Ferromagnetism: Theory and Applications

Ferromagnetism 201
2d
D
E
Ec
EcLGD
F
I
k
L
m
M
N
N1
N2
p
pc
P
Ps
q
unit cell length
electric displacement
electric field
coercive field
coercive field within the LGD theory
without accounting for domains
free energy
current
Boltzmann constant
distance between electrodes of a capacitor
anion mass (within the model)
anion mass density (within the model)
number of anions (cations) in a unit
volume
number of anions (cations) in the atomic
chain
number of the atomic chains in a crystal
pressure
pressure of change of the form of the
short-range potential energy of anions
within the model
polarization
spontaneous polarization
anion (cation) charge (within the model)
Q
S
Tat
Tc
U
V
w
a0
a, b
aat
ad
as
e0
vf
vs
x
charge of a capacitor
area of the capacitor electrode
atomic temperature
phase-transition temperature
potential energy of an anion due to its
short-range interaction with the cations
(within the model)
voltage
polarization probability density distribution
coefficient in the dependence of a on the
temperature
coefficients of expansion of the potential
energy U and the free energy F in series
of powers of P2
atomic value of a
contribution to a due to dipole–dipole
interactions
contribution to a due to short-range interactions
permittivity of vacuum
volume of ferroelectric
volume of the voltage source considered
as a large capacitor
viscosity coefficient
Ferromagnetism
N Magnani, Università di Parma, Parma, Italy
& 2005, Elsevier Ltd. All Rights Reserved.
Introduction
As far as isolated atoms are concerned, only two
different magnetic behaviors exist: diamagnetism
and paramagnetism. The latter property is only
present if the considered atoms possess a nonzero
permanent magnetic dipole, and is responsible for
the weak attractive force experienced by a
paramagnetic substance in an applied magnetic field.
However, the most interesting magnetic properties of
condensed matter arise when the interaction between
the individual elementary moments of each atom is
considered. In particular, ferromagnetism concerns
the situation where the moments have the tendency
to align, one parallel to another, even in the absence
of any external magnetic field, thus leading to the
possibility of obtaining very large values of the
magnetization with very little values of the applied
magnetic field (spontaneous magnetization). Only a
few elements in the periodic table (Fe, Co, Ni, Gd,
Dy, and Tb) order ferromagnetically, but an extremely large number of compounds and alloys do.
From the practical point of view, apart from their
high values of the magnetic susceptibility and relative
permeability, one of the most interesting properties
of ferromagnetic compounds is the possibility of retaining a large amount of the induced magnetization
when the external field is completely removed. This
makes it possible to develop devices such as permanent magnets (i.e., magnetized bodies which produce
a constant magnetic field in a given volume of space
without the need to continuously supply electrical or
chemical energy), memory storage devices, magnetic
circuits, etc.
The Molecular Field Approach to
Ferromagnetism
At the beginning of the twentieth century, Pierre
Weiss interpreted the tendency of ferromagnets to
spontaneous magnetization by introducing a ‘‘molecular field’’ responsible for the ordering of the
elementary magnetic moments, and he assumed
that this field is linearly proportional to the bulk
202 Ferromagnetism
where g is the Landé factor and
2J þ 1
2J þ 1
1
x
coth
x coth
BJ ðxÞ ¼
2J
2J
2J
2J
½2
½4
with
gmB JlM
;
kB T
t¼
kB T
nlðgmB JÞ2
t < tC
1
Spontaneous
magnetization
0.5
y = BJ (x)
y = tx
1
2
3
x
In order to investigate the possibility of spontaneous
magnetization, the H ¼ 0 case is considered; eqn [3]
can be rewritten as
x¼
t = tC
t > tC
0
is the Brillouin function; the magnetization of a ferromagnet can then be obtained by substituting in
eqn [1] the applied magnetic field H with H þ lM,
leading to the implicit equation
M
gmB J
gmB J
¼ BJ
Hþ
lM
½3
ngmB J
kB T
kB T
m ¼ tx ¼ BJ ðxÞ
1.5
y
magnetization (total magnetic moment per volume
unit). It may be recalled that the magnetization of a
set of identical noninteracting ions, each having total
angular momentum J, in a magnetic field H can be
expressed as
gmB JH
M ¼ ngmB JBJ
½1
kB T
½5
A graphical solution of eqn [4] is given in Figure 1;
if tXtC ¼ ðJ þ 1Þ=3J; only the trivial solution x ¼ 0
is present, while otherwise there is also a possible
solution with xa0. This means that there exists a
critical temperature TC (Curie temperature), below
which a ferromagnetic compound can have a nonzero magnetization in the absence of an applied
magnetic field. The Curie temperature and saturation magnetization for several ferromagnetic elements are listed in Table 1. It is worth noting that
eqn [4] can be expressed in terms of J, m ¼
MðTÞ=Mð0Þ and t ¼ ðJ þ 1Þ=ð3JÞ T=TC only. In the
classical limit (J-N), BJ ðxÞ is substituted by the
Langevin function coth x 1=x, and the function
mðtÞ is the same for all ferromagnetic materials; the
latter statement is often referred to as the law of
corresponding states.
Figure 2 shows the temperature dependence of
the magnetization measured for nickel (Ni), together
with the curve derived from eqn [4] considering a
pure-spin moment J ¼ S ¼ 1=2. The overall agreement is fairly good, and can be further improved
Figure 1 Graphical solution of eqn [4] (see text for details).
Table 1 Curie temperature, saturation magnetization at roomtemperature, and magnetic moment per formula unit (at 0 K) of
ferromagnetic elements
Element
TC ðKÞ
MS (106 A m 1)
mB /formula unit
Cobalt
Iron
Nickel
Gadolinium
Terbium
Dysprosium
1388
1043
627
293
221
85
1.42
1.71
0.48
1.72
2.22
0.61
7.10
9.34
10.0
Data from: (1) Kittel C (1962) Introduction to Solid State
Physics, 7th edn. London: Wiley; (2) Jiles D (1991) Introduction
to Magnetism and Magnetic Materials. London: Chapman and
Hall; (3) Elliott JF, Legvold S, and Spedding FH (1954) Some
magnetic properties of dy metal. Physical Review 94: 1143; (4)
Hegland DE, Legvold S, and Spedding FH (1963) Magnetization
and electrical resistivity of terbium single crystals. Physical
Review 131: 158; (5) Heller P (1967) Experimental investigations of critical phenomena. Reports on Progress in Physics
30: 731.
1
0.8
m 0.6
0.4
Experimental
Calculations
0.2
0
0.2
0.4
0.6
0.8
1
t
Figure 2 Experimental (diamonds) and calculated (lines)
value of m ¼ MðT Þ=Mð0Þ vs. t ¼ T =TC for nickel (J ¼ S ¼
1=2). Experimental data are taken from Weiss P and Forrer R
(1926) Aimantation et phenomene magnetocalorique du nickel.
Ann. Phys. 5: 153. The full black line is calculated by means
of eqn [4]; dashed lines take into account the spin-wave
excitations at low T and the critical behavior near the Curie
point.
Ferromagnetism 203
by the use of more accurate models (the lowtemperature decrease due to spin waves and the
critical behavior near TC are shown as dashed lines
for comparison).
Above TC , the sample is in a paramagnetic phase
(disordered moments), and its magnetic susceptibility
can be calculated by noting that BJ ðxÞCðJ þ 1Þ
x=ð3JÞ for small values of x, leading to
w
M
C
¼
H T lC
with their average values:
"
Hi ¼ Si X
#
Jij /Sj S þ 2mB H
½9
j
The case now considered is that of a ferromagnetic body, with all the relevant exchange constants
Jij ¼ J40 for simplicity. Since
½6
M ¼ 2mB
X
/Sl S
½10
l
where C can be identified with the paramagnetic
Curie constant (w ¼ C=T if no ferromagnetic order
exists, i.e., l ¼ 0) and TC ¼ lC. This very simple
model gives a fairly accurate description of the experimental magnetic susceptibilities of ferromagnetic
compounds at high temperatures.
It must be noted that lM can be as large as several
times the maximum magnetic field which can be
produced in a standard laboratory, and is about four
orders of magnitude larger than the dipolar interaction due to the other magnetic ions in the crystal. The
physical origin of this huge molecular field was not
clear until the development of quantum mechanics
made it possible to point out the role of the exchange
interaction (a purely quantum effect and a direct
consequence of Pauli’s principle) in determining the
magnetic spin alignment. The exchange interactions
between a collection of N spins can be described by
the Heisenberg–Dirac Hamiltonian, which has the
form
HH2D ¼ X
Jij Si Sj
½7
i;j
where the sign of the exchange constants Jij depends on whether the two spins labeled i and j
are coupled ferromagnetically (positive) or antiferromagnetically (negative), and the sum usually involves
only the z nearest neighbors since the interaction
strength drops very rapidly as the distance increases.
In the presence of an applied magnetic field H, the
Zeeman term
HZ ¼ 2mB
X
H Si
½8
i
must also be considered (a gyromagnetic factor of
2 was taken, in the hypothesis that pure spin
moments are being dealt with). The molecular field
approximation consists in reducing the N-body
Hamiltonian HH–D þ HZ to N one-body Hamiltonians Hi, by replacing all the spin operators except Si
is the total magnetization of the sample, eqn [9] can
be rewritten as
Hi ¼ 2mB Si ½H þ lM
½11
with
l¼
zJ
4m2B N
½12
Equation [11] immediately shows that the mean-field
analysis based on the Heisenberg–Dirac exchange
Hamiltonian provides an a posteriori justification to
Weiss’ molecular-field hypothesis; as one might expect, the accuracy of this approximation is higher
when z and J are larger, since in this case the role of
the fluctuations Si /Si S is practically negligible
and the substitutions made in eqns [7] and [8] are
well-grounded.
One word of caution is required before the end
of this section. The existing theories of magnetism may roughly be divided into two groups, dealing
respectively with localized and itinerant magnetic moments. The Heisenberg approach which was
mentioned before belongs to the former; to give a
meaning to eqn [7], it must be explicitly assumed
that each ion in the crystal carries on it a well-defined
magnetic moment Si due to the electrons which are
bound to it. Although this approach is very useful
when dealing with certain substances and alloys
and for studying critical magnetic behaviors, its
validity for metals is questionable due to the presence of conduction electrons, not tied to a specific
ion but belonging to the whole crystal. On the
other hand, band models considering the role of
itinerant electrons have managed to solve several
flaws of localized theories, such as the noninteger
values of the magnetic moment per atom experimentally measured in most ferromagnets (Table 1), which
could not be explained within a localized-moment
framework.
204 Ferromagnetism
Magnetostatic Energy and
Demagnetizing Field
One may consider a sample of a magnetic substance,
composed of n magnetic atoms per unit volume, each
carrying an elementary moment l, and imagine applying a magnetic field which is strong enough to
align all these dipoles along the same directions. As
a result of this process, the sample will display a
saturation magnetization
MS ¼ njlj
N
½13
(it may be recalled that the magnetization is defined
as the total magnetic moment per volume unit).
The energy required to obtain this configuration is
Z
½14
E ¼ m0 H dM
where H indicates the magnetic field and M is the
magnetization vector. In the absence of dissipative
processes, an equivalent amount of magnetic potential energy is stored within the sample.
In turn, any magnetized body produces a magnetic
field in its surrounding space, as one can immediately witness by means of a compass needle. In close
analogy with the charge polarization of a dielectric
material, one can describe this field as being generated by ‘‘free’’ magnetic poles (i.e., positive poles
not neutralized by the presence of a negative pole in
their immediate neighborhood and vice versa) on the
surface; in addition to this, the presence of a magnetic field generated by an external source is usually
considered. Outside the sample, the magnetic induction B and the magnetic field H are always proportional (as B ¼ m0 ½H þ M and M ¼ 0) and their flux
lines are oriented from the north (N) to the south (S)
poles. On the other hand, the total magnetic field
H int inside the sample is found by summing the external field, which is parallel to the magnetization,
and the contribution from the free poles, which is
antiparallel to it (in fact, the flux lines of H always
begin on N poles and end on S poles except when the
magnetic field is due to electric currents, in which
case they are closed and continuous). The free-poles
contribution is also called demagnetizing field, and
can be represented as
H d ¼ N d M
H
½15
where N d is, in general, a second-order tensor whose
components depend on the geometry of the studied
system; in some particular cases and/or for special
directions of the magnetic field, M and H are collinear, and one can write a simpler relation where Nd
S
Figure 3 Magnetic field generated inside and outside a uniformly magnetized sphere. The magnetization vector M is directed from the S to the N pole; the resulting demagnetizing field
(inside the sphere) is antiparallel to M.
is a dimensionless number called demagnetizing factor. For example, a homogeneous spherical sample
(Figure 3) has Nd ¼ 4p=3, hence
Hd ¼ 4p
M
3
½16
Demagnetizing factors for samples of various
geometries and for different field directions can be
found in specialized textbooks.
Magnetic Anisotropy
Another contribution to the energy balance, that is,
magnetic anisotropy, will be dealt with now. While
the meaning of such a locution is quite simple (the
magnetic behavior of a material depends on the
direction), it is not easy to understand the physical
reasons behind its very existence. In fact, magnetic
anisotropy may be produced by several different
mechanisms, which include (but are not limited to)
magnetocrystalline (or crystal-field) anisotropy,
shape anisotropy, stress anisotropy, annealing in a
magnetic field, etc. For the sake of simplicity, the
considerations made in the following will be referred
to a uniformly magnetized single-crystal sample.
From a phenomenological point of view, the effect
of magnetic anisotropy may be taken into account by
adding to the energy balance an extra term EA (anisotropy energy), which depends on the relative orientation of the magnetization vector with respect to
the crystallographic axes. As shown by the Russian
Ferromagnetism 205
physicist Akulov, the free anisotropy energy per unit
volume can always be expressed by an infinite series
!
þN X
3 X
3
3
X
X
EA ¼
y
ci1 ;i2 ;y;in ai1 ai2 yain
n¼1
¼
3
X
i1 ¼1 i2 ¼1
c i ai þ
i¼1
þ
in ¼1
3 X
3
X
ci;j ai aj
ci;j;k ai aj ak
ci;j;k;l ai aj ak al þ ?
½17
i¼1 j¼1 k¼1 l¼1
where a1 , a2 , and a3 are the direction cosines of the
magnetization vector with respect to the Cartesian
axes x, y, and z, and several coefficients may equal
zero for symmetry reasons; for example, all the terms
of the sum with odd n vanish if eqn [17] is invariant
under inversion of the magnetization. In most
relevant cases, eqn [17] can be rewritten as
þN
X
Kn An ða1 ; a2 ; a3 Þ
½18
n¼1
for example, a ferromagnetic crystal with cubic lattice symmetry has A0 ða1 ; a2 ; a3 Þ ¼ 1; A1 ða1 ; a2 ; a3 Þ ¼
a21 a22 þ a21 a23 þ a22 a23 ; A2 ða1 ; a2 ; a3 Þ ¼ a21 a22 a23 ; etc. The
coefficients Kn , named anisotropy constants, can be
determined by magnetization measurements; a knowledge of a reliable set of anisotropy constants for a
given sample can lead to a direct phenomenological
interpretation of the magnetic processes and is useful
to make comparisons between different systems.
For lattices with uniaxial symmetry (e.g., cylindrical, hexagonal, and tetragonal), compact expressions
for the anisotropy energy can be derived as a function of the polar and azimuthal angles y and j, which
define, respectively, the angle between the magnetization vector M and the z-axis (coincident with the
symmetry axis), and the angle between the projection
of M on the xy-plane and the x-axis. In this case,
EA ¼
þN
X
n¼0
½21
þN
X
Kn sin2n y
½22
n¼0
3 X
3 X
3 X
3
X
EA ¼
þ K3 sin6 y þ K03 sin6 y cos 4j þ ?
EA ¼
i¼1 j¼1 k¼1
þ
EA ¼ K1 sin2 y þ K2 sin4 y þ K02 sin4 y cos 4j
(m ¼ 4) for a tetragonal system. For a perfectly
cylindrical system (m-N), the anisotropy on the
xy-plane vanishes and one is left with
i¼1 j¼1
3 X
3 X
3
X
(m ¼ 6) for a hexagonal lattice and
sin2n y
X
KðlÞ
n cosðlmjÞ
½19
0plon=m
As for the cubic phase, the K0 term is an isotropic
energy shift, and will therefore be dropped in the
following. The value of m depends on the particular
symmetry, leading to
EA ¼ K1 sin2 y þ K2 sin4 y þ K3 sin6 y
þ K03 sin6 y cos 6j þ ?
½20
If no applied field is present, the sample magnetization will have the tendency to stay along a direction
where the anisotropy energy is lowest; this is called
in brief ‘‘easy magnetization direction’’ (EMD). This
feature will be discussed in detail later, but it may be
immediately inferred that the larger the anisotropy,
the larger the magnetic field required to rotate the
magnetization direction. Therefore, anisotropy is a
key property to take into account when designing or
choosing suitable magnetic compounds for particular
applications.
In the following section, the most important physical mechanisms giving rise to magnetic anisotropy
are discussed in brief.
Magnetocrystalline Anisotropy
Exchange interaction in a ferromagnet can be seen
as a spin–spin coupling which aligns the spin moments along the same direction. On the other hand,
the orbital moments of any magnetic ion in the lattice preferably align along given crystallographic directions due to the crystal-field potential, which
reflects the local crystal symmetry. Magnetocrystalline anisotropy is generated by the spin–orbit interaction, which favors mutual alignment between
the spin and orbital moments; if this coupling is relatively strong, as for rare-earths and actinides,
moment directions which are farther from the crystallographic easy-axis will have a higher cost in
energy. Although several types of anisotropy exist,
only magnetocrystalline anisotropy is intrinsic to the
material.
In terms of anisotropy constants, subsequent terms
of eqn [18] will contain increasing powers of the
ratio of the spin–orbit energy to that of the crystal
field. As the spin–orbit interaction is weak for 3d
metals, the series is generally truncated after the
second or fourth order, and only terms in K1 and K2
are included. In principle, this is not possible for 4f
elements; however, in the case of intermetallic rareearth–transition-metal alloys, approximate expressions can be found as linear combinations of
generalized Brillouin functions, with appropriate
Anisotropy constant, Kn
206 Ferromagnetism
Plateau
Toward TC
0
Temperature
Figure 4 Typical temperature behavior of anisotropy constants
(arbitrary units are used).
coefficients related to the decomposition in terms of
spherical harmonics of the crystal-field potential at
the rare-earth site. On these grounds, it is usually
possible to consider Kn ¼ 0 for nX4. The typical
temperature dependence of anisotropy constants is
shown in Figure 4. On changing the temperature, the
anisotropy constants of different order can have different relative variations; as a result of this, the EMD
of a sample may spontaneously change, giving rise to
a spin reorientation transition (SRT) induced by
temperature even in the absence of a magnetic field.
Shape Anisotropy
The free poles on the surface of a perfectly spherical
and homogeneous sample give rise to a demagnetizing field which is collinear and linearly proportional to the magnetization vector, as described by eqn
[16]. In more general terms, however, one must refer
to eqn [15] to find the correct value and direction of
the demagnetizing field corresponding to a given
magnetization vector. The relatively simple case of an
elongated rotation ellipsoid with two axes of equal
length is considered; for symmetry reasons, one has
0
N>
B
Nd ¼ @ 0
0
0
0
0
N>
B
¼@0
0
N>
0
0
0
0
N> þ DN
0
which corresponds to eqn [22], the only nonzero anisotropy constants being K0 ¼ m0 M2 N8 =2 and
K1 ¼ m0 M2 DN=2. The extrinsic character of shape
anisotropy can be immediately noticed, as the calculated K1 strongly depends on the shape and bulk
magnetization of the sample. While shape anisotropy
is usually smaller than crystal-field anisotropy for
ferromagnetic rare-earth elements and alloys, they
may be of the same order of magnitude for transition
metals such as iron and cobalt.
Stress Anisotropy
Applying mechanical stress treatments such as lamination, rolling, or extrusion to a magnetic sample
can lead to significant crystal deformations. Stress
anisotropy may then be generated due to the
magnetoelastic coupling between the lattice and the
magnetic moments present in the system (magnetostriction).
Anisotropy Induced by Magnetic Annealing
Heat treatments performed under the Curie temperature of the sample in the presence of an applied
magnetic field (‘‘magnetic annealing’’) may result in
the generation of uniaxial anisotropy with respect to
the field direction. The induced anisotropy is, in
general, rather small, although larger effects may be
obtained when a structural transition takes place
during the annealing.
1
C
0 A
N8
N>
where z# is the unit vector along the direction of the
elongated axis of the ellipsoid. The associated energy
can be calculated as
Z
DN
2 N>
2
cos y
þ
E ¼ m0 H d dM ¼ m0 M
2
2
N8 DN 2
¼ m0 M2
sin y
½25
2
2
Magnetic Domains
1
C
A
½23
where DN ¼ N8 N> According to eqn [15] and
knowing that all homogeneous bodies delimited by
second-order surfaces magnetize uniformly, one can
write
H d ¼ N d M ¼ N> M DNðM z# Þ#z
½24
As has been mentioned in the introduction, the situation of complete alignment of the elementary
magnetic dipoles present in a ferromagnetic substance is energetically favorable due to the exchange
interaction. However, it is easy to verify that a piece
of iron, taken off the shelves and kept below its Curie
temperature, is basically unmagnetized and does not
produce any magnetic field. Thermal disorder should
obviously not be expected to affect the net magnetization significantly, since the exchange interaction
responsible for the alignment of the moments is
several orders of magnitude larger than the dipole–
dipole interaction in a paramagnet. On the other
Ferromagnetism 207
hand, if one sticks to the molecular-field approach,
one must be aware that the solution M ¼ 0, H ¼ 0
satisfies eqn [3] both above and below TC .
In order to gain some physical insight on this
point, the energy of a magnetized body, under the
effect of an external magnetic field H, is considered:
Z
E ¼ m0 ðH þ H d Þ dM
½26
In the simple case that the magnetization is uniform
all over the sample and that the demagnetizing field
can be expressed as H d ¼ Nd M, eqn [26] becomes
Z
Z
E ¼ m0 H dM þ m0 Nd M dM
¼ m 0 H M þ m 0 Nd
M2
2
(a)
½27
The first term on the right-hand side is the familiar
expression for the Zeeman energy, and can be minimized by aligning the magnetization vector with the
external field and maximizing its modulus. The second term is the self-energy of the magnetized body
due to the free poles, and can be minimized by minimizing the net magnetization. On the other hand,
this term also represents the amount of energy which
is stored in the external magnetic field generated by
the sample; therefore, if the applied field H equals
zero, it can be inferred that in the lowest energy state,
the magnetic field generated by the sample must be as
small as possible. From the microscopic point of
view, the energy balance results from a competition
between exchange coupling (which lowers the total
energy if two spins are parallel) and the dipole–dipole interaction (which raises the total energy if two
spins are parallel). As a result, the sample is divided
in several macroscopic regions, each one having uniform magnetization, called domains. Within a single
domain, the spins are parallel to one another; in turn,
spins belonging to two different domains can have
very different directions (Figure 5) in order to minimize the bulk magnetization.
One may naively expect that the subdivision in
domains occurs as in Figure 6a (i.e., all the spins
belonging to domains A and B are arranged along the
easy-axis direction), and that spins belonging to different neighboring domains are antiparallel to one
another. This configuration certainly minimizes the
anisotropy energy, but at a large cost of exchange
energy for the two antiparallel spins situated at the
interface. In general, it is less expensive to have the
spins arranged as in Figure 6b; the 1801 spin rotation
is performed in several different steps within a region
between domains A and B, known as ‘‘Bloch wall.’’
(b)
(c)
Figure 5 Schematic view of the dipole arrangements in (a) a
paramagnet (or ferromagnet above TC ), (b) a magnetized (saturated) ferromagnet, and (c) an unmagnetized ferromagnet
(divided into domains). While the moments in the paramagnetic
phase are disordered at the atomic level, the domain size is
macroscopic (even several hundred micrometers).
For the sake of clarity, consider the case of a simple
cubic crystal with lattice constant a, with a nearestneighbor exchange constant J and EMD along
[1 0 0]. The energy balance may be estimated by
assuming that the rotation is performed within the
xz-plane in N equal steps, each of an angle p=N; then
the total exchange energy per unit area of the Bloch
208 Ferromagnetism
wall is
The Magnetization Curve
N
X
Sl Slþ1 ¼ J
N
X
l¼0
l¼0
p
N
ðN þ 1Þ
p
cos
a2
N
½28
while, according to eqn [18] with ay ¼ 1, az ¼ cos y,
and ax ¼ cosðp=2 yÞ ¼ sin y, the anisotropy energy
per area unit is
N
X
K1 sin2
l¼0
lp
lp aK1
cos2 ¼
N
N
N
8
½29
(strictly valid for N4k). Eex and EA are opposite in
sign, and the absolute value of both grows with N;
the actual size of the Bloch wall then results from a
competition of the two, being smaller when the anisotropy dominates and larger when the exchange
interaction dominates. In the large-N limit
Eex CJS2 p2 =ðNa2 Þ, and minimizing the total energy
gives an approximate domain wall width of
sffiffiffiffiffiffiffiffi
2J
½30
c ¼ NaC2pS
aK1
typically a few hundred lattice constants for iron.
Dom
ain A
The Hysteresis Cycle
When an external magnetic field is applied to a
ferromagnetic sample in a demagnetized state in
order to bring its resulting magnetization to saturation, one obtains the magnetization curve shown
in Figure 7. Two distinct and coexistent processes
must be considered in order to understand this
(Figure 8): boundary motion and domain rotation.
In the former case, those domains which are favorably oriented with respect to the applied field grow
in size at the expense of the others, without changing their overall magnetization direction; this process
is dominant at low fields. At intermediate fields,
the process of sudden rotation of the magnetization
of unfavorably oriented domains to the easy-axis
direction(s) nearest to that of the applied field also
becomes significant. At a given value of the magnetic
field (labeled HS), saturation is finally achieved
(M ¼ MS , defined in eqn [13]), mainly by coherent
rotation of the domain magnetization toward the
applied field direction. For H4HS, the MðHÞ curve
is flat since the applied field is already strong enough
Dom
"easy" axis
ain B
Dom
ain A
(b)
Bloc
Domain wall motion
M
(a)
Magnetization
saturated
EA ¼ a
From the macroscopic point of view, ferromagnets
may be considered at a first glance as magnetic materials with extremely large susceptibility and permeability. In practice, as both of them are strongly
dependent not only on the temperature, but also on
the applied field and on the past history of the sample, they are not very useful parameters and it is quite
simple to describe the physical properties of a ferromagnet by means of M versus H (or B versus H,
being B ¼ m0 ðH þ MÞ) plots, whose main characteristics will be discussed in the following sections.
Reversible coherent
rotation of domains
¼ JS2
S2 cos
Boundary motion +
irreversible domain : rotation
Eex ¼ J
h wa
ll
Dom
ain B
Figure 6 Schematic view of the dipole arrangements at the
boundary between two domains (Bloch wall).
H1
H2
Hs
H
Figure 7 ‘‘Virgin’’ magnetization curve of a ferromagnet. The
main mechanisms by which the magnetization process advances
for several applied field values are indicated.
Ferromagnetism 209
Easy magnetization
direction
M
MR
Magnetic
field
HC
H
H=0
Figure 9 Hysteresis cycle of a ferromagnet. Remanence (MR )
and coercivity (HC ) are indicated on the graph axes.
H = H1
H = H2
H = Hs
Figure 8 Schematic view of the main mechanisms by which the
magnetization process advances for several applied field values.
to align all the elementary dipoles in the sample
along its direction.
If the value of the applied field is now reduced to
zero, it is noticed that the obtained MðHÞ curve does
not coincide with the previous one apart from a small
region near HS. In particular, the magnetization at
zero applied field does not go back to zero (retentivity), and its value MR is called remanence or
remanent magnetization. In order to force the
magnetization to zero, it is necessary to apply a
magnetic field HC in the opposite direction; this value
is called coercivity or coercive field. (Some authors
make use of a distinction between remanence/coercivity (indicating the sample behavior after it has
reached saturation) and remanent magnetization/coercive field (relative to a magnetization curve reaching an arbitrary value MoMS ).) Raising the
reverse field strength to HS leads to the saturation
of the magnetization in the opposite direction as before. Reversing the field once again, one obtains the
so-called hysteresis loop (Figure 9).
While ‘‘hard’’ magnetic materials (i.e., those displaying a large coercivity) are generally useful as
permanent magnets or for memory-storage purposes
(in order to avoid demagnetization due to stray
magnetic fields), ‘‘soft’’ ones may be good candidates
for the realization of power transformers or electromagnets. In the former case, apart from retentivity,
coercivity, and Curie temperature, another figure of
merit which is often used in practice is the maximum
energy product BHmax , which is the maximum value
of the product B H in the demagnetizing quadrant
(M40, Ho0) of the hysteresis curve. This corresponds to the energy stored in the considered material within a magnetic circuit operating at an
optimized workpoint, and should not be confused
with the energy lost due to the irreversible processes
during one hysteresis loop, that is,
I
B dH
½31
Magnetization Rotation and Anisotropy
Most of the energy loss during the hysteresis cycle
occurs at low fields as a result of irreversible domain
boundary motion, due to inhomogeneous microstrains (dislocations) which obstructs the rotation of
the magnetic moments. On the other hand, processes
involving domain rotation are governed by a competition between the anisotropy and the Zeeman
energy EZ ¼ m0 H M: To study the role of anisotropy in determining the shape of the magnetization
curve, assume that the boundary motion of domain
walls can be achieved with a negligibly small magnetic field. In the case of a uniaxial sample, MðHÞ can
be found by minimizing
X
EA þ EZ ¼
Kn sin2n y m0 H M
½32
n
210 Film Growth and Epitaxy: Methods
Let y be the angle between the magnetization vector
and the c-axis, which will be considered also as the
EMD, for simplicity. If H is applied along the c-axis,
both EA and EZ are minimized with M8H and saturation is reached immediately when the external applied field is equal to the demagnetizing field. If H is
applied perpendicularly to the easy-axis, differentiating eqn [32] and knowing that M ¼ MS sin y leads to
HMS ¼
X
n
2nKn
M
MS
2n1
½33
an implicit expression for the magnetization curve. By
putting M ¼ MS , one can immediately find
HS ¼
and
2 X
nKn
MS n
M2S
dM
¼
dH H¼0 2K1
½34
½35
Similar calculations can be made for any other
orientations of the field and of the EMD, leading to
the remarkable result that, in principle, complete saturation cannot be achieved if H is not parallel to an
‘‘extremal’’ direction (i.e., one which minimizes or
maximizes the anisotropy energy).
See also: Diamagnetism; Magnetic Domains; Magnetic
Interactions; Magnetic Materials and Applications; Paramagnetism.
PACS: 75.50. y
Further Reading
Ashcroft NW and Mermin ND (1976) Solid State Physics. Philadelphia: Saunders College.
Bozorth RM (1951) Ferromagnetism. Princeton, NJ: Van Nostrand.
Brailsford F (1966) Physical Principles of Magnetism. London:
Van Nostrand.
Cullity BD (1972) Introduction to Magnetic Materials. Reading,
MA: Addison-Wesley.
Hubert A and Schaefer R (1998) Magnetic Domains. Berlin:
Springer.
Mattis DC (1965) The Theory of Magnetism. New York: Harper
and Row.
Nomenclature
B
H
M
T
m
magnetic induction (T)
magnetic field (A m 1)
magnetization (A m 1)
temperature (K)
magnetic moment (mB )
Film Growth and Epitaxy: Methods
F Lévy, Ecole Polytechnique Fédérale de Lausanne,
Lausanne, Switzerland
& 2005, Elsevier Ltd. All Rights Reserved.
Introduction
The growth of thin films is achieved through many
methods resulting from the combination of various
steps and devices as:
source material and its handling,
transport from the source to the substrate,
* growth substrate and its conditioning, and
* suitable environment in a reactor chamber.
*
*
The methods of deposition and growth can be
classified according to the source and transport. In
the most frequent cases where the material is transported through the vapor phase, two main groups are
distinguished: physical vapor deposition (PVD) and
chemical vapor deposition (CVD). Besides vapor
transport, films can grow from liquid or solid phases.
The key process parameters determine the source,
transport, and growth steps as well as the properties
of the deposited films. The substrate temperature is
the most important parameter. Depending on the
method, the pressure, the energy delivered to the
growing film (applied power density, electrical bias,
bombardment, etc.), the chemistry and the environment considerably influence the growth and properties of the films. The functionality of the film
meets the requirements of surface engineering or of
sophisticated physical and chemical exigencies in
microelectronics and photonics, for example. It leads
to the distinction between normal polycrystalline
thin films and epitaxial single crystalline films.
The growth of epitaxial thin films by PVD and
CVD needs special techniques and equipment. These
methods are therefore described in separate sections.
The thicknesses of the functional thin films and coatings grown by thin-film techniques, range from a few