Magnetic Relaxation in Nanoscale
Granular Alloys
Daniel Hans Ucko
UCL
University College London
A thesis submitted in accordance with the requirements o f the University of London for the degree o f Doctor of Philosophy
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Abstract
Magnetic nanoscale granular alloys are o f current interest from both a
fundamental and a technological viewpoint. At a fundamental level,
nanocrystalline magnetic materials exhibit fme-particle magnetic behaviour that is
significantly different from the bulk behaviour. At a practical level, they are used
as nanocrystalline soft magnetic alloys for electromagnet cores and transformers,
and for magnetoresistive sensors and devices.
Primarily to address the fundamental issues, a systematic series of
magnetic relaxation measurements were performed on a suite o f nanostructured
iron-silver and iron-copper-silver alloys. Several experimental techniques (DC
Magnetisation, Vibrating Sample Magnetometry, AC Susceptibility, Mossbauer
Spectroscopy and Muon Spin Relaxation) were applied in order to observe
superparamagnetic relaxation at a wide range of measurement timescales.
Structural tests by X-ray Diffractometry and Rietveld analysis showed that the
samples had grain sizes o f 4 - 20 nm, and dispersion fractions o f magnetic
material o f 5 - 40 % o f the whole sample volume.
Superparamagnetic relaxation transitions or blocking temperatures were
observed to be a function o f measurement time For a low concentration
sample, FenCuogAgyg, the blocking temperatures, after correction to compensate
for the particle size distribution and interaction effects, were found to vary from
28 K to 69 K over the measurement time range 100 s to 0.1 ms. For more
concentrated samples, e.g. Fes2Ag4 8, blocking temperatures at all timescales were
higher than room temperature.
From the relaxation behaviour, it was concluded that the magnetic grains
experience strong inter-particle interactions that raise the effective energy barrier
against spontaneous magnetic relaxation. These were shown to be mainly dipolar
Acknov^gements
The fact that this thesis is written at all is due to the help and efforts of many
people around me. I would like to take this opportunity to thank them for their
support.
Firstly, I must thank my supervisor Quentin Pankhurst for his support and
scientific guidance throughout my project. I am also very grateful to Luis
Fernandez Barquin for his outstanding scientific help both here in the UK and
during the time I have been fortunate enough to spend visiting the Universidad de
Cantabria, Santander, Spain.
I would also like to thank Mark Ellerby for his technical and scientific
help, always making me challenge my assumptions. I owe thanks to Andrew Steer
and Rebeca Garcia Calderon for helping me in obtaining some of the X-ray and
magnetic results presented in this thesis. Outside CMMP, I owe thanks to James
Lord, Marianne Odlyha and Kevin Reeves and for help with my pSR, DSC and
ED AX experiments respectively.
During my first year at UCL I was much helped by Quang Bui, Neil
Cohen, and Glyn Forster in finding my feet. I am also grateful to Louise Affleck
and Andrew Steer, my magnetism colleagues, who have provided me with a
support network and a scientific exchange of ideas during my PhD. Furthermore I
am grateful to Shusaku Hayama and Jonathan Wasse for experimental help and
support.
Outside o f the scientific dimension, this PhD would have been a much less
interesting experience had I not been in such a lively and vibrant group as CMMP.
I am also particularly grateful for the company and support I received fi*om fiiends
like Jon Allen, Louise Affleck, Simon Armitage, Eamonn Beime and Shu
Hayama.
Finally, as the dedication in the fi*ont o f the thesis indicates, I am grateful
most o f all to my parents, without whom (literally) none o f this would have been
Chapter 1 : Introduction
Chapter 2 : Magnetic Relaxation and Superparamagnetism_______________ 11
1. The domain theory of ferromagnetism_______________________________ 11
2. The Langevin theory of paramagnetism______________________________ 11
3. Weiss theory of ferromagnetism_____________________________________15
4. Ferromagnetic domains___________________________________________ 16
5. Single-domain particles___________________________________________ 17
6. Cubic anisotropy________________________________________________ 18
7. The energetics of single-domain systems______________________________ 19
8. Theory of superparamagnetism_____________________________________21
9. Blocking Temperature____________________________________________ 24
10. Real superparamagnetic systems___________________________________26
11. Relaxation time calculation_______________________________________ 26
12. Interparticle interactions________________________________________ 27
13. Conclusions____________________________________________________ 31
Chapter 3 : Sample preparation and experimental details__________________33
1. Sample preparation______________________________________________ 33
2. Magnetic measurement techniques__________________________________36
3. Structural measurement techniques_________________________________45
4. Conclusions_____________________________________________________ 48
Chapter 4 : Structural analysis and resu lts_____________________________ 49
1. FeA g__________________________________________________________ 53
2. FeCuAg________________________________________________________ 60
3. Alloying of FeCu with milling tim e__________________________________67
4. FeAg and FeCu annealing_________________________________________ 69
Chapter 5 : Magnetic measurements,___________________________________78
1. FeA g__________________________________________________________ 79
2. FeCuAg________________________________________________________ 88
3. Conclusion____________________________________________________ 104
Chapter 6 : Muon Spin Relaxation (juSR)_____________________________ 105
1. Experimental methods____________________________________________106
2. jiSR and Superparamagnetism_____________________________________ 110
3. Zero-field Results_______________________________________________ 113
Chapter 7 : Discussion and conclusions_______________________________ 140
1. Inter granular interactions and blocking temperatures_________________ 141
2. Blocking temperatures in real superparamagnetic systems_____________ 143
3. Discussion_____________________________________________________ 144
4. Summary and conclusion_________________________________________150
Bibliography_______________________________________________________152
List o f figures______________________________________________________159
Chapt^ 1 : IntroducBon
The aim o f this project was to investigate magnetic relaxation effects in nanoscale
granular alloys. Nanoscale magnetic/non-magnetic composite materials have, over
the last decade, emerged as an important technological area o f interest, in both the
development o f nanocrystalline soft magnetic alloys such as alloys o f one or more
non-magnetic elements with Fe-B, and magnetoresistive alloys like Fe-Cu and Fe
Ag [Wang et al., 1994; Peng et al, 1994], and also ternary Fe-Co-Ag and
Fe-Cu-Ag [Nash et al., 1998; Pierre et al., 1995; Cohen et al., 1999].
The reasons for this study are twofold; the problem is a scientific
intellectual exercise as well as containing many opportunities for practical
benefits fi’om the point o f view o f the materials studied. While having its basis in
a practical problem, this is primarily a study o f relaxation behaviour in
unconventional samples. This project stems fi'om earlier work showing that
nanoscale granular alloys exhibit giant magnetoresistive properties as well as
superparamagnetism [Peng et al., 1994; Cohen, 1998].
The problem of single-domain particles was a significant scientific
challenge in the 1930s and 40s. The possibility of isolated particles o f uniform
magnetisation were first suggested by Frenkel and Dorfinan in 1930, but it was
only after Landau and Lifshitz published their domain theory in 1935 that these
two theories were compared [Brown, 1969]. The definitive theory o f single
domain particles was further developed in the 1940s [Kittel, 1946, 1949; Stoner
and Wohlfarth, 1948; Kittel, 1949]. This led to the discovery by Néel in 1949 of
the spontaneous, thermally activated change in the direction o f the magnetic
moment o f single-domain magnetic particles.
The subject o f single-domain particles and their relaxational behaviour has
continued to get much attention since then [Kittel 1946, 1949; Brown, 1959, 1963,
1969, 1979; Dormann, 1981] and magnetic relaxation in fme-particle magnetic
systems is still the subject o f much study today [Khanna and Linderoth, 1991;
Hanson et al., 1995; loune et al., 1993; Dormann et al., 1997]. In particular these
studies have concentrated on model systems, in which the physics o f fine-particle
idealised systems of, for example, y-Fe2 0 3 particles in an insulating matrix, with
the particles dispersed enough for one to be able to ignore effects due to inter
particle interactions [Dormann et al., 1997; Hansen et al., 2000]. Studies have also
been made on a-Fe particles suspended in carbon [Bodker et ah, 1998], suspended
in a zeolite matrix [Lazaro et al., 1996] and in Fe-substituted cement [Le Callonec
et al., 1997]. The question of interparticle interactions has received attention in a
well-defined system o f increased volume fractions o f magnetic particles, which
has applications for instance in the field o f magnetic recording media [Inoue et ah,
1993; Morup and Tronc, 1994; Pedersen et ah, 1997; Dormann et ah, various,
1999; Hansen et ah, 2000].
In view o f this, it was thought interesting to consider the relaxation
behaviour o f magnetic particles in a metallic matrix, rather than an insulating host,
in relation to how interparticle interactions would manifest themselves in such a
system. Previous work has been done on the relaxation behaviour in Fe-M
materials, where M is a metallic non-magnetic host [Peng et ah, 1994; Dimitrov et
ah, 1995; Pedersen et ah, 1997]. After this project had started, other research was
focused on this problem [Bewley et ah; 1998, Jackson et ah, 2000; Lopez et ah,
1998], but the systems studied still possessed notable differences from the
materials studied in this thesis. The systems covered in this thesis have a higher
volume fraction than any o f the three systems mentioned above, and the ones that
show superparamagnetic relaxation most readily are furthermore ternary alloys
rather than binary copper-cobalt and iron-silver.
Another reason for the choice o f sample class was the desire to probe
superparamagnetic relaxation with Muon Spin Relaxation (pSR). pSR has been
used to study relaxation phenomena such as spin glasses before [Uemura et al.,
1980, 1985, Wu et al., 1993], but at this project's start no pSR study had been
made o f a Fe-M system. As this project progressed other studies of Cu-Co
[Bewley et al., 1998] and Fe-Ag [Jackson et al., 2000] were published, for much
more dilute systems than the ones studied here.
A notable feature o f the iron-silver and iron-copper-silver systems are the
immiscibility o f the constituents [Hansen et al., 1958]. It was known that, in the
single-domain particles in a granular silver matrix [Cohen, 1998]. Samples were
made using high-energy ball milling to obtain metastable alloying; a convenient
method in that the sample preparation can be done in a laboratory setting and
yields a powder, which is a versatile form for scientific characterisation. The
technique o f mechanical alloying has become an important technological tool in
materials physics [Maurice et al., 1990]. With the development of high-energy
ball milling, a higher impact energy became available, and samples could be made
in a matter o f days rather than weeks. Also, with the higher impact energy comes
the ability to break down samples to even smaller grain sizes, even creating
amorphous systems given the right starting elements [Schultz et al., 1994].
The samples were found to be below the threshold size (33 nm) for single
domain particles to be energetically feasible [Brown, 1969]. Since
superparamagnetic relaxation is a frequency-dependent effect, a proper study of
the relaxation behaviour of a superparamagnet can only be done by using
techniques with a selection o f measurement timescales. This project attempts to
combine a study o f superparamagnetism by Muon Spin Relaxation with data
obtained by the more conventional techniques Mossbauer Spectroscopy, AC
Susceptibility and DC Magnetisation. Analysis o f the transition temperatures, or
'blocking temperatures,' for different measurement fi-equencies by plotting the
inverse temperature against the natural logarithm of the measurement frequency
allows one to obtain the energy barrier against magnetic reversal and the
characteristic time parameter.
For a system with non-negligible interactions, the Néel-Arrhenius model
[Néel, 1949] is usually modified to a phenomenological Vogel-Fulcher model
[Shtrikman et al, 1983], but lately more involved analysis o f the effects of
interparticle interactions have been made [Dormann, Fiorani and Tronc, 1999;
Hansen et al, 2000]. Comprehensive studies of this kind are more usually done on
model systems [Dormann et al., 1997] but recently more concentrated systems
have been given this analysis in an attempt to understand the effects o f
interparticle interactions [Dormann, Fiorani et al., 1999; Spinu et al., 1999; Duan
et al., 2000]. The work presented in this thesis is a contribution to this continuing
This thesis contains seven chapters. In chapter 2 we shall consider the
underlying theory of superparamagnetism, starting with the concepts of
paramagnetism and ferromagnetism, and then considering the domain theory. We
shall consider the single-domain particle, and the relaxation o f this particle. After
this, we shall consider an assembly o f relaxing single-domain particles, and the
nature and effects o f interparticle interactions in this system.
High-energy ball milling is a good way to make metastable alloys of
otherwise immiscible constituents. Chapter 3 contains descriptions of the
experimental methods and procedures used in this project, for samples preparation
and analysis o f data. The data obtained by the methods described in this chapter
are described in chapter 4, 5 and 6.
The samples’ physical properties were investigated by analysing X-ray
diffraction patterns by Rietveld profile matching [Rodriguez Caijaval, 1993] as
well as by fitting hysteresis loops to a Langevin function convoluted with a log
normal particle size distribution. Chapter 4 contains structural data and analysis,
demonstrating the physical make-up of the system as well as the microstructural
effects o f annealing and time-dependent ball milling.
The magnetic relaxation properties o f the samples were evaluated using
DC Magnetisation, AC Susceptibility, Vibrating Sample Magnetometry and
Mossbauer Spectroscopy. Chapter 5 contains the data obtained from these
techniques. This chapter also contains a comprehensive record of all blocking
transitions seen in this system.
Chapter 6 is dedicated to the Muon Spin Relaxation experiments
performed during this project. The muon data are considered from both a
phenomenological and an analytical viewpoint.
Chapter 7 contains the discussion and conclusion section o f the thesis. The
blocking temperatures obtained are collated and compared in an effort to obtain
parameters o f the relaxation behaviour. The interactions of the system are
discussed and quantified, and possibilities o f future work arising from the results
Chapter 2 : Magnetic Relaxation and Superparamagnetism
In this chapter the physics o f ferromagnetism and paramagnetism, ferromagnetic
domains, the criteria for single-domain particles to be energetically favourable,
and superparamagnetism will be discussed. The physical behaviour o f a single
domain system in its blocked state, the effects of interparticle interactions and
further complications, (for instance when one is dealing with a sample with a
distribution o f magnetic grain sizes and hence a distribution o f cluster moments,)
will also be covered.
1. The domain theory o f ferromagnetism
To explain the difference between ferromagnetism and paramagnetism, it is
necessary to consider them on a microscopic scale. It is known macroscopically
that the permeability // (where B = jujuqH) of ferromagnets is much higher than
that o f paramagnets. In 1893, Ewing suggested that the difference between
magnetised and demagnetised ferromagnets is that in the demagnetised state, the
magnetic moments are randomly aligned. This, however, does not tell the full
story. Instead, Weiss correctly hypothesised in his domain theory o f 1907 that
magnetic moments were in permanent existence and that they were arranged in
regions called ‘domains,’ in which they were aligned. A typical domain contains
lO’^ to 10^* atomic moments [files, 1991]. The direction o f alignment varies from
domain to domain, and is preferentially aligned along certain crystallographic
easy magnetic axes. So, why are the moments inside a domain aligned and
ordered? This is explained by Weiss’ mean field theory, which was developed
from the Langevin theory of paramagnetism. We shall discuss this in detail, since
it is particularly relevant to superparamagnetism.
2. The Langevin theory o f paramagnetism
In materials with unpaired electrons, there is a net permanent magnetic moment.
The energy o f such a moment can be expressed as
where m is the vector sum o f the electron spin and electron orbital components
[files, 1991], and H is the applied field. In a paramagnetic system o f moments,
their orientations tend to he thermally randomised. Classical Boltzmann statistics
can be used to express the probability o f any electron occupying an energy state
E. If kTis the thermal energy, then
/?(£')= exp (2.2)
By considering the number o f moments # in a unit volume, and their
orientations, one can get the resultant magnetisation per unit volume [files, 1991],
M
=r
m cos 6dn =Nm ^ cos 6 sin 6 exp lUQÏnH cos 6
kT de
£ sin 6 exp ju^mH cos 6 d e
(2.3)
which, when integrated, gives
M = Nm coth
kT MpmH = Nm L
Mo^ H
kT (2.4)
where L(jM)mH/kT) is the Langevin function, which can be expanded as an infmte
series. When ^m H /kT « 1, the Langevin function L(x) tends to x = fjjomH/kT, so
the entire expression reduces to
M =
2>kT (2.5)
which leads to the Curie law % = M /H = C/T[files, 1991].
This relation is only valid in the paramagnetic region, and is often
critical temperature, which is the Curie point for materials that undergo a
ferromagnetic transition,
Weiss theorised that the variation o f paramagnetic susceptibility observed
experimentally in different materials could be explained by an interaction between
the moments via an interaction field [Weiss, 1907], Since paramagnets are
magnetically homogeneous locally, the magnetic moment per unit volume is equal
everywhere. This does not hold for ferromagnets, where there is a domain
structure. However, for paramagnets it is possible to express the interaction field
as
He - ocM (2.6)
where a is a (so far) undefined parameter. The total field is thus
H,oi = H + He = H + ccM (2.7)
Though this is a variation on the Langevin model, a Curie-type law should still be
obeyed, such that
^ ^ (2 .8)
H ., T
and H = Htot - ocM, so
r r
%= - (2.9)
^ T - a C T - T
This is the Curie-Weiss law (figure 2.1). What Weiss proved with this was
that a paramagnetic solid with localised, interacting atomic moments will obey
this law, down to the critical temperature Tc. This means that the energy o f the
E = -/u^m • ( ! { + o M)
and the magnetisation as a function o f field will follow
(2.10)
M = M coth /UqM • {H + oM )
kT
kT
fÀ^m • { ! ! + oM) (2.11)
which is a variation on the Langevin function. In other words, the interactions
raise the paramagnetic susceptibility. It must be emphasised that below Tc the
behaviour is very different. Although a number o f paramagnetic materials follow
the Curie law, most metals do not. The Weiss theory further assumes that the
moments are localised. The Weiss theory does not work for most metals because
the magnetic moment is provided by the outer electrons, which are comparatively
mobile in a metallic solid. The Curie-Weiss law does work well for some metals
like nickel, for reasons that are still not clear.
Curie-Weiss law
T
Temperature (arbitrary units)
3. Weiss theory o f ferromagnetism
In a ferromagnet^ at temperatures well below the magnetic moments within
domains are aligned parallel to each other. For localised moments, we can use an
interaction o f the form used to explain paramagnetism, i.e. an effective field, to
explain the alignment o f ferromagnetic moments within domains below Tc.
Consider the interaction between two magnetic moments, nti and ntj. The
moment will experience a field Hdp which we will assume is in the direction of
nip such that
He\j=Jiimy (2.12)
Thus, the total exchange interaction field will be the vector sum of all the
interactions with other moments
Hc\ = 'LJ\jntj (2.13)
This can be radically simplified if one considers that, if the interactions between
all the moments are identical and thus independent o f displacement between
moments, then all the Jÿ are identical, such that
HQ = a 'L m j (2.14)
where the a is the same as used earlier for the mean-field interaction. If this is so,
then within a domain
He = a(Ms - m\) « oMs (2.15)
where Ms is the spontaneous magnetisation. Then the interaction energy is
E^ = - f M i C x m \ * M ^ . (2.16)
This is the original formulation o f the Weiss theory. It is not entirely
realistic since the assumption that all moments interact equally is not true.
However, within a domain it is a reasonable approximation to make this
assumption.
When there is no applied field, and if there are no constraints on the
direction of w, we arrive at an analogous Langevin equation
= coth
\ kT
kT
(2.17)
which leads to a solution of perfectly ordered magnetic moments as T approaches
zero. Ms decreases with temperature up to the Curie point where it is zero.
The Weiss model can also be considered in terms of nearest-neighbour
interactions, but that is less interesting for understanding superparamagnetic
behaviour, which focuses on the behaviour of domains.
4. Ferromagnetic domains
In a ferromagnet of finite size, a domain (collection o f coupled parallel magnetic
moments) produces surface free poles, giving rise to magnetostatic energy
In order to reduce Emzg, the spin distribution must be altered, which modifies the
complete parallel spin arrangement. Consequently, exchange energy Eq, and
magnetocrystalline anisotropy energy £a are increased^. Stability is obtained by
minimising the total energy
E = E^^^+E^+E^. (2.18)
^ There is another term, magnetoelastic energy Ex, but this is often ignored since it
Landau and Lifschitz showed in 1935 that the existence o f domains is a
consequence o f energy minimisation. The existence o f domains was empirically
realised earlier from Weiss’ theories, since ferromagnetic materials are not, by
default, magnetised to saturation. The only explanation for this was in postulating
the existence o f domains. A single-domain particle has large magnetostatic
energy, but the division o f the magnetisation into localised regions (domains)
lowers the magnetostatic energy. If the decrease is more than the energy needed to
form a domain wall, multidomain structures will arise.
5. Single-domain particles
Domain walls separate magnetic domains in order to minimise the total energy,
which consists o f the magnetostatic term as well as the exchange and anisotropy
energies. The relative magnitudes o f these energy components determine the
domain structure and shape. Reducing the dimensions o f the crystal by a factor of
I reduces the number o f domains by a factor o f V/ [Chikazumi, 1997]. The energy
cost o f forming a domain wall set against the magnetostatic energy results in an
optimum domain size, and a corresponding minimum multidomain crystal size,
below which a single-domain structure becomes energetically preferable. This
occurs when the energy cost due to the formation o f a domain wall is higher than
the energy gain caused by dividing the single-domain grain into a multidomain
structure.
The static, ‘blocked’ state for a single-domain system is quite unlike that
for a multidomain ferromagnet. With the advent of single domain structures, there
is a great increase in the coercive field where » K/ 3Ms for uniaxial
symmetry [Dormann et al., 1997]. Also, a single-domain system’s magnetic
behaviour is more strongly affected by its domain surface structure than a
multidomain system is. This is because while domain walls can move, shrink or
grow, the effective “domain wall” for a single-domain grain is the physical
surface o f the grain, which is fixed in position and unchangeable. The only
process o f magnetisation that remains is rotation.
For simple anisotropies like uniaxial or cubic systems, the anisotropies can
energies are the main sources o f anisotropy, in fine particles the surface
contribution becomes significant too. For a fine particle system, the anisotropy
energy is a sum o f the contributions^
E = E + E_ + E, (2.19)
where E^ is the magnetocrystalline energy, E^ is the magnetostatic energy, and E^
is the surface energy.
Since this thesis deals with iron-silver and iron-copper-silver, it is
important at this point to note that iron exhibits cubic anisotropy, not uniaxial
anisotropy as discussed above.
6. Cubic anisotropy
Iron in its BCC state (a-Fe) has cubic magnetic anisotropy. This means that the
magnetic moments are most easily aligned along the sides o f the BCC unit cell,
[100], [010] or [001] (see figure 2.2).
Figure 2.2. The a-Fe BCC magnetic unit cell.
The anisotropy energy o f a cubic system can be expressed in terms of the
direction cosines (au %) of the magnetisation vector with respect to the three
cube edges. The cubic anisotropy energy can be expressed as a polynomial series
o f the direction cosines a\, % and «3, because o f the high symmetry o f the cube.
Terms including odd powers of a\ must vanish, since a change in sign of any of
the Oj should bring the magnetisation vector to a direction equivalent to the
original direction. Also, the expression must be invariant to any interchange o f i ,j
and k in a\. The expression for the cubic magnetocrystalline anisotropy is
E = V
+ ^ (s in ^ 0 + sin"* ^sin^ <p)
+ — sin^ ^sin^ 2^sin^ 2<p + ...
16
(2.20)
with volume V and anisotropy constants Ku Ki and Ku For a-iron at 20°C, K\ =
4.72x10"^ Jm'^, and K2 = -0.075x10"^ Jm'^ [Chikazumi, 1997]. The value for K^, is
even lower. This means that, effectively, could be adequately described using
only the first term. Doing this, it is easy to see that for [111] is higher than E^
for [1 0 0], so [1 0 0] is an easy axis, as are [0 1 0] and [0 0 1].
7. The energetics o f single-domain systems
The magnetostatic energy per unit volume of a single-domain grain can be
obtained through a simple derivation. The energy per unit volume o f a dipole of
magnetisation M in a magnetic field H is
E = -/u ^ \H » d M . (2.21)
Taking into account the demagnetising field generated by M, = -N^M can
be put into the integral to make
which, for a spherical particle (#d = V3), becomes
(2.23)
X
2h
(a) Multi-domain particle (b) Single-domain particle
Figure 2.3. Magnetisation M distribution for a high anisotropy particle separated by a domain wall with width 2h, with the single domain case (right).
The critical size for a single-domain particle to be energetically feasible
has been calculated by considering a multidomain particle o f the same size
[Brown, 1969], as shown in figure 2.3(a). The critical size would thus be one
where the energy for a multidomain system exceeds that for a single-domain
configuration. The calculation makes use o f the continuum approximation, which
assumes that the spatial distribution o f lattice spins can be replaced by a
continuous magnetisation, and compares the free energy terms exchange energy
We, anisotropy energy W^ and magnetostatic self-energy W^- For a system with
cubic anisotropy, these are defined as
K = \ c I [{Va, Y + { Va , Y + { Va , Y F
= a : , j { a f a l + « 2 « 3 + )dv
j M * H ' d v ,
(2.24)
where C and K\ are positive constants, and where H is the magnetising force due
The result depends on the degree o f anisotropy within the material, but we
shall not concern ourselves with the low-anisotropy situation since a-Fe is a cubic
(high anisotropy) system [Chikazumi, 1997]. Adding all components together
gives us an upper bound of the free energy F, and eventually yields an expression
for the critical radius Vq [Brown, 1969]
3;r[c(r,+ 2 M .V )]^
where Ms is the saturation magnetisation and cr is an anisotropy parameter which
for cubic anisotropy comes to 0.785392. For BCC a-iron, rc comes to 167 Â. This
is the critical radius, so the critical diameter or crystallite thickness for single
domain particles would be about 33 nm [Brown, 1969].
8. Theory o f superparamagnetism
The anisotropy energy is proportional to the volume o f the particle V, to a first
approximation. For cubic anisotropy, the associated energy Eb = K] V/4 [Brown,
1979]. For positive cubic anisotropy, as in a-iron and BCC iron-copper, a particle
must cross over an energy barrier in order to change the orientation of its
magnetisation. With a decreasing particle size the anisotropy energy decreases,
and may become comparable to or lower than the thermal energy kT. Thus the
energy barrier for magnetic relaxation may be overcome, and the single-domain
particle can behave like a paramagnetic moment. Thus the entire system can
behave like a paramagnet if the particle size/energy barrier is small enough.
Superparamagnetic behaviour thus occurs within a defined particle size range. If
too small, surface effects become disproportionate and the superparamagnetic
model can no longer be applied to describe the relaxation. The upper limit is, in
theory, the critical radius for a single-domain particle, but in practice, the
temperature required for relaxation to be observable may be so high that it
The observation o f superparamagnetic relaxation is dependent upon the
experimental measurement time tm compared to the relaxation time r associated
with the magnetic relaxation. If the system is seen as being static, i.e.
ferromagnetic for Fe-Ag and Fe-Cu-Ag. If Tm»T, the system is seen as
fluctuating, i.e. superparamagnetic. For a single-domain particle the anisotropy
energy is given by equation (2.19), F = E^ + .
i. Magnetocrystalline Anisotropy
For cubic symmetry, as in a-Fe, the magnetocrystalline energy can be written as
E = V + (K, /4)sin" 20 + sin^ Osin^ 2cp
+ (K^ /16)sin^ ^sin^ 2cp + ...
(2.26)
where 0 and ç are defined in the ordinary spherical co-ordinate system [Dormann
et al., 1997]. For bulk a-iron at 20°C, K\ = 4.72x10^^ Jm'^, and Kj = -0.075x10'^
Jm'^ [Chikazumi, 1997]. Thus K2 and higher order Ks are small enough to be
negligible for the purposes o f magnetocrystalline energy calculations.
ii. Magnetostatic Anisotropy
For single-domain particles this energy is related to the components of
magnetisation in the x, y and z directions and can be expressed exactly for an
ellipsoidal particle. The magnetostatic energy per unit volume for a spherical
particle is [Dormann et al., 1997]
j^mag = - M)A^d/\M ^ dM = jjqNaM^ 12. (2.27)
where M is the magnetic moment o f the entire particle. For an ellipsoid of
revolution, Dormann [Dormann et al., 1997] gives the magnetostatic ^dependent
E mag — { N , - N ^ ) s m ‘ 9 . (2.28)
iii. Surface Anisotropy
As particles become smaller the surface-to-volume ratio rises, and the surface
contribution to anisotropy becomes more important. For a cubic system this
becomes, written by surface unit
{E ^ \= K ,c o s ^ e ' (2.29)
where 6' is the angle between m and the perpendicular for the surface [Dormann
et al., 1997]. Simplistically, this should vanish for a perfect sphere. But if one
considers an ellipsoid, the surface energy is given by
E^=K^F (e)Ss\n^e (2.30)
where S is the particle surface and e is the ellipticity (1-6^/a^, where 2a and 2b are
the lengths o f the major and minor axes o f the ellipse) [Dormann et al., 1997]. In
this case, F(e) is
. 1 ( 4 -3 /g ^ arcsing + (3/g^ - 2)eyl\-e^ n
0 ' / Ï 2^ *
^ a r c s m e + e V l - e
This reduces to approximately for small values o f e. This
calculation, by Néel [Néel, 1949], is only valid to a first approximation.
iv. Conclusions
As seen above, the anisotropies for fine particles do not in general have a simple
analytical form. In order for a proper analysis o f anisotropies to be performed, the
system studied has to be known to a very high degree. However, from relaxation
T= To exp [E^lkT] (2.32)
is sufficient, where the main influence on r comes from the exponential [Dormann
et al., 1997]. So, while Eb and % (the characteristic time constant) can be obtained
experimentally, it is usually very difficult to obtain the individual anisotropies
from them.
9. Blocking Temperature
It is known that t is temperature dependent, so the temperature at which r = is
called the blocking temperature 7b. By its very nature, Tg is not well defined,
since it is dependent upon the measurement timescale of the technique used to
observe superparamagnetic behaviour.
For cubic magnetic anisotropy, the energy barrier Eq = K\ K/4, where V is
the particle volume and K\ is the anisotropy defined earlier [Brown, 1979]. This
makes the Néel relaxation relation
T~ To exp {K\ VIAkT) (2.33)
or, for relaxation frequency v,
v= vb exp(-A'iF/4Â:7) (2.34)
This is strictly speaking only correct for a single magnetic particle, in isolation. If
we have a system with a distribution o f moment orientations, it can be treated
statistically. Consider an assembly of single-domain particles, each having a
magnetic moment //. In the case o f superparamagnetic relaxation, where Eb < kT,
at a temperature T, and an applied field H, and assuming thermodynamic
equilibrium has been achieved, there will be a Boltzmann distribution of
orientations o f // with respect to H, just like the case o f classical paramagnetism
[Jacobs and Bean, I960]. The average fraction o f the moment in the direction of
~ \ ( u * H I /j H)eKç{-f i » H lkT)dÇl
^ - (2.35)
where d/2 is a solid angle and Qx^{-fi*H!kT) is proportional to p{àf2), i.e. the
probability that // lies within d/2 relative to H. The result of the integration is,
again, the Langevin function
^ = (2.36)
which reduces to
H ZkT
and (2.37)
i^ = \ - — ioT i j H » k T .
f i f i H
The difference between the above and Langevin paramagnetism is that ju is not the
moment o f a single atom, but the moment o f a single-domain particle. This
treatment also ignores any anisotropy. The cubic anisotropy of a-Fe, for instance,
will affect the Boltzmann distribution o f magnetic alignment with respect to H.
The magnetisation curve will thus no longer be a simple Langevin function.
Considering uniaxial anisotropy, where E = KVsir^9 - pHcosO [Dormann,
1981], the low-field approximation will depend on KVIkT. For > 0, the
approximation now becomes
^ ÎOT K V « k T
3kT
and (2.38)
For cubic anisotropy, however, the initial magnetisation o f the assembly remains
at juH/3kT for all Eb, and l-kUjijH in the approach to saturation [Jacobs and Bean,
I960], Thus, in the approach to saturation the system follows a l / / f law, in which
the magnetisation varies linearly with MH for high / / values.
10. Real superparamagnetic systems
In real systems that exhibit superparamagnetic relaxation, things are more
complex. For one thing, there will be a distribution in particle volumes, which
leads directly to a distribution in r values. This broadens the ‘blocking region’
around Th. Also, as discussed earlier in this chapter, there are many contributions
to the anisotropy energy: magnetocrystalline, magnetostatic, shape, stress and
surface. Notably, the relatively large surface to volume ratio o f single-domain
particles means that the magnetic moment \m\ will be different from the bulk
material value.
Usually in real superparamagnetic system at temperatures comparable to
the blocking temperature, there will be a fraction o f the moments that are not
relaxing at a frequency detectable by the measurement technique in question. This
component is given as It is difficult to determine this experimentally in most
techniques. Also, the value o f M„r will vary with [Dormann et al., 1997]. Most
theoretical models neglect surface anisotropy energy (as already mentioned) as
well as interparticle interaction anisotropy energy (see below). This is because, to
consider these fully, an advanced knowledge o f the main parameters, e.g. particle
size, shape and spatial distribution, is required.
11. Relaxation time calculation
The energy barrier against magnetic relaxation Eb and characteristic time
parameter tq in equation (2.32), t = roexp[£B/^T], can be determined
experimentally, but without a good knowledge o f the anisotropies it is difficult to
relate Eb or tq to these. Neel made the first relaxation time r calculation in 1949.
He assumed that the particle spins were rigidly coupled and that synchronous
rotation o f the spins occurs when the magnetic moment m is reversed. For
T= roNCxp {KVIkT) (2.39)
with
3 \ 2 r M i G K . / K f 7 Ï T
Brown, however, criticised this calculation because the system is not
strictly gyromagnetic. His modified relaxation time formula requires substantial
algebra and use o f the Gilbert and Fokker-Planck equations [Dormann et al, 1997,
Brown, 1969, 1979].
Brown assumed that the relaxation time r = tyJX\, where X\ is the smallest
non-vanishing eigenvalue o f the Sturm-Liouville equation [Dormann et al., 1997,
Brown, 1969]. For cubic positive anisotropy, and for a = KV/kT> 10, the simplest
form o f X\ is
„ M Æ > 9.-JÏ
A, = a
1 + 77;
exp \
(2.41)
where rfr is the damping constant in the Fokker-Planck equation. For small a, X\ =
2, which is a good approximation for a < 5. As opposed to the uniaxial symmetry
approach, the high and low energy cases do not overlap [Dormann et al., 1997].
There is not yet a unique Brown formula for rin cubic anisotropy.
12. Interparticle interactions
All fine-particle systems experience some degree o f interparticle interactions,
whose strength depends on the volume concentration Cy. Magnetic dipolar
Fe-Cu-Ag, where both the magnetic particles and matrix are metals, so are RKKY
interactions. The three types o f interaction are thus dipolar, RKKY and exchange.
Exchange interactions are very powerful, o f the order o f 1000 T, but only
within a ferromagnetic material. The exchange interaction does not penetrate
paramagnets more than a few atomic spacings [Hernando, 1999]. Exchange
interactions between particles are not thought to be significant for magnetic
particles in a diamagnetic matrix [Kemeny, 1999].
RKKY interactions [Kittel, 1949] get their name from the names o f the
authors Ruderman, Kittel, Kasuya and Yosida. These interactions occur in dilute
solutions o f a magnetic metal in a non-magnetic matrix (such as Fe in Ag). The
interaction caused is between the magnetic ions, and is due to a magnetisation of
the conduction electrons in the vicinity o f the magnetic ions. In the compounds
dealt with in this thesis, the magnetic ions are all in the magnetic particles. The
RKKY effect is dominant over small intergranular separations o f 1 nm or less, but
for larger distances, the dipolar interaction is likely to be the most dominant,
provided the grains are large enough, with more than 50 atoms [Altbir et al.,
1996].
Dipolar interactions are due to the moment o f the magnetic grains in
themselves. This interaction diminishes as l/t/^ÿ, where d\^ is the intergranular
distance. All these interactions modify the energy barrier Es. If a relaxation in m
does not change the overall anisotropy energy, then the interparticle interactions
can be said to be constant for the assembly as a whole.
As discussed, a single-domain magnetic particle will fluctuate thermally at
a characteristic time according to the Néel relation
r= To exp(EB/A:T). (2,42)
For an assembly o f non-identical particles a distribution of relaxation times r will
result. Since the Néel relation is only fully correct for a single particle it is often
modified to the Vogel-Fulcher (VF) equation,
where Tq is a phenomenological parameter, aimed at compensating for the
elevated blocking temperatures caused by interparticle interactions, which would
otherwise yield a lower result for tq than is realistic. The parameters tq and Eb can
be obtained experimentally, and while expressions for them exist, the system
needs to be known to a very high degree before these expressions can be used.
The variety of possible interparticle interactions (dipolar, exchange,
RKKY) makes the analysis o f inteiparticle interactions difficult. The fact that, in
the superparamagnetic state, the moments are fluctuating at a frequency v does
not simplify matters. Considering the system at a sufficiently high temperature,
such that the collective state is not present can allow a theoretical analysis for a
system o f a sufficiently narrow distribution of particle sizes, and hence moments.
Dormann et al. have managed to obtain a model for interparticle
interactions in a uniaxial system, which we shall refer to as the DBF model
[Dormann et al., 1999]. Only dipolar interactions, judged the most dominant in
most superparamagnetic systems, were considered in this work, though other
interactions should behave in roughly the same way.
Consider two particles with magnetic moments nt\ and imj as shown in
figure 2.4. These moments are aligned along the unit vectors u\ and i#j
respectively, with |iw,| = M\Vi and \ntj\ = Mj^, where M is the magnetisation at a
temperature T and V is the particle volume. The line between the centres o f the
particles is parallel to the unit vector fÿ, and the scalar distance between the
particles is d\y
Particle i sees a field due to the dipolar interaction with particle j. The
energy corresponding to the field is E\^ = - • U{M\V\, where = (MjFj /
Jjj^)[3(«j • i*ij) fjj - Wj]. These equations assume that m for each particle is located
in the centre o f the particle.
The exchange interactions inside the particle are much stronger than the
dipolar interactions due to other particles here, so the spins remain nearly parallel
in the core o f the particle (surface effects are neglected here). This is a good
X
Figure 2.4. The two-particle interparticle interaction model co-ordinate system.
Dormann obtained a statistical expression for the energy barrier with
regards to dipolar interactions, based on the above equations
{E,\^^=M^V'Za^L{M^Vaj/kr)
(2.44)in which M is the average magnetisation, Vis the average volume, L is a Langevin
function and ûtÿ = (FjMij^)(3cos^<fjj - 1), where is an angle parameter o f ry
[Dormann et al., 1997]. The model yields, for particles o f uniaxial anisotropy, the
total energy barrier for a superparamagnetic system o f inhomogeneous blocking
as
= ^BO + n\a\M mL[aiM mV/kT\ (2.45)
where Ebo is the energy barrier for a non-interacting particle, and the second term
represents the interparticle interactions. In this «j is the average number of NN, a\
~ Cv/V2 where Cv is the volume concentration of magnetic particles in the sample.
According to this theory, interparticle interactions in a system o f particles with
barrier Eg. This considers only nearest-neighbour (NN) interactions. L is a
Langevin function coth a - l/« , with a = [a\M^x\rVlkT\. For a weakly interacting
system this becomes
Eb = Ebo + n\{a\hf'mVflQkT) (2.46)
and for medium-strength interactions or low temperatures one obtains [Dormann
et al., 1999]
Eb « ^Bo - n\kT + n\{a\M^^Vf • (2.47)
For strong interactions the system enters into a collective state, with spin glass
like homogeneous freezing. For systems this dense the relaxation time ceases to
obey any modified Arrhenius-Néel relation and instead follows a power law
r = r o [r g /(r f-r g )r (2.48)
where Tg is the glass transition temperature and 7f is the freezing temperature. The
exponent v depends on the law governing the phase transition. This spin glass
relation adequately describes the collective state o f particle assemblies, though it
is difficult to determine the value o f v experimentally.
13. Conclusions
In this chapter, we have discussed the domain concept in ferromagnets, and the
basic concept o f paramagnetism. We have seen that there is a critical radius for
single-domain particles, which is a first step in determining whether or not the
samples studied are potential superparamagnets. We have discussed the
superparamagnetic effect, and the nature o f the energy barrier against spontaneous
magnetic relaxation. We have also discussed the nature and effects o f exchange,
RKKY and dipolar interactions in fme-particle assemblies, their role in the
can now consider how superparamagnetic effects manifest themselves in the
Chapter
3 : Sample preparation and experimental details
The focus of this thesis, in terms o f the systems studied, is on silver and
iron-copper-silver mechanically alloyed samples. There are many reasons behind the
choice o f these samples. One reason for choosing iron-silver and
iron-copper-silver was that previous work had been done on these samples to investigate the
Giant Magnetoresistance behaviour in these samples. [Cohen et al., 1997; Cohen,
1998] The relatively low volume fractions of magnetic material meant these
samples were considered the most suitable ones in which to study
superparamagnetic effects. Iron and silver are immiscible to a high degree
[Hansen and Anderko, 1958], and with the right ratio of iron and silver,
nanometric clusters o f iron should be created in a silver matrix when mechanically
alloyed. A similar structure has been obtained by annealing a completely alloyed
sample in order to obtain re-crystallisation into clusters [Peng et al., 1994].
A further reason for the choice o f iron-silver was the desire to use pSR to
probe superparamagnetism in the samples (see chapter 6). Silver has almost no
nuclear magnetic moment, and therefore no perceivable pSR relaxation signal.
This makes an M-Ag sample, where M is a. magnetic element or compound, ideal
for studying exclusively the behaviour of M in a dilute solution.
It was soon discovered that the mechanical alloying o f iron and silver does
yield a nanoscale distribution o f particles in a silver matrix. However, the particle
sizes were still relatively large, and any superparamagnetic relaxation behaviour
only occurs at temperatures above room temperature. Not wishing to alter the
metastable sample by annealing it, these regions were inaccessible. Adding a
small amount o f copper to the mixture o f an iron-silver sample does, however,
decrease the particle size and make the relaxation phenomena more accessible.
This was the reason for turning to ternary iron-copper-silver alloys.
1. Sample preparation
Iron and silver are almost totally immiscible, and will not even alloy when melted
together. In the study o f immiscible or partially miscible metal alloys, sample
however, often be created by arc melting [Bewley and Cywinski, 1998], melt
spinning [Lopez et al., 1998], sputtering [Jackson et al., 2000] or mechanical
alloying [Schultz and Eckert, 1994], just to name a few techniques.
In this project, the majority o f samples were created by mechanical
alloying in a high-energy ball mill. The workings o f a high-energy planetary ball
mill can be seen in figure 3.1. The Fritsch Pulverisette 7 high-energy ball mill
used here consists o f a turntable with attachments for two milling pots o f Cr-Ni
steel or Syalon, a compound consisting o f Si3N4 and some dopant containing
aluminium. Because syalon has a lower density than Cr-Ni steel, it is thought that
the syalon ball mill will have lower milling energy than the Cr-Ni steel one.
M illing bowl
Turntable
M illing bowl
Figure 3.1. A sketch o f the workings o f a planetary ball mill.
These milling pots each contain 6-7 balls o f the same material as the pot.
When the turntable turns, the pots rotate as shown in the figure, which causes
collisions that result in cold welding and fracturing o f crystallites. This initially
produces layered powder particles on a microscopic scale, but further milling
create a metastable alloy [Schultz and Eckert, 1994]. If the constituents are
sufficiently miscible, an amorphous alloy will be the result.
Elemental powders with micron scale crystallites were weighed to achieve
the correct stochiometry and mixed in a pot. The pot was sealed inside a Saffron
glove box under an argon atmosphere in order to prevent oxidation. The results of
the milling depend on the ratios of initial components and their miscibility. For
instance, FezoCugo, when milled for 60 hours, alloys completely, whereas Fe2oAggo
milled for the same time yields a nanoscale polycrystalline granular mixture. To
prevent over-heating, the samples were only milled for two hours continuously
(heating the pots to 40-50°C), after which a cooling period o f one hour would
follow before the next two-hour milling period. As such, a 70 hour milling would
take approximately AYz days to complete.
One sample presented in this thesis was prepared by arc melting. The arc
furnace used was an Edmund Bühler Arc-Melting apparatus. The electrode was
made of tungsten and water-cooled. The electrode had a maximum possible
current o f 400 A, and the generator could generate 18kW. The tungsten tip was
brought near a water-cooled copper trough. The atmosphere inside the chamber
was 0.5 bar Argon. The tungsten electrode was first “fired” onto a piece of
zirconium (which melts under the beam) to clear the atmosphere o f impurities.
The zirconium absorbs oxygen particularly well, as well as other impurities. Zr
melts at 2128 K, but the actual temperature the arc could induce was proportional
to the electrical resistance. Silver, for example, is a very good conductor and does
thus not attain such high temperatures.
Because the trough was water-cooled, the cooling time o f the samples was
short. The alloy composition after melting varied with gas pressure. In addition,
the actual composition o f the sample can vary due to mass loss while under the
arc beam. In the case o f the FeAg samples, a small loss o f silver through the
arc-melting process was expected, thus leaving a higher ratio o f iron to silver. By
moving the arc in a rolling motion, the melt could be mixed in a way that ensured
homogeneity.
In following the procedure detailed above, it was seen that the FeAg
unclear what caused this; perhaps studying the effect o f melting pure silver would
give an answer. It is unlikely that this is an effect o f the iron in the mixture.
2. Magnetic measurement techniques
In this project, being in the main an examination o f magnetic relaxation effects, a
thorough investigation o f the magnetic properties can only be realised by
investigating the relaxation behaviour through a variety o f measurement
timescales (see chapter 2). Apart from the more traditional techniques o f DC
Magnetisation and Mossbauer Spectroscopy, pSR has also been used in an
attempt to explore an intermediate timescale.
i. VSM - Tm = 1-10 s
The Vibrating Sample Magnetometer, usually abbreviated as VSM, works by
moving a magnetic sample rapidly up and down between the poles o f a magnet.
The magnetisation is detected by induction coils placed between the poles and
around the sample. This has a measurement timescale on the order o f 1-10
seconds. A VSM is, simply put, a gradiometer that measures the difference in
induction in a space with and without a sample present. This leads to a direct
measurement o f the bulk magnetisation M. Important considerations when using a
VSM are the packing o f the sample for powder samples, the alignment o f the
sample, and demagnetisation effects. As the space between the poles is only a few
centimetres, the samples cannot easily be made large enough to wholly eliminate
demagnetisation effects. Within the context o f this project, the VSM has been
used to obtain magnetisation curves and hysteresis loops. Some of these have later
been analysed via a Langevin function (see chapters 2 and 4).
The sample is mounted on a rod between the poles o f a magnet, where also
a small pick-up coil is placed, as shown in figure 3.2. The sample vibrates
sinusoidally at a fixed frequency o f 66 Hz in order to minimise interference from
the mains frequency (50 Hz). An e.m.f. is generated in the pick-up coils that is
proportional to the sample magnetisation. As the sample vibrates, a voltage is
