Raw text from my Diplomarbeit
appears as the change of few percent in electrical resistivity 
ρ(θ) depending
on the angle θ = \(j,M) between the direction of the electrical current j(r)
and the orientation of the samples magnetisation M(r) at any point r in the material.
Extreme cases are collinear and perpendicular orientation with maximal ρll or minimal resistivity
ρT. A cos2-term can well describe the variation between those extremes
[8].
�(#) = �? + (�k − �?) cos2 # = �? + �� cos2 #,
= �k + (�? − �k) sin2 #, (2.1)
= �? cos2 # + �k sin2 #.
Resistivity r [W]
-180 -90 0 90 180
Angle q between current and magnetisation [°]
r∥
r⊥
-180° -90° 0° 90° 180°
Figure 2.1: Electrical resistivity �(#) as a function of the angle
- = \(~j, ~M ) between the direction of the electrical current ~j(~r) and the
orientation of the samples magnetisation ~M (~r).
2.1.1 Formalism of the AMR
The AMR ratio �AMR is defined as the normalized variation in resistivity
and is a good measure for the size of the effect2. It can be obtained directly
2In many publications the AMR ratio is also given as ��/�ave where �ave = 1
3�k + 2
3�?
is the average value for truly demagnetized polycrystalline bulk material. In this case
2.1. Anisotropic magnetoresistance 5
from experiment by measuring the resistance parallel and perpendicular to
the magnetisation.
�AMR =
(Rk − R?)
Rk
=
�R
Rk
,
=
(�k − �?)
�k
=
��
�k
.
(2.2)
Taking the z-axis along the direction of magnetisation, the resistivity can be
written in form of a tensor as
ˆ�ik = 0@
�? −�H 0
�H �? 0
0 0 �k
1A
. (2.3)
With ~uM being the unit vector in the direction of the magnetisation, Ohm’s
law can be given in the form of
~E
= ˆ�ik
~j
= �?
~j + �� (~uM ·~j)~uM + �H ~uM×~j
(2.4)
The diagonal elements are the resistivities along or perpendicular to the magnetisation
while the off-diagonal elements ±�H represent the spontaneous or
anomalous Hall effect, which is small for permalloy and shall not be discussed
here.
2.1.2 Fundamentals of the AMR
An exhaustive quantum mechanical description of the effect is rather lengthly.
Solely an overview on the microscopic origin of the anisotropic magnetoresistance
as well as a short introduction to the basic principles of ferromagnetism
in iron and nickel will be given. This should give a qualitative understanding
of the nature of the AMR (see also [11], [12] and [13]). A more extensive
description can be found in [8].
Ferromangetism in transition metals
Exchange interaction In a model assuming localized electrons one can describe
the interaction between the electron spins with the Hamiltonian introthe
magnetisation of the domains is randomly orientated throughout the three dimensions.
In thin films where the magnetisation is only in-plane the expression changes to
�ave = 1
2�k + 1
2�?.
6 2. THEORY
duced by Heisenberg [14]:
HHeisenberg = −2A �1�2. (2.5)
For a positive exchange constant (A > 0) the two spins �1 and �2 will
energetically prefer a parallel orientation, which leads to a ferromagnetic
spin lattice. A negative exchange constant (A < 0) will cause the spins to
orientate antiparallel and promote an antiferromagnetic spin lattice.
This exchange interaction can issue from divers sources: either directly
from a relevant overlap of the electron orbits or indirectly via interaction with
electrons of diamagnetic atoms situated between the atoms of the (anti-) ferromagnetic
lattice. This is a good model for antiferromagnetic manganous
oxide (Mn2+O2−). Additionally the conduction electrons can act as mediator
between the spins as proposed by M. A. Rudermann, C. Kittel, T. Kasuya
and K. Yosida. RKKY-interaction can play a relevant role in the ferromagnetism
of rare earths [15].
In transition metals the spins of the 3d-like electrons are responsible for
the ferromagnetism. In those materials these electrons cannot be considered
as localized since they hybridize with the 4s-like electrons and form a half
filled conduction band. Here the exchange interaction between the quasi free
electrons lowers the energy of the majority electrons (") and rises that of the
minority electrons (#) as shown in Fig. 2.2.
a)
10 20 30 10 20 30 0
EF
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Energy (Ry)
D(E) [103/Ry]
- "
b)
0.4
0.3
0.2
0.1
0.5
0.6
0.7
0.8
0.9
Energy (Ry)
0 10 20 30 10 20 30
EF
s,p-band
d-band
D(E) [103/Ry]
- "
Figure 2.2: Density of states in nickel divided into majority (") and minority
(#) spin states according to J. Callaway and C. S. Wang [16]
a) From self-consistent band structure calculations
b) Schematic illustration of the s,p- and d-bands
2.1. Anisotropic magnetoresistance 7
Domain patterns In a macroscopic ferromagnetic structure the magnetisation
will generally not point homogeneously in one direction, but divide into
domains in which the magnetisation is parallel, divided by domain walls,
where the magnetisation changes direction. The average over many domains
eventually sums up to zero, so that an iron nail may seem without magnetisation.
This is due to a series of energy terms, that oppose and balance:
Etotal = Eexchange + Estray-field + Eanisotropy + EZeeman + Eother. (2.6)
The exchange energy, as described by the Heisenberg Hamiltonian, tends to
parallelize the spins and therefore prefers a homogeneous magnetisation. This
is in contradiction to the second term, that represents the energy of the stray
field, which would maximize for a homogeneous magnetisation. The third
term denotes the symmetry of the lattice, that makes certain directions of
magnetisation more favourable than others. The face-centred cubic lattice of
nickel, for example, prefers magnetisations in one of the four [111] directions,
i. e. the space diagonals. The Zeeman energy arises from the interaction
with the applied magnetic field and tends to align the magnetisation with
~B
ext. The last term contains other energies like the magnetostrictive energy,
which can be neglegted here. As a result, a magnetic object may have a
magnetisation as in Fig. 2.3, where the stray field is minimized and seven
domains result with a near zero average magnetisation.
a) b) c)
Figure 2.3: Possible domain pattern in a 2×4 µm2 permalloy element of
25 nm thickness:
a) Measured magnetisation with a magnetic force microscope (MFM).
b) Calcutated MFM image from simulation.
c) Micromagnetic computer simulation of the magnetisation.
8 2. THEORY
Origin of the AMR
Spin-orbit coupling The dependence of the resistivity on the angle between
the magnetisation and the current as illustrated in Fig. 2.4 is due to electron
scattering from the 4s,p-band to the 3d-band connected by a spin flip. This
additional scattering channel is opened by the spin-orbit interaction which
contributes to the Hamiltonian in the form of
Hspin-orbit = K L S = K Lz Sz + K2
L+ S− + K2
L− S+. (2.7)
Here the generators and annihilators L± = Lx ± iLy and S± = Sx ± iSy,
if applied to an electron wave function, can increase or decrease the orbital
quantum number or flip the spin. The operator product L+ S− turns a majority
spin p"-electron wave function into that of a d#-electron.
Unlike the rather isotropically widely spread s,p-states, the more localized
d-states have a strong orbital anisotropy. This accounts for an anisotropic
scattering cross section for interband scattering with spin flip, that is bigger
for s,p-electrons with a momentum parallel to the orbit of the empty
3d-state and therefore, due to spin-orbit coupling, bigger if parallel to the
magnetisation [4].
a) b)
Figure 2.4: Illustration of the origin of the anisotropic magnetoresistance.
The direction of the sample’s magnetisation is connected to the spin of
the 3d-electrons. The anisotropic 3d-orbits have a bigger cross section for
an electron current in the direction of the magnetisation.
a) If the magnetisation is perpendicular to the current direction
( ~M ?~j), then the resistivity is low.
b) For parallel orientation (Mk~j) the resistivity is higher.
Since the s,p-band has a few times higher group velocity
vg(EFermi) = 1
~
@E(k)
@k and a smaller effective mass m? = ~2( @2E(k)
@k2 )−1 at
the Fermi energy than the d-band, as one can see from band structure
2.1. Anisotropic magnetoresistance 9
calculations and photoelectron spectroscopy [18, 16, 19], mostly s,p-like
electrons contribute to the conductivity in permalloy. It is assumed, that
the high density of states D#d(EFermi) of the minority spin d-electrons at the
Fermi energy is responsible for the short mean free path of the minority
s,p-conduction electrons, which results in a high resistivity for the minority
s,p-subband. Thus mostly majority spin s,p-electrons contribute to the
conductivity and therefore the spin-orbit interaction plays a relevant role,
since it scatters the main conduction electrons into the biggest reservoir of
empty states at the Fermi energy.
r (B)
r^
r | |
| |
^
Bext[mT]
Figure 2.5: Resistivity of a multi-domain ferromagnetic structure in dependence
of an applied magnetic field ~Bext. The red curve shows the
behaviour with the magnetic field in the direction of the current (k). The
blue curve corresponds to the perpendicular case (?).
Applying magnetic fields As described above, a macroscopic ferromagnetic
structure consists of domains with different directions of magnetisation.
These domains can be aligned by an applied magnetic field ~Bext. The strength
of the magnetic field ~Bsat, that is needed to saturate the alignment varies
with material, form, direction, and temperature of the ferromagnetic structure.
Although the alignment of the domains is not brought about by simultaneously
rotating the magnetisation in each domain, a quasi continuous
alignment process of the magnetisation of the sample is possible. The domain
walls are shifting in such a way, that the domains with a magnetisation in
direction of ~Bext grow on cost of those pointing in other directions.
10 2. THEORY
A typical AMR signature is depicted in Fig. 2.5. Starting from a demagnetized
multi-domain state at zero field, the magnetisation becomes aligned
as the field strength rises. An external magnetic field ~Bext either parallel
or perpendicular to the current results in a rise or fall of the resistance,
respectively.
In ferromagnetic microstructures certain domain configurations like that
in Fig. 2.3 are energetically favourable. In these cases, the domain configuration
can switch from one state to another, so that the magnetisation process
is a series of quasi continuous (reversible) domain-wall movements separated
by (irreversible) configuration changes. The shape of the ferromagnetic microstructure
can induce hard and easy axes of magnetisation, e.g. directions
in which the structure can easily be saturated and othes in which the magnetic
field strength for saturation Bsat is much higher. In cases of extreme
shape anisotropy quasi single-domain microstructures can be produced. In
thin permalloy films with a thickness of about or less than t . 100 nm the
magnetisation is preferably in-plane and the domain walls are N´eel walls,
where the magnetisation between the domains is rotating in the plane of
magnetisation.
2.1.3 Magnitude of the AMR
a)
1
2
3
4
5
60 70 80 90 50
Concentration of
nickel in iron x [%]
b)
1
2
3
4
5
40 60 80 20
Film thickness t [nm]
c)
2
100 200
4
6
8
Temperature T [K]
Figure 2.6: Magnitude of the AMR-Ratio in NixFe1−x alloys:
a) Depending on the concentration x of nickel at room temperature
[2].
b) Depending on the film thickness in Ni80Fe20 at 4 K [20].
c) Depending on the temperature in Ni80Fe20 [4].
The size of the effect usually given in percent as �AMR = �R
R changes
with material, temperature, shape, and many other parameters. Especially
2.2. Magnetic-force microscopy 11
in thin films the resistance R is influenced by grain size, surface quality, and
film thickness and therefore varies with deposition rate, groth temperature,
heat treatment, and vacuum quality. In Fig. 2.6 the variation with three
main parameters is shown. For room temperature measurements Ni80Fe20
permalloy films with a thickness of t & 20 nm still show a reasonable effect.
![{\displaystyle \Delta _{AMR}[\%]\;\;{\overset {\underset {\mathrm {def} }{}}{=}}\;\;{\frac {\Delta \rho }{\rho _{0}}}\cdot 100}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ecec8568e0174832cce50140d55cc00bb1b0465a)
