
(4.63)
∂A
A
54

Numerical Methods
Integration of Ampere’s law gives for the radial field
H
( )
⎧ ρ
⎪
j : 0≤ρ<
r
⎪ 2
. (4.64)
ϕ ρ =⎨ 2
⎪
r
j : r ≤ρ
⎪⎩ 2ρ
This function is plotted in Figure 4.3, and shows a linear increase, reaching its maximum at
the wire’s surface and declines proportional to 1/ρ outside.
H ϕ [A/m]
0.10
0.08
0.06
0.04
0.02
0.00
0
0 10 20 30 40 50
ρ [mm]
Figure 4.3: Hϕ for an infinite long wire is plotted over the distance from the middle axis.
The second curve shows the error of the numerical calculation.
The second curve in Figure 4.3 shows the relative error of the FEM calculation compared to
(4.64). The error is negative and increases according to amount with increasing ρ . The model
with finite length used for the FEM calculation is the reason for that. The finite length has a
greater effect for larger radial distances. The discontinuities of the error curve are due to the
Adaptive Cross-Approximation Technique (see Section 4.3). For small ρ we have a very
good match of the numerical and analytical calculated fields.
4.4.7 Example: Coil
A coil as shown in Figure 4.4 is the second example. For the coil wire we have again the same
7 -1 -1
parameters as for the straight wire ( σ= 5.961⋅10 Ω m , I = 1 mA , r = 2 mm ). The radius
of the coil is r′ = 20 mm , the length is l = 100 mm , and it has N = 20 windings. The model
additionally has two straight leads with lengthl L = 80 mm . Similar to the straight wire we can
now calculate the difference of the electric potential by analytical means:
-10
-8
-6
-4
-2
Error [%]
55

V
2l + N2πr′ ≈ = 3.57 µV
σrπ L
2 2
Numerical Methods
(4.65)
With the FEM we get V 2 = 3.81µV . The difference is again due to the smaller cross sectional
area of the meshed model.
H z (r=0) [A/m]
0.20
0.15
0.10
0.05
0.00
0
-100 -50 0 50 100 150 200
z [mm]
Figure 4.4: The model of the coil (left). The color code shows the linear decrease of the
electric potential inside the wire. The chart (right) shows the magnetic field of a solenoid in
z-direction at the middle axis. The two dotted lines identify the position of the solenoid.
The second graph shows the relative error of the coil’s magnetic field calculated with FEM
calculation compared to the solenoid’s field.
The magnetic field for a perfect solenoid (hollow cylinder with infinitesimal thin nappe) with
length L, face current density J and radius r′ can be calculated exactly along the middle axis
(z-axis):
⎛ ⎞
J L−z z
Hz( z)
= ⋅ ⎜ + ⎟.
(4.66)
2 ⎜ 2 2 2 2
( L− z) + r′
z + r′
⎟
⎝ ⎠
When we now approximate the coil by such a solenoid and set the face current density equal
to
NI
J = , we have the possibility to compare the field of the solenoid H z with that of the
L
calculated field of our model. The run of the magnetic field along the middle axis of the
solenoid is shown in Figure 4.4. In addition the relative error of the FEM calculated coil
compared with the solenoid is given. So the quantitative values are again consistent.
10
8
6
4
2
Error [%]
56

4.5 Time integration
Numerical Methods
Using the LLG equation, the time evolution of the magnetization can be calculated. The
discretization of realistic problems leads to an initial-value problem (4.2) with up to a million
unknowns.
To solve such problems the free solver CVODE is used [29] [30], which offers iterative and
direct multistep methods. Multistep methods make use of the past values of the solution.
For our purposes the iterative backward differentiation formula (BDF) seemed to fit best to
solve our problem, in order to achieve fastest convergence times. The nonlinear system is
solved using a Newton method, which requires only a few iterations. In order to improve the
performance, CVODE allows preconditioning. By declaring an estimated Jacobian matrix the
convergence time can be drastically decreased.
CVODE is capable of choosing the time step size automatically. For rapidly changing
processes a smaller time step is chosen to increase accuracy.
Figure 4.5 schematically shows the complete simulation cycle. First the mesh model and the
material parameters are read. Then the boundary matrices for the FEM/BEM methods for the
stray field and the current field are set up. Afterwards the initial magnetization is read and set,
and the time integration starts.
CVODE solves the LLG equation, and requests the right hand side of the LLG equation for
certain points of time. CVODE passes the actual time and the magnetic polarization J i of
each node to the LLG calculation routine, which calculates the right hand side of the LLG
equation. All magnetic fields which contribute to the total effective magnetic field H eff are
calculated for each nodal point. Then the effective field is inserted in the right hand side of the
LLG equation, which leads to new functions f i of time and magnetic polarization of the i-th
node. These functions are transmitted back to the solver, which decides to modify the
magnetic polarization. If the desired precision is not fulfilled the new magnetic polarizations
are passed to the LLG calculation routine again, and the cycle is repeated. Otherwise the
solver chooses the next time step and the system time is increased. The whole procedure is
repeated until a user chosen final time is reached.
57

Time Integration
CVODE
∂J
∂t
i
= f ( t,
J , J ,..., J )
i 1 2 N
Numerical Methods
START
Read geometry (FE mesh)
and material parameters
Calculate the B matrix for
the hybrid FE/BE method
Set the initial magnetization
t, J1, J2,..., J N
f1, f2,..., f N
Increase time (t)
t < tfinal
END
Hexch + Hani + HS+ Hcurr + Hext
�������������
H
Figure 4.5: Schematic diagram of the simulation cycle. Adapted from [31].
eff
LLG
f1, f2,..., f N
σ ( J )
Φ, j
H ext () t
58

5 Read Head DesignEquation
5.1 Bias Schemes
Read Head Design
Section (Next)
For AMR and GMR reading heads, magnetic layers with a uniform predefined magnetization
direction are necessary. Besides a single domain state is desirable, to avoid Barkhausen noise
due to moving domain walls. Therefore the magnetizations of these layers have to be forced
into a certain direction by biasing. The most important bias schemes used in MR read heads
are explained below.
5.1.1 Demagnetizing Factor
The field of a magnetized body, the stray field, or also called demagnetizing field H S , acts on
the magnetization itself. This field is proportional to the magnetization M and opposite
directed within the body. So the demagnetizing field can be written as
N =− H M (5.1)
S d
Here N d is the demagnetizing factor, which is generally a tensor and only depends on the
shape of the body. Principal axis transformation leads to three demagnetizing factors a N , N b ,
N c for each axis one respectively. For their sum
Na + Nb + Nc
= 1
(5.2)
always holds. The longest axis corresponds to the smallest demagnetizing factor. It is also the
preferred direction for the magnetization, because then the stray field energy has its minimum
(see (2.37)).
A GMR element basically exists of two ferromagnetic layers, which should be magnetized
perpendicular in respect to each other in ground state. Moreover the free layer should be able
to rotate freely. Therefore the shape of these layers has to be almost quadratic (with an aspect
ratio close to 1) in order not to discriminate a magnetization direction.
59

Read Head Design
On the other hand the signal field decays with distance from the recording layer (see Section
3.4) and therefore a higher effective (average) signal field is reached for smaller sensor
heights. Thus a compromise must be found. Usual width-to-height ratios are between 1.2 and
1.5.
5.1.2 Hard Bias
To align the magnetization of the free layer hard magnets are used (see Figure 2.4). Usually
the GMR element is placed between two hard magnets, whose static magnetic field effects the
perpendicular alignment in respect to the pinned layer. The permanent magnetic fields also
influence the magnetization of the pinned layer. Therefore the biasing field for the pinned
layer has to overcome the hard bias field.
5.1.3 Exchange Bias
Exchange bias is based on the exchange anisotropy, which was discovered by Meiklejohn and
Bean 50 years ago [32]. They investigated small Co particles coated with its
antiferromagnetic oxide CoO. A field cooling process, i.e. heating up above the Néel
temperature and cooling under applying a magnetic field, leads to a shift of the hysteresis
loop, which is shown in Figure 5.1. The exchange bias effect can be characterized by the
displacement of the hysteresis loop, denoted by H ex .
Figure 5.1: The shift of the hysteresis loop after a field cooling treatment of a
ferromagnetic/antiferromagnetic system [19] and the temperature dependence of the
exchange bias field for NiFe(4-8nm)/Cu(2.2nm)/CoFe(2nm)/IrMn(15nm) spin valves with
different free layer thicknesses [33].
60

Read Head Design
The exchange bias field depends on the thicknesses of the antiferromagnet and the
ferromagnetic layer. If the antiferromagnet is thicker than the ferromagnetic layer the
exchange bias field decreases with the thickness of the ferromagnet t p like
H
J
1
t p
− [34].
ex
ex = . (5.3)
µ 0 Ms, ptp Here J ex is the exchange bias interaction parameter [J/m 2 ]. This kind of dependence leads to
the conclusion, that the exchange anisotropy is an interface effect. The maximum exchange
bias field, which can be achieved for usual layer thicknesses, is about 0.05 T [20]. The
exchange bias field decreases linearly with temperature [33]. Figure 5.1 shows the
temperature dependence of the exchange bias field for NiFe(4-8nm)/Cu(2.2nm)/CoFe(2nm)/
IrMn(15nm) spin valves with different free layer thicknesses.
The exchange bias effect is used to align the magnetization of the pinned layer perpendicular
to the recording layer. The advantage of this bias scheme is that the free layer is not
influenced by the antiferromagnet. Unfortunately the stray field of the pinned layer causes an
unwanted torsion of the free layer’s magnetization, which leads to a displacement of the
equilibrium state towards the antiparallel state. The result is an asymmetric sensor curve.
5.1.4 Synthetic Antiferromagnet
To reduce the asymmetry of the transfer curve of the spin valve, a stronger exchange bias is
required. The limitation of the exchange bias field leads to an alternative spin valve design
[35]. Instead of the direct coupling of the antiferromagnet with the pinned layer, an additional
ferromagnetic layer is introduced which is separated by a very thin (0.7-0.9 nm) nonmagnetic
layer (mostly Ru) as shown in Figure 5.2. This layer structure is also called a synthetic
antiferromagnet, because the pinned layer is antiferromagnetically coupled with the additional
ferromagnetic layer by interlayer exchange coupling (IEC).
61

Read Head Design
Figure 5.2: A comparison (left) of a conventional spin valve and a spin valve using a
synthetic ferromagnet to fix the pinned layer [20]. The interlayer exchange constant (right)
for two Ni80Co20 layers as function of the Ru spacer layer thickness [32].
The IEC effect has its origin in quantum interference of electron states [36]. The exchange
coupling constant J IEC [J/m 2 ] depends on the nonmagnetic layer thickness t NM like
⎛ tNM
⎞
sin ⎜2π +Φ
Λ
⎟
J = J
⎝ ⎠
(5.4)
IEC
0 2
tNM
as shown in Figure 5.2. Λ , Φ , and J 0 are constants, depending on the composition of the
layers. The advantage of spin-valves with a synthetic antiferromagnetic pinned layer is the
very large exchange field of IEC up to 0.5 T. This is an increase of a factor 10 compared to
the direct exchange coupling with an antiferromagnet. So the stability against external fields
is much better due to the higher anisotropy. Moreover the demagnetizing fields of the two
ferromagnetic layers Fp,1 and Fp,2 cancel each other partially, which leads to a reduced
influence on the free layer.
5.1.5 Sense Current
The sense current flowing through the GMR sensor generates a magnetic field. This field also
influences the pinned layer magnetization. For conventional simple spin valves there is one
current direction which is more favorable. The main part of the current flows through the Cu
layer, due to its higher conductivity compared to the other layers. So the current field either
adds up with the demagnetizing field of the pinned layer or weakens the field acting on the
62

Read Head Design
free layer (see Figure 5.3). Unfortunately the more favorable current direction causes a
decline of the exchange bias field.
Figure 5.3: The effect of the sense current direction on the free layer (FL). For the first
case (left) the current field Hcurr partially cancels the stray field HS of the pinned layer (PL)
and weakens the exchange bias field Hexch. For the other current direction (right) the stray
field and the exchange bias field are supported by the current field.
Shield
µ → ∞
Hcurr
I
HS
FL Cu PL
Hexch
Hcurr
I I
I
Hmirr
FL Cu PL
Shield
µ → ∞
Figure 5.4: The layers of the GMR element also see mirror currents due to the high
permeability of the shields. The field of these mirror currents is able to reduce the field of
the original current at the free layer and the pinned layer.
Hcurr
Hcurr
FL Cu PL
Hmirr
I
HS
Hexch
63

Read Head Design
In presence of shields with high permeability the free layer sees an additional mirror current,
which must also be considered, especially for small gap distances. These mirror currents
reduce the current field as shown in Figure 5.4.
5.1.6 Crystalline Anisotropy
Similar to the shape anisotropy the crystalline anisotropy can be used for biasing. For
example a field cooling treatment of a material can generate an intrinsic anisotropy.
Nevertheless this bias scheme is not often used in GMR sensors, because the free layer should
be able to rotate freely and the pinned layer is normally biased by exchange anisotropy.
5.2 Shielding
To avoid influence of external fields and neighboring bit transitions, shields with high
permeability enclose the GMR element. Shields normally consist of Permalloy (Ni80Fe20).
The relative permeability of Permalloy is in the order of
4
10 .
The gap distance between both shields is limited to 60 nm for CIP spin valves and 20 nm for
CPP spin valves (see Section 2.8.3). In order to have an improvement of the read back signal
the gap distance should not exceed the bit length.
5.3 Thermal Stability
The sense current leads to a heating of the GMR element. The properties of the GMR sensor
are temperature dependent. For instance the exchange bias field decreases with temperature
(see Figure 5.2). Moreover the noise increases. To achieve thermal stability a constant sensing
current is applied instead of a constant voltage. Then the change in resistance is determined
by measuring the change in voltage. The heat production of the current limits the current
density. The heat dissipation is especially a problem for GMR sensors working in CIP mode,
because for the CPP mode the shields are used as leads. Therefore the electric conductance
between GMR sensor and shields is larger, which normally also leads to a larger thermal
conductance due to the Wiedemann-Franz Law.
64

FEM Simulations
6 FEM SimulationsEquation Section (Next)
6.1 Model
The model of the read head used in the following calculations is very simplified, because of a
shortage of detailed information. Despite cooperation with hard disk manufacturers, their
information was very restricted. Nevertheless we were able to simulate the qualitative
behavior and to prove the functionality of our code. The whole read head model is separated
into two parts: A conductor model and a magnetic model.
6.1.1 Magnetic Model
A simple GMR read head basically exists of five parts, which define the magnetic behavior:
The free layer, the pinned layer, the hard bias, the shields, and finally the antiferromagnet for
fixing the pinned layer. The magnetic net moment of an antiferromagnetic material is zero.
Apart from exchange coupling the antiferromagnet does not contribute to the magnetic
behavior. Therefore the antiferromagnetic part can be neglected, if the exchange coupling is
taken into account by adding the exchange bias field to the effective field within the pinned
layer.
The geometry of the FEM model for the calculations is shown in Figure 6.1. The free layer
and the pinned layer dimensions are h GMR = 80 nm in height and w GMR = 100 nm in lateral
FL
PL
direction. Their thicknesses are t = 5 nm and t = 2 nm , and their spontaneous
FL
polarizations are J = 1.2 T , which is the average of a composite free layer (NiFe/CoFe) and
s
J = 1.885 T (CoFe) respectively. Both layers are assumed to have no anisotropy and are
PL
s
Cu
separated from each other by a t = 2.7 nm thin nonmagnetic Cu layer. The two hard bias
magnets have uniaxial anisotropy in lateral direction
HB
5 3
K 1 = 2.5⋅ 10 J/m and a spontaneous
HB
polarization of J = 1T.
Their lateral distance from the GMR layer structure is 10 nm. The
s
exchange bias field is always set to H ex = 0.05 T , unless otherwise noted. This corresponds
65

FEM Simulations
approximately to the exchange coupling with the pinned layer for a 8 nm IrMn
antiferromagnet. The two shields are assumed to be made of Permalloy (NiFe) with
Shields
spontaneous polarization J = 1T,
and their dimensions are 50 nm× 100 nm× 50 nm . The
s
gap between the two shields is 60 nm. The Gilbert damping constant for the whole model is
always 0.1 unless otherwise noted. The exchange constants are
PL
A
−11
= 1.2⋅ 10 J/m ,
HB
A
are summarized in Table 6.1.
−11
= 1.0⋅ 10 J/m , and
Shields
A
FL
A
−11
= 1.3⋅ 10 J/m ,
−11
= 1.3⋅ 10 J/m . All magnetic properties
The GMR element of the model is meshed with a size of 5 nm. The mesh of the shields has a
general size of 50 nm, but it is refined near the gap and near the track region.
Figure 6.1: The magnetic model used for the FEM calculations. The mesh size for the
shields (green) is 50 nm, and is refined to 5 nm near the track and near the spin valve
layers. The hard bias magnets (purple), the free layer (dark blue), and the pinned layer
(light blue) are also meshed with triangle size of 5 nm.
z
y
x
66

FEM Simulations
Magnetic Material Js [T] K1 [J/m 3 ] A [J/m] Hex [T] Α
Free Layer 1 0 1.3E-11 0 0.1 *
Pinned Layer 1.885 0 1.2E-11 0.05 *
0.1 *
Hard Bias 1 2.5E5 1.3E-11 0 0.1 *
Shields 1 0 1.0E-11 0 0.1 *
Table 6.1: The values of the spontaneous polarization, the anisotropy constant, the
exchange constant, and the antiferromagnetic exchange coupling field, and the Gilbert
damping constant, which are assumed for our magnetic model. The asterisk-marked values
are valid unless otherwise noted.
6.1.2 Conductor Model
For the current model we have to consider all conductive parts of the model. This also
includes the antiferromagnet or capping layers. The conductor model is shown in Figure 6.2.
However, only the most important layers are taken into account for our simulations. So we
neglect the thin capping layers, although they make a minor contribution to the conductance.
Thus our conductor model contains the free layer, pinned layer, nonmagnetic spacer layer
(Cu), the antiferromagnetic layer, and parts of the leads.
Figure 6.2: The FEM conductor model is shown. This simplified model shows the four
main layers of a GMR element, which contribute most to the conductance: The free layer
(dark blue) the nonmagnetic Cu layer (orange), the pinned layer (light blue), and the
antiferromagnet (yellow). Additionally the high conductive leads (pink) are shown.
x
y
z
67

FEM Simulations
The sense current through the spin valve is applied at the lateral surfaces of the conductor
model and has a value of 3 mA.
The effective conductivities of the individual parts are estimated from the measured
conductivities in Figure 6.3 [37]. These conductivities differ much from the bulk values,
because the electron mean-free path is larger than the layer thicknesses.
Figure 6.3: The effective conductivities for a typical spin valve taken from [37].
In our simulation the free layer has a conductivity of
PL 6 −1 −1
σ = 3.5⋅10 Ω m , the Cu layer
FL 6 −1 −1
σ = 310 ⋅ Ω m , the pinned layer
Cu 6 −1 −1
σ = 4.5⋅10 Ω m , the antiferromagnet
AF 6 −1 −1
Lead 7 −1 −1
σ = 110 ⋅ Ω m and finally the leads σ = 110 ⋅ Ω m . Here the conductivities of free
layer, pinned layer and the Cu layer are not constant, because it is assumed that these layers
contribute to the GMR effect. Their maximum change in conductivity, the GMR ratio, was
assumed to be 10%. So the effective GMR ratio of the whole layer structure is smaller,
because the sense current is shunted by the antiferromagnet. All values assumed for the
conductor model are listed in Table 6.2.
Conductor σ [Ω -1 m -1 ] Type MR-Ratio Magnetic Material 1 Magnetic Material 2
Free Layer 3.0E6 2 (GMR) 0.1 Free Layer Pinned Layer
Cu Layer 4.5E6 2 (GMR) 0.1 Free Layer Pinned Layer
Pinned Layer 3.5E6 2 (GMR) 0.1 Free Layer Pinned Layer
Antiferromagnet 1.0E6 0 - - -
Lead 1.0E7 0 - - -
Table 6.2: The properties for all parts of the conductor model. Here the conductivities, the
type of magnetoresistance, and the magnetic materials, which influence the conductivity,
are given. (Compare with Table 4.1).
68

6.2 Results
FEM Simulations
In the following the results of the simulations are presented. Naturally the model described
above of a read head is not perfect at all. It was not the target to design a perfect read head,
but to be able to simulate the read back process. Although the used read head model is not
technically mature, the results of the simulations show qualitative accordance with reality.
6.2.1 Equilibrium State
Ideally the magnetizations of the pinned and the free layer should be perpendicular to each
other in equilibrium state, so that the magnetization of the pinned layer shows in z-direction,
and that of the free layer in y-direction. Then the total resistance would be 32.87 Ω and the
potential difference 98.6 mV. In reality such a perfect configuration is not achievable.
The equilibrium states of our model for the two different current directions are shown in
Figure 6.4. For the more favorable current direction the magnetizations are almost
perpendicular to each other in the center of the GMR element. Unfortunately the free layer
magnetization is twisted downwards due to the demagnetizing field of the pinned layer.
Similarly the pinned layer is rotated in the same direction due to the hard bias. Both effects
lead to an asymmetric transfer curve (see Section 6.2.2).
Figure 6.4: The normalized magnetization of the free layer and the pinned layer in
equilibrium state for the two different current directions. The color code gives the zcomponent
of the normalized magnetization, red corresponds to the positive and blue to the
negative z-direction. Thus the red arrows represent the pinned layer magnetization. The
right hand side gives the configuration for the more favorable current direction, because
here the current field cancels partially the demagnetizing field of the pinned layer.
69

FEM Simulations
If the current flows in the more favorable direction the exchange bias field is weakened (see
Section 5.1.5), which can also be seen in Figure 6.4, when considering the torsion of the
magnetization of the pinned layer.
6.2.2 Transfer Curve
To get the transfer curve of the spin valve, a triangle pulsed external field as shown in Figure
6.5 is applied. Its amplitude is 0.1 T and its period time is 8 ns. As we will see later in Section
6.2.4, the changing of the external field is slow enough, so that the GMR sensor is always
near equilibrium.
H z [T]
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
0 2 4 6 8 10
Time [ns]
Figure 6.5: The external applied field, which is applied in z-direction to evaluate the
transfer curve of the GMR sensor.
70

Output [V]
0.108
0.107
0.106
0.105
0.104
0.103
0.102
0.101
-0.10 -0.05 0.00 0.05 0.10
H z [T]
FEM Simulations
ΔR/R [%]
Current in pos. y-direction
Current in neg. y-direction
Current in pos. y-direction, Hex = 0.1T
1
0
-1
-2
-3
-4
-0.10 -0.05 0.00 0.05 0.10
Hz [T]
Figure 6.6: The output voltage (left) and the relative change in resistance (right, in respect
to the equilibrium state) over the external applied field in z-direction for both current
direction and a larger exchange bias field.
The corresponding outputs for both current directions and a stronger exchange bias are given
in Figure 6.8. A positive external field leads to a parallel alignment of the magnetizations of
both layers, resulting in a low resistance. If the external field is applied in negative z-
direction, the resistance increases, because the magnetic configuration approaches the
antiparallel state. However, if the magnetic field overcomes the exchange bias field, the
pinned layer magnetization is not fixed anymore and rotates towards negative z-direction. So
for large negative external fields we have again a parallel state and therefore low resistance.
As expected the favorable current direction shows less asymmetry in change of resistance
around the equilibrium state than the opposite current direction. Moreover the GMR sensor is
more sensitive, because the slope of the transfer curve at zero field is larger. The output
voltage for this case is also higher, because here the magnetization vectors are more likely
aligned antiparallel (see equilibrium state in Figure 6.4), leading to larger resistance.
In case of a stronger exchange bias field, H ex = 0.1T , we have a much better transfer curve
for higher external fields. (Unfortunately such high exchange fields are rather unrealistic. To
increase the bias field for the pinned layer, synthetic antiferromagnets, as described in Section
71

FEM Simulations
5.1.4, are used in modern spin valves.) The transfer curve and the slope at zero point do not
differ much compared to the low exchange bias field. However, for large fields we have a
much better sensor performance in respect to sensitivity and symmetry. The higher voltage in
equilibrium is due to the larger angle between both magnetizations. The large exchange bias
field reduces the deflection due to the hard bias, while the free layer magnetization stays
unmodified, because of the widely unchanged demagnetizing field of the pinned layer.
6.2.3 Influence of Shields
Now the shields are taken into account. It is expected that the shields have an influence on the
transfer curve due to the mirror currents (see Section 5.1.5). Indeed the shields have a large
influence. Again the transfer curve is evaluated by applying the external field shown in Figure
6.5 on the GMR element. The shields are not exposed to the external field. The result is given
in Figure 6.7.
Output Voltage [V]
0.109
0.108
0.107
0.106
0.105
0.104
0.103
0.102
-0.10 -0.05 0.00 0.05 0.10
H z [T]
ΔR/R [%]
without shields
with shields
-0.10 -0.05 0.00 0.05 0.10
H z [T]
Figure 6.7: The transfer curves for the spin valve with and without shields (for both
Hexch = 0.1 T).
The presence of shields generally leads to a greater output voltage, particularly at zero field.
The origin of this effect is based on mirror currents. They reduce the current field which
works against the demagnetizing field of the pinned layer. Therefore the demagnetizing field
2
1
0
-1
-2
-3
-4
-5
72

FEM Simulations
has a greater influence. The result is a more likely antiparallel state, which is equivalent to a
higher total resistance.
This proves that the shield also have an influence on the read head behavior and cannot be
neglected in micromagnetic simulations.
6.2.4 Relaxation
The micromagnetic behavior of the GMR sensor depends on the Gilbert damping constant.
Starting from an equilibrium state with an external field of H ext = 0.03 T in positive z-
direction, the relaxation to the new equilibrium state when suddenly switching off the field, is
shown in Figure 6.8.
Output Voltage [V]
0.1060
0.1055
0.1050
0.1045
0.1040
0.1035
0.1030
0.0 0.2 0.4 0.6 0.8
Time [ns]
α = 0.1
α = 0.2
α = 0.3
α = 0.5
α = 1
Figure 6.8: The output signal over time for a relaxation from equilibrium state with
Hext = 0.3 T to the equilibrium state without field for different Gilbert damping constants.
For large Gilbert damping constants we have strong damping, resulting in a quite large
relaxation time (larger than 0.4 ns). New read heads work with a read frequency up to 2 GHz.
Therefore such large relaxation times would weaken the read back signal. Fortunately realistic
damping constants are about α ∼ 0.1,
which lead to an undercritical damping, resulting in
damped oscillations. Smallest relaxation times are achieved for critical damping with
damping constants around 0.3.
73

6.2.5 Signal of a Perfect Transition
FEM Simulations
To calculate the signal of a perfect transition a track width of 120 nm, a data layer thickness
of 12 nm, and a spontaneous polarization of J s = 0.44 T was assumed. The read back signal
was investigated for both kind of transitions ( →← and ←→).
Figure 6.9 and Figure 6.10 show the read back signal of the read head with and without
shields for the different kind of transitions. The head velocity was 20 m/s. For this calculation
the Gilbert damping constant was 0.3 and 1.0 for the GMR sensor and the shield respectively.
The signal peaks for the full read head simulation (with shields) are more localized compared
to the simulation without shields. The asymmetric transfer curve leads to a large difference in
amplitude height for both cases. Again the output voltage is shifted up for the read back signal
with shields due to mirror currents, as we have already seen in Figure 6.7.
Output Voltage [V]
0.1070
0.1065
0.1060
0.1055
0.1050
0.1045
0.1040
-300 -200 -100 0 100 200 300
x [nm]
ΔR/R [%]
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
-1.6
-300 -200 -100 0 100 200 300
without shields
with shields
x [nm]
Figure 6.9: The read back signal and the relative change in resistance for a perfect →←
transition.
74

Output Voltage [V]
0.1074
0.1072
0.1070
0.1068
0.1066
0.1064
0.1062
0.1060
x [nm]
FEM Simulations
0.1058
-300 -200 -100 0 100 200 300
ΔR/R [%]
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-300 -200 -100 0 100 200 300
without shields
with shields
x [nm]
Figure 6.10: The read back signal and the relative change in resistance for a perfect ←→
transition.
6.2.6 Read Back Signal
Finally the read back signal was calculated. As replacement for the data layer, a bit pattern as
shown in Figure 6.11 was used. It consists of 30 bits with perfect transitions, which are
alternately magnetized in positive and negative x-direction. The assumed bit length was
60 nm, which is equal to the gap of the read head model. The track width was 120 nm, which
is larger than the free layer width. The thickness was 12 nm and the flying height of the head,
i.e. the distance between ABS and the data layer was 10 nm. The spontaneous polarization
was 0.44 T.
Figure 6.12 shows the read back signal for a read head with and without shields. According to
the transfer curve the output signal for the whole read head (with shields) is again shifted up,
due to mirror current effects. The shields lead to an increase of the amplitude. Therefore the
read back signal is much better, although the signal for single perfect transitions was lower
than that without shields (see previous section). This fact can be traced back to the wider
peaks in the absence of the shields, so that the signals of consecutive transitions partially
cancel each other.
75

Output Voltage [V]
0.1080
0.1075
0.1070
0.1065
0.1060
0.1055
0.1050
FEM Simulations
Figure 6.11: The model of a perfect bit pattern. It consists of 30 perfectly written bits and
is used for our read back signal calculations. The track width is 120 nm and the bit length
60 nm. The layer thickness is 12 nm and the spontaneous polarization is 0.44 T. The bits
are alternately magnetized in positive and negative x-direction.
0.1045
0 100 200 300 400
x [nm]
ΔR/R [%]
1.0
0.5
0.0
-0.5
-1.0
-1.5
0 100 200 300 400
without shields
with shields
x [nm]
Figure 6.12: The read back signal for the periodic bit pattern calculated for the GMR
sensor and for the whole read head (with shields).
76

7 OutlookEquation
Section (Next)
Outlook
This work was a first step to take arbitrary current distributions into account in micromagnetic
simulations. The program solves the LLG equation and simultaneously calculates the current
distribution, which can be influenced by the magnetization. So it can be used for read back
signal calculations. Of course it is also possible to calculate the read back signal of
perpendicular recording layers. Here the additional SUL has to be taken into account. Spin
valves with synthetic antiferromagnet can also be calculated, if the interlayer exchange
coupling is introduced.
Possible enhancements of this program are the consideration of eddy currents [41] [42] or
finite temperatures. To take fluctuations at finite temperatures into account, a stochastic,
thermal field H th is added to the effective field in the LLG equation [38]
∂J α ∂J
=−γ J× ( Heff + Hth) + J × . (7.1)
∂t J ∂t
s
Future hard disks will make use of spin valves operating in CPP mode. Therefore simulations
of CPP spin valves become more and more interesting. A current perpendicular to the layer
structure leads to spin transfer, which has to be taken into account by an additional term,
called spin torque term [39]
∂J α ∂J
Cj
=−γ J× H ′
eff + J× + J× J × J. (7.2)
∂t J ∂t
J J′
2
s s s
Here J ′ is the magnetization of a reference layer, which polarizes the electrons. The polarized
current transfers the spin to the actual magnetic polarization J and causes an additional
motion of J. j is the local current density. For CPP spin valves the magnetization is transferred
from the pinned layer to the free layer or vice versa.
The application of this program is not restricted to read head simulations. It can also be
applied, for example, on magnetic tunnel junctions (MTJs), which are used in MRAMs
(Magnetic Random Access Memory).
77

Appendix A
The theorem
Appendix A
∫grad f dV = � ∫ f dA
(A.1)
V ∂V
is often used in literature, but it is normally limited to continuously differentiable functions.
Nevertheless this relation is also frivolously applied for functions with first order
singularities, as for example
1
f () r = with V
r−r ′
′∈ r . (A.2)
We were not able to find any proof, which shows the validity of the above theorem for this
function. Thus it is given here by our own.
3
Let us assume any constant vectorc
∈ � and start from the equality
divcf= c grad f . (A.3)
Integration over the whole volume V gives
divcfdV = � cgrad
f dV . (A.4)
∫ ∫
V V
If f has continuous first order partial derivatives we can apply the Divergence Theorem
resulting in
� ∫ c f dA= ∫cgrad
f dV . (A.5)
∂V
V
c is constant and therefore it can be brought in front of the integral.
c� ∫ f dA= c∫grad
f dV . (A.6)
∂V
V
Last equation holds for any c, therefore c can be cancelled, and the theorem (A.1) is verified.
78

Appendix A
In case we have the function as given in (A.2), we have to divide the volume V into a small
sphere with radius ε around the singularity r ′ denoted with Kε ( r ′ ) and the rest of V. Then
the theorem (A.1) can be written as
∫ grad f dV + ∫ grad f dV = �∫ f dA− �∫ f dA+ � ∫ f dA.
(A.7)
V / Kε( r′ ) Kε( r′ ) ∂V ∂Kε( r′ ) ∂Kε(
r′
)
For the volume V / Kε ( r ′ ) with the outer surface ∂ V and the inner surface ∂Kε ( r ) the
theorem holds, because the singularity is excluded. To prove the theorem, we only have to
show, that
∫ ∫
grad f dV = � f dA
. (A.8)
Kε( r′ ) ∂Kε(
r′
)
The equality is valid, if both sides vanish for ε → 0 . Without loss of generality, it is sufficient
to show the equality for the first vector component. Then the integrand of the left hand side is
( f ) = 3
grad x
x − x′
, (A.9)
r−r′ which is asymmetric around r ′ . Therefore the left hand side in (A.8) vanishes. For the right
hand side we have
1
⎛ ⎞
4πε
= ∫ dA = f dA ≥ ⎜ f d ⎟
Kε( ′ − ′ ∫ ⎜ ∫ A
r r
⎟
∂ r ) ∂Kε( r′ ) ⎝∂Kε( r′
) ⎠
� � � . (A.10)
So this side also vanishes, if ε→ 0 , q.e.d.
x
79

Appendix B
Appendix B
To calculate the magnetic field of conductive materials (4.50) has to be solved:
Δ H =−curl
j. (B.1)
In addition boundary conditions are needed to make this problem solvable. The derivation of
the boundary conditions for a surface point r is given here. We have to calculate
∂ ∂
∂n ∂n
out in
( H () r − H () r ) = lim ( H( r+ nε) −H( r−nε) )
ε→0
. (B.2)
n denotes the unit vector normal to the boundary face in point r. We start from the Biot-
Savart Law (2.46)
1 r± nε−r′ Hr ( ± nε ) = ×
4π ∫ j
V r± nε−r′ Now we separate a half sphere
Volume V.
3
dV ′ . (B.3)
1/2
Kη () r with radius η (η � ε ) and midpoint r from the
1 r± nε− r′ 1 r± nε−r′ Hr ( ± nε ) = × dV ′ + × dV ′
4π ∫ j
4π
∫ j
3 3
1/2 1/2
V / Kη ( r) r± nε− r′ Kη
( r)
r± nε−r′ (B.4)
In this half sphere the current density j is assumed to be constant. If j is a continuous function,
1/2
the mean value theorem tells us that there exists a point r ′′ ∈ K for which
± η
1 r± nε− r′ 1 r± nε−r′ Hr ( ± nε ) = × dV ′ + ( ′′ ) ×
dV ′
4π ∫ j jr
4π
∫
. (B.5)
3 ±
3
1/2 1/2
V / Kη ( r) r± nε− r′ Kη
( r)
r± nε−r′ The integrand of the second integral can be expressed as gradient of a scalar term
1 r± nε−r′ 1 1
Hr ( ± nε ) = × dV ′ + ( ′′ ) × ∇′
dV ′
4π ∫ j jr
4π
∫
. (B.6)
r± nε−r′ 3
±
1/2 1/2
V / Kη ( r) r± nε−r′ Kη
( r)
80

Appendix B
Moreover the second integral can be transformed into a surface integral by applying the
theorem, proved in Appendix A
1 r± nε−r′ 1 1
Hr ( ± nε ) = × dV ′ + ( ′′ ) ×
d ′
4π ∫ j j r
4π
� ∫
A . (B.7)
r n r r± nε−r′ 3
±
1/2 1/2
V / Kη ( r) ± ε− ′
∂Kη
( r)
Now we are able to calculate the difference in (B.2). Due to the difference and ε→ 0 the first
integral and the rotund surface of the half sphere do not contribute ( η � ε ), which leads to
1 dA′ 1 dA′
Hr ( + nε) −Hr ( −nε ) = jr ( ′′ ) × − ( ′′ ) ×
π ∫ jr
+ ε− ′ π∫ . (B.8)
r n r r−nε−r ′
+ −
4 4
Cη( r) Cη(
r)
Here Cη( r ) denotes a circle with radius η and radius r. Derivation in direction n gives
∂ 1 εdA′
∂n4π C r+ nε−r′ ( Hr ( + nε) −Hr ( −nε ) ) =− ( jr ( ′′ + ) + jr ( ′′ −)
) ×
3
∫
η ( r)
. (B.9)
The substitution ρ= r−r ′ and the simultaneous consideration that n is perpendicular to
( − ′ )
r r lead to
∂ 1 ε
( Hr ( + nε) −Hr ( −nε ) ) =− ( jr ( ′′ + ) + jr ( ′′ −)
) × n
ρdρdϕ. (B.10)
3/2
∂n4π ρ +ε
Calculating the integral yields
2π
η
∫∫ 2 2
0 0(
)
∂ out in jr ( ′′ + ) + jr ( ′′ ) ⎛
− ε ⎞
( H () r − H () r ) = n×
⎜1− ⎟.
(B.11)
∂n 2 ⎜ 2 2
η +ε ⎟
⎝ ⎠
Now the limit can be calculated in (B.2)
∂
∂n
out in j( r′′ + ) + j( r′′
−)
H r − H r = n×
. (B.12)
2
( () () )
And finally, for η→ 0 we get the result
∂
∂n out in
( )
H () r − H () r = n× j() r . (B.13)
In case of an interface between two volumes V 1 and 2
interface j 1 and j 2 respectively, we have a jump for the normal derivation of H
V with different current densities at the
81

∂
∂n 2 1 2 1
( H () r − H () r ) =− n× ( j () r −j()
r )
Appendix B
. (B.14)
82

List of Figures
List of Figures
Figure 1.1: The original drawing of Oberlin Smith (1888) and Valdemar Poulson’s
telegraphone [2].................................................................................................................7
Figure 1.2: The first computer disk drive RAMAC. The pneumatically driven actuator
arm and some of the 50 disks are visible [4].....................................................................8
Figure 1.3: The development of bit sizes and storage densities over the last years [7]. ........10
Figure 2.1: a) Illustration of the damped precession of the magnetic polarization J
around the effective field Heff, taken from [15]. b) Influence of the Gilbert
damping constant on the field rise time in magnetic read heads. Head field
strength as a function of time for different Gilbert damping constants within the
head [43]. The dashed line depicts the current profile. The solid lines give the
head field for α = 1, 0.5, 0.1, and 0.02. The fastest field rise time is achieved with
intermediate damping (α = 0.5).......................................................................................20
Figure 2.2: The spin-split bands of a ferromagnet [18]..........................................................21
Figure 2.3: Giant magnetoresistance in Fe/Cr multilayers [18]. ............................................22
Figure 2.4: Schematic model of a spin-valve GMR head (CIP mode) with exchangepinned
layer and longitudinal hard bias [19]...................................................................23
Figure 2.5: Change in resistance of a spin valve compared to AMR (NiFe) vs.
magnetization orientation [19]. .......................................................................................23
Figure 2.6: The two different modes of a spin valve. The current is applied either
longitudinal (CIP) or perpendicular (CPP) to the layer structure [7]. .............................24
Figure 2.7: Potential landscape inside the FM/NM/FM active part of a spin-valve, for
the two spin channels in the parallel state (a,b) and in the antiparallel state (c),
indicating the different origins of scattering [20]............................................................25
Figure 2.8: A comparison of longitudinal recording and perpendicular recording [7]. .........27
Figure 3.1: A schematic 2-dimensional model of a read head for longitudinal
recording..........................................................................................................................30
Figure 3.2: The magnetic potential at the ABS under Karlquist approximation....................31
Figure 3.3: The x- and the z-component of the normalized head field of a read head
with 60 nm gap, and sensing layer thickness 5 nm, 10 nm below the ABS....................33
Figure 3.4: The effective signal field as function of the relative position of the read
head in respect to a perfect transition (Js = 0.5 T)...........................................................35
Figure 3.5: These two figures show the x-component of the magnetization (Js = 0.5 T)
of the data layer after a write process with head velocity 10 m/s (left) and 20 m/s
83

List of Figures
(right). Red color represents magnetization in positive x-direction and blue color
magnetization in negative x-direction. The scale is in µm, thus the bit length is
approximately 40 nm. The data layer thickness is 12 nm. ..............................................35
Figure 3.6: The effective fields of the recording layers (shown in Figure 3.5) as
function of the head’s position. The dashed lines indicate the boundaries of the
recording layer model......................................................................................................36
Figure 3.7: The simplified read head model (left) for perpendicular recording. In the
right picture the mirror heads are indicated, which replace the SUL with its high
permeability [22]. ............................................................................................................37
Figure 3.8: The x- and the z-component of the normalized head field for perpendicular
recording..........................................................................................................................38
Figure 3.9: a) Application of Ampere’s Law on the dashed path, b) application of
Gauss’s Law on the dashed box. .....................................................................................39
Figure 4.1: Typical basis functions of two neighboring nodes of a 2-dimensional mesh......43
Figure 4.2: The left figure shows a set of 10 nodes after renumbering and the
clustering. The right figure shows the associated boundary matrix structure. The
large off-diagonal blocks, corresponding to far field interactions of two clusters,
can be approximated by low-rank matrices [25]. ............................................................49
Figure 4.3: Hϕ for an infinite long wire is plotted over the distance from the middle
axis. The second curve shows the error of the numerical calculation.............................55
Figure 4.4: The model of the coil (left). The color code shows the linear decrease of
the electric potential inside the wire. The chart (right) shows the magnetic field of
a solenoid in z-direction at the middle axis. The two dotted lines identify the
position of the solenoid. The second graph shows the relative error of the coil’s
magnetic field calculated with FEM calculation compared to the solenoid’s field. .......56
Figure 4.5: Schematic diagram of the simulation cycle. Adapted from [31]. ........................58
Figure 5.1: The shift of the hysteresis loop after a field cooling treatment of a
ferromagnetic/antiferromagnetic system [19] and the temperature dependence of
the exchange bias field for NiFe(4-8nm)/Cu(2.2nm)/CoFe(2nm)/IrMn(15nm) spin
valves with different free layer thicknesses [33].............................................................60
Figure 5.2: A comparison (left) of a conventional spin valve and a spin valve using a
synthetic ferromagnet to fix the pinned layer [20]. The interlayer exchange
constant (right) for two Ni80Co20 layers as function of the Ru spacer layer
thickness [32]...................................................................................................................62
Figure 5.3: The effect of the sense current direction on the free layer (FL). For the first
case (left) the current field Hcurr partially cancels the stray field HS of the pinned
layer (PL) and weakens the exchange bias field Hexch. For the other current
direction (right) the stray field and the exchange bias field are supported by the
current field. ....................................................................................................................63
Figure 5.4: The layers of the GMR element also see mirror currents due to the high
permeability of the shields. The field of these mirror currents is able to reduce the
field of the original current at the free layer and the pinned layer. .................................63
84

List of Figures
Figure 6.1: The magnetic model used for the FEM calculations. The mesh size for the
shields (green) is 50 nm, and is refined to 5 nm near the track and near the spin
valve layers. The hard bias magnets (purple), the free layer (dark blue), and the
pinned layer (light blue) are also meshed with triangle size of 5 nm..............................66
Figure 6.2: The FEM conductor model is shown. This simplified model shows the four
main layers of a GMR element, which contribute most to the conductance: The
free layer (dark blue) the nonmagnetic Cu layer (orange), the pinned layer (light
blue), and the antiferromagnet (yellow). Additionally the high conductive leads
(pink) are shown..............................................................................................................67
Figure 6.3: The effective conductivities for a typical spin valve taken from [37]. ................68
Figure 6.4: The normalized magnetization of the free layer and the pinned layer in
equilibrium state for the two different current directions. The color code gives the
z-component of the normalized magnetization, red corresponds to the positive and
blue to the negative z-direction. Thus the red arrows represent the pinned layer
magnetization. The right hand side gives the configuration for the more favorable
current direction, because here the current field cancels partially the
demagnetizing field of the pinned layer. .........................................................................69
Figure 6.5: The external applied field, which is applied in z-direction to evaluate the
transfer curve of the GMR sensor. ..................................................................................70
Figure 6.6: The output voltage (left) and the relative change in resistance (right, in
respect to the equilibrium state) over the external applied field in z-direction for
both current direction and a larger exchange bias field...................................................71
Figure 6.7: The transfer curves for the spin valve with and without shields (for both
Hexch = 0.1 T). ..................................................................................................................72
Figure 6.8: The output signal over time for a relaxation from equilibrium state with
Hext = 0.3 T to the equilibrium state without field for different Gilbert damping
constants. .........................................................................................................................73
Figure 6.9: The read back signal and the relative change in resistance for a perfect
→← transition................................................................................................................74
Figure 6.10: The read back signal and the relative change in resistance for a perfect
←→ transition................................................................................................................75
Figure 6.11: The model of a perfect bit pattern. It consists of 30 perfectly written bits
and is used for our read back signal calculations. The track width is 120 nm and
the bit length 60 nm. The layer thickness is 12 nm and the spontaneous
polarization is 0.44 T. The bits are alternately magnetized in positive and negative
x-direction........................................................................................................................76
Figure 6.12: The read back signal for the periodic bit pattern calculated for the GMR
sensor and for the whole read head (with shields). .........................................................76
85

Bibliography
Bibliography
[1] O. Smith, Electrical World, September 8, 1888.
[2] S. E. Schoenherr, Recording Technology History, http://history.acusd.edu/gen/
recording/notes.html, University of San Diego, January 27, 2005.
[3] A. S. Hoagland, “History of Magnetic Disc Storage Based on Perpendicular
Recording”, IEEE Trans. Magn., vol. 39, no. 4, pp. 1871-1875, July 2003.
[4] B. Hayes, “Terabyte Territory”, American Scientist, vol. 90, no. 3, pp. 212-216,
May/June 2002.
[5] R. Hunt, “A magnetoresistive readout transducer”, IEEE Trans. Magn., vol. MAG-7,
no. 1, pp. 150-154, March 1971.
[6] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G.
Creuzet, A. Friederich, J. Chazelas, “Giant Magnetoresistance of (001)Fe/(001)Cr
Magnetic Superlattices“, Phys. Rev. Lett., vol. 61, no. 21, pp. 2472-2475, November
1988.
[7] Hitachi Global Storage Technologies, Hitachi Research and Technology, http://www.
hitachigst.com/hdd/research/, April 22, 2005.
[8] E. E. Fullerton, D. T. Margulies, M. E. Schabes, M. Carey, B. Gurney, A. Moser.,
M. Best, G. Zeltzer, K. Rubin, H. Rosen, M. Doerner, “Antiferromagnetically
coupled magnetic media layers for thermally stable high-density recording”, Appl. Phys.
Lett., vol. 77, no. 23, pp. 3806-3808, December 2000.
[9] E. E. Fullerton, D. T, Margulies, N. Supper, H. Do, M. Schabes, A. Berger, A.
Moser, “Antiferromagnetically Coupled Magnetic Recording Media”, IEEE Trans.
Magn., vol. 39, no. 2, pp. 639-644, March 2003.
[10] C. T. Rettner, S. Anders, T. Thomson, M. Albrecht, Y. Ikeda, M. E. Best, B. D.
Terris, “Magnetic Characterization and Recording Properties of Patterned Co70Cr18Pt12
Perpendicular Media”, IEEE Trans. Magn., vol. 38, no. 4, pp. 1725-1730, July 2002.
86

Bibliography
[11] H. Katayama, M. Hamamoto, J. Sato, Y. Murakami, K. Kojima, “New
Developments in Laser-Assisted Magnetic Recording”, IEEE Trans. Magn., vol. 36, no.
1, pp. 195-199, January 2000
[12] P. Vettiger, M. Despont, U. Drechsler, U. Dürig, W. Häberle, M. I. Lutwyche, H. E.
Rothuizen, R. Stutz, R. Widmer, G. K. Binnig, “The ‘Millipede’ – more than one
thousand tips for future AFM data storage”, IBM J. Res. Dev., vol. 44, no. 3, pp. 323-
340, May 2000.
[13] P. Mohn, Magnetism in the Solid State: An Introduction, Springer Verlag, Berlin, 2003.
[14] W. Bailey, P. Kabos, F. Mancoff, S. Russek, “Control of Magnetization Dynamics in
Ni81Fe19 Thin Films Through the Use of Rare-Earth Dopants”, IEEE Trans. Magn., vol.
37, pp. 1749-1754, July 2001.
[15] M. Kirschner, Exchange Bias in Ferro-/Antiferromagnetic Bilayers, Diploma Thesis,
Working Group Magnetic Materials and Micromagnetics, Vienna University of
Technology, 2003.
[16] N. F. Mott, “The Electrical Conductivity of Transition Metals“, Proc. Roy. Soc. (Lond.)
A, no. 153, pp. 699-717, 1935.
[17] N. F. Mott, “Electrons in Transition Metals“, Adv. Phys., no. 13, pp. 325-422, 1964.
[18] S. Blundell, Magnetism in condensed matter, Oxford Univ. Press, New York, 2001.
[19] S. X. Wang, A. M. Taratorin, Magnetic Information Storage Technology, Academic
Press, San Diego, 1999.
[20] R. Coehoorn, Novel Magnetoelectronic Materials and Devices, Lecture Notes, 2003.
[21] M. M. Aziz, Signal and Noise Properties of Longitudinal Thin-Film Disk Media,
Master Thesis, Electronic and Information Storage Systems Research Group, University
of Manchester, 1999.
[22] H. Muraoka, Y.Sugita, Y. Nakamura, “Simplified Expression of Shielded MR Head
Response for Double-Layer Perpendicular Medium“, IEEE Trans. Magn., vol. 35, no. 5,
pp. 2235-2237, September 1999.
[23] T. A. Davis, Algorithm 8xx: a concise sparse Cholesky factorization package,
http://www.cise.ufl.edu/research/sparse/ldl/, January 5, 2004.
[24] D. R. Fredkin, T. R. Koehler, “Hybrid Method for Computing Demagnetizing Fields“,
IEEE Trans. Magn., vol. 26, no. 2, pp. 415-417, March 1990.
87

Bibliography
[25] S. Kurz, O. Rain, S. Rjasanow, “The adaptive cross-approximation technique for the
3D boundary-element method”, IEEE Trans. Magn., vol. 38, no. 2, pp. 421-424, March
2002.
[26] D. M. Mount, ANN Programming Manual, http://www.cs.umd.edu/~mount/ANN/,
University of Maryland, 2005.
[27] D. A. Lindholm, “Three-Dimensional Magnetostatic Fields from Point-Matched
Integral Equations with Linearly Varying Scalar Sources”, IEEE Trans. Magn. , vol. 20,
no. 5, pp. 2025-2032, September 1984.
[28] S. Pissanetzky, “Comments on ‘Analytical Computation of Magnetic Vector Potential
for Tetrahedral Conductors’”, IEEE Trans. Magn., vol. 29, no. 2, pp. 1282-1283, March
1993.
[29] S. D. Cohen, A. C. Hindmarsh, “CVODE User Guide“, Lawrence Livermore National
Laboratory report, UCRL-MA-118618, 1994.
[30] S. D. Cohen, A. C. Hindmarsh, “CVODE, a stiff/nonstiff ODE solver in C”,
Computers in Physics, vol. 10, no. 2, pp. 138-143, March/April 1996.
[31] P. Speckmayer, Electron Microscopy and Micromagnetic Simulation of FePt and CoPt
Nanoparticles, Diploma Thesis, Working Group Magnetic Materials and
Micromagnetics, Vienna University of Technology, 2004.
[32] W. H. Bean and C. P. Meiklejohn, “New Magnetic Anisotropy”, Phys. Rev., vol. 102,
pp. 1413, 1956.
[33] S. Tanoue, K. Tabuchi, “Magnetoresistance characteristics of NiFe/Cu/CoFe/IrMn
spin valves”, J. Vac. Sci. Technol. B, vol. 19, no. 2, pp. 563-566, March 2001.
[34] J. Nogués, C. Leighton, I. K. Schuller, “Correlation between antiferromagnetic
interface coupling and exchange bias“, Phys. Rev. B, vol. 61, pp. 1315, 2000.
[35] H. A. M. Van den Berg, W. Clemens, G. Gieres, G. Rupp, W. Schelter, M. Veith,
“GMR sensor scheme with artificial antiferromagnetic subsystem”, IEEE Trans. Magn.,
vol. 32, no. 5, pp. 4624-4626, September 1996.
[36] P. Bruno, “Theory of interlayer exchange coupling”, Phys. Rev. B, vol. 52, no. 1, pp.
411-439, July 1995.
[37] A. T. McCallum, S. E. Russek, “Current density distribution in a spin valve
determined through in sito conductance measurement”, Appl. Phys. Lett., vol. 84, no.
17, pp. 3340-3342, April 2004.
88

Bibliography
[38] W. F. Brown Jr., Micromagnetics, Wiley, New York, 1963.
[39] S. J. Slonczewski, “Current-driven excitation of magnetic multilayers”, J. Magn. Magn.
Mater., vol. 159, no. 1-2, pp. L1-L7, June 1996.
[40] R. L. Comstock, Introduction to Magnetism and Magnetic Recording, Wiley, New
York, 1999.
Parts of this work have been published:
[41] G. Hrkac, T. Schrefl, M. Kirschner, F. Dorfbauer, D. Suess, O. Ertl, J. Fidler,
“Three-dimenisonal micromagnetic finite element simulations including eddy currents“,
J. Appl. Phys., vol. 97, no. 10, May 2005, in press.
[42] G. Hrkac, T. Schrefl, O. Ertl, M. Kirschner, D. Suess, J. Fidler, “Influence of Eddy
Currents on Magnetization Processes in Sub-Micron Permalloy Structures”, IEEE
Trans. Magn., in press.
[43] O. Ertl, T. Schrefl, D. Suess, M. E. Schabes, “Influence of the Gilbert damping
constants on the flux rise time of write head fields“, J. Magn. Magn. Mater., vol. 290-
291, part 1, pp. 518-521, April 2005.
[44] M. E. Schabes, T. Schrefl, D. Suess, O. Ertl, “Dynamic Micromagnetic Studies of
Anisotropy Effects in Perpendicular Write Heads”, IEEE Trans. Magn., in press.
[45] T. Schrefl, M. E. Schabes, D. Suess, O. Ertl, M. Kirschner, F. Dorfbauer, G.
Hrkac, J. Fidler, “Partitioning of the Perpendicular Write Field into Head and SUL
Contributions”, IEEE Trans. Magn., in press.
89

Danksagung
Ich möchte all jenen meinen Dank aussprechen, die mich bei der Diplomarbeit unterstützt
oder motivierend auf mich eingewirkt haben.
Insbesondere möchte ich mich bei meinem Betreuer Doz. Thomas Schrefl für seine ständige
Bereitschaft, mir bei Fragen oder Anliegen weiterzuhelfen, bedanken. Weiters seien Florian
Dorfbauer, Gino Hrkac, Markus Kirschner und Dieter Süss erwähnt, die mir viele hilfreiche
Tipps zur Diplomarbeit gaben. Generell möchte ich allen danken, die zum ausgezeichneten
Arbeitsklima innerhalb der Arbeitsgruppe Mikromagnetismus beitrugen.
Chee Lim vom Samsung Advanced Institute of Technology möchte ich für die (wenigen)
Informationen zu aktuellen Leseköpfen danken.
Meinen Eltern möchte ich danken, dass sie mir durch ihre finanzielle Unterstützung das
Studium ermöglicht haben.
Meiner Freundin Katrin danke ich für ihre Unterstützung während meiner Diplomarbeit.
90

Lebenslauf
Otmar Ertl
Reinprechtsdorferstraße 20/16
1050 Wien
Tel.: +43/650/9461912
PERSÖNLICHE ANGABEN
AUSBILDUNG
AUSZEICHNUNGEN
� Staatsangehörigkeit: Österreich
� Geburtsdatum: 22. Februar 1982
� Geburtsort: Vöcklabruck
� Eltern: Mag. Karl Ertl und Renate Ertl
� 1988 bis 1992 Volksschule Vöcklabruck
� 1992 bis 2000 BRG Schloss Wagrain, Vöcklabruck
� Juni 2000 Ablegung der Reifeprüfung mit ausgezeichnetem
Erfolg
� seit Okt. 2000 Studium der technischen Physik an der TU Wien
� Juni 2002 Abschluss des 1. Studienabschnitts mit
ausgezeichnetem Erfolg
� 2. Platz (1999) und 1. Platz (2000) beim Bundeswettbewerb der
österreichischen Mathematikolympiade
� Bronzemedaille (1999) und Silbermedaille (2000) bei der
internationalen Mathematikolympiade
91