Charles University in Prague Faculty of Mathematics and Physics
DOCTORAL THESIS
RNDr. Simona Kubíčková
Nanoparticles based on 3d metal oxides – correlation of structure and magnetism
Institute of Physics CAS, v.v.i.
Supervisor of the doctoral thesis: doc. RNDr. Jana Kalbáčová Vejpravová, Ph.D.
Study program: Physics
Specialization: Physics of Nanomaterials (4F13) Prague 2015
First and foremost I would like to thank to my supervisor Jana Vejpravova who has supported me during last 9 years, shared her knowledge and discussed my work with me.
Thanks to her I could finish my thesis after the birth of my daughter.
Second, I will be thankful to my co-supervisor Daniel Niznansky who introduced me to the measurements of Mössbauer Spectroscopy and showed me the disputable problems in measurements and data analysis of this widely used technique.
I would like to thank to my co-workers Barbara Pacakova and Alice Mantlikova for selected experimental work, consultation with data analysis and support. I also thank our colleagues Maria del Puerto, Morales, Gorka Salas, Marzia Marciello and Petr Brazda for preparation of the magnetic nanoparticles and Dominika Zakutna and Andrea Ardu for measurements of High-Resolution Transmission Electron Microscopy.
Finally, I would like to especially thank to my whole family and my husband for the support and encouragement, especially during the final stages of writing this thesis.
I declare that I carried out this doctoral thesis independently, and only with the cited sources, literature and other professional sources.
I understand that my work relates to the rights and obligations under the Act No.
121/2000 Coll., the Copyright Act, as amended, in particular the fact that the Charles University in Prague has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 paragraph 1 of the Copyright Act.
In Prague 13.10.2015 signature of the author
Název práce: Nanočástice na bázi oxidů 3d kovů – korelace struktury a magnetismu Autor: RNDr. Simona Kubíčková
Katedra / Ústav: Fyzikální ústav AV ČR, v.v.i.
Vedoucí disertační práce: doc. RNDr. Jana Kalbáčová Vejpravová, Ph.D., Fyzikální ústav AV ČR, v.v.i.
Abstrakt: Dizertační práce řeší korelaci magnetické odezvy nanočástic oxidů železa s jejich vnitřní strukturou. V rámci práce byly srovnány výsledky několika vzájemně se doplňujících charakterizačních metod s cílem určení krystalinity zkoumaných nanočástic. Hlavní důraz byl kladen zejména na objasnění vzniku tzv. „spin-canting“
(sklonění spinů) efektu v nanočásticích. Stěžejní metodou pro tuto problematiku byla Mössbauerova spektroskopie ve vnějším magnetickém poli (IFMS). Měření IFMS provedená na sérii vzorků lišících se svou vnitřní strukturou vedla k závěru, že IFMS nepředstavuje jedinou výlučnou metodu pro studium povrchových efektů v nanočásticích neboť nedokáže rozlišit, zda je spin-canting povrchový či objemový efekt. Kromě této problematiky byla IFMS využita taktéž pro studium specifické fáze oxidu železa – ε-Fe2O3 a jejího chování ve vnějším magnetickém poli.
Klíčová slova: nanočástice oxidů železa, magnetismus, Mössbauerova spektroskopie, sklonění spinů
Title: Nanoparticles based on 3d metal oxides – correlation of structure and magnetism
Author: RNDr. Simona Kubíčková
Department: Institute of Physics CAS, v.v.i.
Supervisor: doc. RNDr. Jana Kalbáčová Vejpravová, Ph.D., Institute of Physics CAS, v.v.i.
Abstract: The thesis is focused on the correlation of the magnetic response of iron oxide nanoparticles (NPs) with their internal structure. Several complementary methods were used and compared that bring insight into the relative crystallinity of the investigated NPs. The main goal was devoted to the elucidation of the origin of the so-called spin canting angle determined by In-field Mössbauer Spectroscopy (IFMS) by examination of samples with different internal structure. It has been observed that the IFMS is not an unambiguous method to study the surface effects in the NPs as the IFMS is sensitive only to the average value of all spins and does not distinguish between the surface and core effects. Moreover, the IFMS was performed on the epsilon phase of the iron(III) oxide NPs in order to inspect the peculiar behavior of this phase in an external magnetic field.
Keywords: iron oxide nanoparticles, magnetism, In-field Mössbauer Spectroscopy, spin canting
1
Content
1 Motivation and aims of the work ... 5
2 Theoretical part ... 7
2.1 Magnetization and magnetic susceptibility ... 7
2.2 Magnetism in non-interacting systems ... 7
2.2.1 Diamagnetism ... 8
2.2.2 Paramagnetism ... 8
2.2.3 Interactions in magnetic materials ... 9
2.2.4 Ferromagnetism... 10
2.2.5 Antiferromagnetism ... 11
2.2.6 Ferrimagnetism ... 11
2.2.7 Magnetism of 3d ions ... 11
2.3 Magnetic nanoparticles ... 14
2.3.1 Theory of superparamagnetism ... 14
2.3.2 Particle size distribution ... 17
2.3.3 Effective anisotropy constant, Keff... 17
2.3.4 Surface effects in nanoparticles ... 17
2.3.5 Interparticle interactions... 20
2.3.5.1 Weakly interacting systems ... 20
2.3.5.2 Strongly interacting systems ... 21
2.4 Iron(III) oxides ... 22
2.4.1 Hematite, -Fe2O3 ... 23
2.4.1.1 Crystal structure ... 23
2.4.1.2 Magnetic structure ... 24
2.4.1.3 Mössbauer Spectroscopy ... 24
2.4.2 Maghemite, -Fe2O3 ... 25
2.4.2.1 Crystal structure ... 25
2.4.2.2 Magnetic structure ... 26
2.4.2.3 Mössbauer Spectroscopy ... 27
2.4.3 Magnetite, Fe3O4 ... 28
2.4.3.1 Crystal structure ... 28
2.4.3.2 Magnetic properties ... 29
2.4.3.3 Mössbauer Spectroscopy ... 29
2.4.4 Epsilon, -Fe2O3 ... 30
2.4.4.1 Crystal structure ... 30
2.4.4.2 Magnetic properties ... 30
2.4.4.3 Mössbauer Spectroscopy ... 31
2
2.4.5 Beta, -Fe2O3 ... 32
2.4.5.1 Crystal structure ... 32
2.4.5.2 Magnetic properties ... 32
2.4.5.3 Mössbauer Spectroscopy ... 33
2.5 Theoretical background of the experimental techniques ... 34
2.5.1 Mössbauer Spectroscopy ... 34
2.5.1.1 Mössbauer effect ... 34
2.5.1.2 Recoilless factor ... 35
2.5.1.3 Nuclear hyperfine interaction ... 36
2.5.1.4 Electrostatic hyperfine interaction ... 36
2.5.1.5 Magnetic hyperfine interaction ... 38
2.5.1.6 Combined electric and magnetic hyperfine interaction ... 40
2.5.1.7 Integrated intensities of the Mössbauer lines ... 41
2.5.1.8 Mössbauer Spectroscopy of nanoparticles ... 41
2.5.1.9 Summary of determined parameters ... 44
2.5.2 Powder X-ray Diffraction (PXRD) ... 45
2.5.3 Typical magnetic measurements of nanoparticles ... 48
2.5.3.1 The temperature dependence of magnetization ... 48
2.5.3.2 Magnetization isotherms ... 49
2.5.3.3 Ac susceptibility ... 51
3 Experimental part ... 53
3.1 Preparation of nanoparticles ... 53
3.1.1 The maghemite nanoparticles ... 53
3.1.1.1 Organic route ... 53
3.1.1.2 Aqueous route ... 54
3.1.2 The epsilon phase ... 55
3.2 Powder X-ray Diffraction ... 55
3.3 Transmission Electron Microscopy ... 55
3.4 High Resolution Transmission Electron Microscopy ... 55
3.5 Mössbauer Spectroscopy ... 56
3.6 Magnetic measurements ... 56
4 Results and discussions ... 57
4.1 Characterization of the samples... 57
4.1.1 Powder X-ray Diffraction ... 57
4.1.1.1 Maghemite nanoparticles obtained by oxidation ... 58
4.1.1.2 Rectangular nanoparticles ... 60
4.1.2 Transmission Electron Microscopy ... 64
3
4.1.3 High Resolution Transmission Electron Microscopy ... 66
4.1.4 Room temperature Mössbauer Spectroscopy ... 68
4.1.5 Magnetic measurements ... 71
4.1.5.1 Temperature dependence of magnetization ... 71
4.1.5.2 Magnetization isotherms ... 73
4.1.5.3 High-temperature measurements ... 77
4.1.6 Quantification of structural and spin disorder ... 79
4.1.6.1 Magnetic response ... 81
4.1.6.2 Correlation of internal structure of a NP with SAR performance 83 4.2 In-field Mössbauer Spectroscopy of nanoparticles with different internal structure ... 86
4.2.1 General results ... 86
4.2.2 Evolution of the effective field, Beff ... 92
4.2.3 Evolution of the FWHM ... 93
4.2.4 The core-shell model applied to the OD15 samples ... 94
4.2.5 Evolution of the spin canting angle ... 98
4.2.6 Refinement considering distribution of the effective field ... 100
4.2.7 Comparison of the A22orig and A22oxid samples ... 100
4.2.8 Overall discussion ... 101
4.3 In-field Mössbauer Spectroscopy of the epsilon phase ... 102
4.3.1 Characterization of the sample ... 102
4.3.2 Results of In-field Mössbauer Spectroscopy... 104
4.3.2.1 General comments ... 104
4.3.2.2 Evolution of the effective field, Beff with increasing Bapp ... 107
4.3.2.3 Magnetization history ... 108
4.3.2.4 Evolution of the quadrupolar shift, 2 ... 109
4.3.2.5 Evolution of the distribution width of the Beff ... 111
Evolution of the spin canting angles, S and ... 112
4.3.2.7 Peculiar behavior of magnetic moments at Td4-sites ... 113
5 Conclusions ... 114
6 References ... 116
7 Attachments... 126
7.1 List of Tables ... 126
7.2 List of Abbreviations ... 130
7.3 Appendix ... 131
7.3.1 Atomic coordinates of individual phases ... 131
7.3.2 In-field Mössbauer Spectroscopy ... 132
4 7.3.3 IFMS on epsilon phase ... 134
5
1 Motivation and aims of the work
The magnetic iron oxide nanoparticles (NPs) have attracted significant attention last few decades due to their promising physical properties that can be utilized in many fields such as biomedicine [1, 2, 3], magnetic recording [4, 5], catalysis [6, 7, 8] etc.
The main goal of the NP-related research is to prepare the highly crystalline NPs with defined size and shape suggesting controlled magnetic response. It has been reported recently, that the magnetic properties of NPs are mostly affected by their internal structure [9, 10, 11], particle size distribution [12, 13] and/or interparticle interactions [14, 15]. Correlation of the magnetic properties with the crystal structure or particle size is not as straightforward as is generally believed.
First, it is usually claimed, that small NPs exhibit high ratio of the surface to volume spins that lower the resulting saturation magnetization, Ms [16, 17, 18]. However, several studies have shown smaller Ms for large NPs [19, 20, 21]. Most of the studies have not taken into account the internal structure of NPs and collective phenomena that affect the resulting magnetic response.
Furthermore, it is common practice to characterize magnetic NPs based upon their particle sizes that can be determined by various complementary methods such as Powder X-ray Diffraction (PXRD), Transmission Electron Microscopy (TEM) or magnetic property measurements with different physical meaning. For instance, the TEM determines only a specific projection of the physical size of a NP, dTEM without any deeper insight into its relative crystallinity. Thus, correlations of the dTEM with the magnetic response of NPs is highly misleading, however it has been widely performed [19, 20, 22].
Finally, it is usually claimed, that the surface disordered spins could be detected by the In-field Mössbauer Spectroscopy (IFMS) through the so-called spin canting angle. However the volume or surface nature of the spin canting has been still debated [9, 16, 18, 23, 24].
The so-called real effects thus play important role in magnetic properties of NPs and their proper understanding is a crucial point in correlation of structural and magnetic parameters of the NPs with their magnetic properties related to their performance in applications. The general motivation of the research presented in the thesis is to significantly deepen knowledge on the real effects in physics of magnetic NPs with relation to biomedical applications. The specific aims of the thesis are summarized as follows:
1) To demonstrate that detailed characterization of the maghemite NPs by several complementary methods is essential for correlation of their structure with magnetic response.
2) To explore the origin of the spin canting phenomenon by measurement of IFMS on maghemite NPs with different internal structure in varying applied magnetic field.
6 3) To measure the IFMS on high-temperature phase of the epsilon phase of iron(III) oxide in order to determine the evolution of the hyperfine parameters in varying applied magnetic field.
The thesis starts with motivation and aims of the work followed by the theoretical part where the key principles of magnetism with emphasis on single-domain NPs are summarized. Afterwards, an overview on crystallography, magnetic properties and hyperfine parameters determined from Mössbauer Spectroscopy (MS) of all five polymorphs of iron(III) oxides are given. Finally, the theoretical background of the used experimental techniques such as PXRD, MS and magnetic property measurements of NPs are discussed.
In the experimental part the details of the preparation routes are summarized together with the experimental details of the measurement techniques used.
The results are divided into three main parts. In the first part, the characterization of the iron(II,III) oxide NPs by several complementary methods is given and the results are further interpreted in terms of relative crystallinity and magnetic properties.
Furthermore, introducing of the parameter of absolute spin alignment and its correlation with the specific absorption rate (SAR) is given. In the second part, the experiments of IFMS carried out on selected samples with different relative crystallinity are discussed and the emphasis is given on the origin of the spin canting phenomenon. In the third part, the scenario of magnetic order in epsilon iron(III) oxide is discussed with focus on results of IFMS.
Finally, conclusions to the presented results are given followed by appendix and references used in the thesis.
7
2 Theoretical part
The theoretical part is divided into four main sections: First, the basic theory of magnetism will be presented, detailed background on magnetism is well described in the following used books [25, 26, 27]. Second, the theory of superparamagnetism (SPM) describing the magnetic properties of ideal system of monodomain NPs will be discussed. It is followed by description of the magnetic behavior of real systems of magnetic NPs. Third, summary on crystal structure, magnetic properties and hyperfine parameters determined from measurements of Mössbauer Spectroscopy (MS) of iron(III) oxides will be presented. Fourth, a brief summary of theoretical background of the principal experimental techniques used will be given.
2.1 Magnetization and magnetic susceptibility
The magnetic moment, is associated with the orbital and the so-called spin motion of an electron in the solid. The moving electron in an orbit produces a magnetic field in a perpendicular direction with respect to the plane of an orbit. A magnetic solid consists of a large number of atoms with , the quantity volume magnetization, M that is the per unit volume, V is then defined as:
V i
M 1 i . (2.1)
The response of the material to an external applied magnetic field, 0H is described by the dimensionless quantity called the magnetic susceptibility:
H M
0
. (2.2)
2.2 Magnetism in non-interacting systems
The perturbed Hamiltonian describing the response of the localized magnetic moments to the externally applied uniform magnetic field, app
B can be written as:
i Z
i
r m B
B e S g L m V
H p H
H app i 2
e 2 B app
1
i e 2 i
0 ( )
). 8 2 (
ˆ ˆ
ˆ , (2.3)
where the non-perturbed Hamiltonian, Hˆ0 consists of the electronic kinetic energy and potential energy (first term in (2.3)), me is the mass of an electron, B the Bohr magneton, the g represents the so-called electron spin g-factor and ri the position of ith electron in an atom. The
L and
S represent the orbital and spin angular momentum of an electron, respectively that sum to the total angular momentum
J :
8
J LS. (2.4)
The magnetic susceptibility can be calculated as a second derivative with respect to magnetic field using the ordinary perturbation theory to the second order:
n i n
n r B m n
e E
E
n gS L B n n
S g L n B E
2 app i e
2
´ n n
2
´ B app
B app
8
´
, (2.5)
where the first two terms describe the paramagnetic response of a material and the third term the diamagnetic response of a material.
2.2.1 Diamagnetism
The so-called Larmor diamagnetic susceptibility in the case of the non-interacting ions with filled electron shell (thus the
L and
S are zero) can be expressed as:
Z
i
m r e V N
1 2 i e
0 2
d 6
, (2.6)
where V is the solid volume, N is number of ions with Z electrons. The d is temperature and field independent. Application of app
B induces a magnetic moment,
that points in opposite direction to the direction of app
B . Diamagnetism occurs in every material but it is only a weak perturbation that can be ignored in most cases.
2.2.2 Paramagnetism
In a paramagnetic material the induced
is aligned in parallel direction with the
app
B . The paramagnetic susceptibility is computed in two different cases. First, only the electron shells that are one electron short of being half filled are considered (
J
= 0). Then the first term in (2.5) vanishes and from the second term the paramagnetic contribution to the d can be written as:
n E E
n gS L V
N
0 n
2 z 2 z
B VV
2 0
. (2.7)
This term is known as the as the Van Vleck paramagnetism and it is positive and temperature independent.
Second, assuming the shell with J 0, (all cases except the one that was discussed above), the first term in (2.5) is non-zero and is much larger than the two previously discussed, thus they can be safely ignored. The magnetization of non-interacting ions can be described as:
9
T k B JB J ng M
B app B J B
J .
, (2.8)
where BJ is the Brillouin function. For the small argument in the Brillouin function (low Bapp and/or high temperatures), the resulting paramagnetic susceptibility is given by:
T C T k
n
B 2 eff 0
3
, (2.9)
where eff gJB J(J 1) is the effective moment and gJ is the Landé g-factor.
This equation is the well-known Curie law and presents a typical behavior of paramagnetic materials – the alignment of magnetic moments is favored by the app
B and opposed by the thermal disorder.
The above susceptibilities were derived for the localized electrons in partially filled ionic shells, the contribution of the conduction electrons in metals are well described in followed references [25, 26, 27].
2.2.3 Interactions in magnetic materials
There are a lot of materials that exhibit a spontaneous magnetization without presence of app
B bellow a specific temperature. Generally, the interaction that is responsible for the magnetic ordering is called the exchange interaction and is described by the so-called Heisenberg Hamiltonian:
j i
S S J H
,
j i ijˆ ˆ
ˆ , (2.10)
where Jij is the exchange constant between ith and jth spin, respectively.
There are several types of the exchange interaction:
a) Direct exchange
The direct exchange interaction proceeds between neighboring magnetic moments that have sufficient overlap of their wave functions. The direct interaction is unfavored in most elements and materials because most of the orbitals are localized and lie very close to nucleus, thus an intermediary is needed in these materials.
b) Superexchange
The superexchange is observed between non-neighboring magnetic ions and is mediated by non-magnetic ion, which is placed between the magnetic ions. This kind of interaction mainly occurs in the iron(III) oxides.
c) RKKY interaction
The RKKY interaction is named after Ruderman, Kittel, Kasuya and Yoshida, who discovered the effect. A localized magnetic moment spin-polarizes the conduction electrons and these conduction electrons then interact with neighboring localized magnetic moment a distance, r away. The interaction is long range and exhibits an oscillatory dependence on the distance between the magnetic moments.
10 The basic magnetic ground states resulting from the interactions described hereinbefore such as ferromagnetism, antiferromagnetism and ferrimagnetism will be now discussed. The illustration of the magnetic ordering in materials is depicted in Figure 2.1.
Figure 2.1: The types of magnetic ordering in the materials with and without the application of an external magnetic field, Bapp.
2.2.4 Ferromagnetism
In a ferromagnetic crystal, the internal magnetic field caused by the exchange energy is so large that till certain temperature, called the Curie temperature, Tc the material is magnetically ordered. Above the Tc the thermal fluctuations become significant to break the ferromagnetic ordering and the material behaves as a paramagnet. This Tc
is material specific and the transition is of a second-order type phase transition.
One of the easiest attempts how to quantitatively analyze the behavior of ferromagnets and their transitions to paramagnetic state is the so-called Weiss model where the following approximation is made – the exchange interaction is replaced by an effective molecular field, Bmf produced by the neighboring spins. The effective Hamiltonian then looks like the Hamiltonian for a paramagnet in a magnetic field Bapp + Bmf. The Tc is the given by:
B 2 eff
C 3k
T n
, (2.11)
where is a constant that parametrizes the strength of the molecular field as a function of magnetization. The susceptibility above the Tc is given by:
11 TC
T C
, (2.12)
which is known as the Curie-Weiss law. The Tc corresponds to the so-called Weiss critical temperature as Tc ~ p.
2.2.5 Antiferromagnetism
Antiferromagnets are magnetically ordered materials where the local magnetic moments cancel each other resulting in no spontaneous magnetization. The simplest model of antiferromagnet consists of two interpenetrating sublattices of identical magnetic structure (see Figure 2.1) - within each sublattice the magnetic moments have the same magnitude and direction, but the net moments of the two sublattices are oppositely directed, summing to zero total magnetic moment. The model is described by the Néel theory of sublattices [28].
Considering the two sublattices the magnetic susceptibility can be calculated in same way as in a ferromagnetic case and the resulting magnetic susceptibility above the critical temperature called the Néel temperature, TN is given by:
TN
T C
. (2.13)
The behavior of the susceptibility below the TN is more complicated than in ferromagnets because it strongly depends on orientation of the app
B with respect to the magnetic moment.
2.2.6 Ferrimagnetism
In the case where the magnetization of the two sublattices is not equal (e.g.
neighboring magnetic ions are not identical), the net magnetization will occur and this phenomenon is called ferrimagnetism. Generally, the molecular field on each sublattice is different, therefore the temperature dependence of the net magnetization and the susceptibility, respectively is quite complicated [27].
2.2.7 Magnetism of 3d ions
In the previous section, the magnetism of localized magnetic moments was briefly described. The iron(II,III) oxides are representatives of the 3d ions, in which the 3d- orbitals are extended further from a nucleus, thus the influence of the crystal environment has to be taken into account. The crystal field theory (CFT) developed in 1930s by H. Bethe and J. H. van Vleck [29] describe this influence. The CFT does not include the covalent effects (the ligand field theory has to be used [30]), however for basic description of the influence of the crystal environment on the 3d ions, the CFT is sufficient.
12 Generally, the crystal field (CF) in the iron(II,III) oxides is an electric field that originates from the negatively charged oxygen anions creating holes for the magnetic ions. The CF lifts the degeneracy of electron orbital states according to its local symmetry. In the 3d ions, only the valence d-orbitals are affected by CF, because p- orbitals are well localized and s-orbitals are spherically symmetric. The five degenerated d-orbitals are present in free ion state, these orbitals can be divided into two groups according to its local symmetry; the dz2 and dx2-y2-orbitals with the lobes lying on the coordination axes that are labeled in the group theory as eg or e-orbitals and the dxy, dxz, dyz-orbitals with the lobes lying between the coordination axes, labelled as t2g or t2-orbitals (see Figure 2.2).
In the iron(II,III) oxide the O2- ions create the tetrahedral, Td and octahedral, Oh-sites for the Fe2+ or Fe3+ ions (see Figure 2.2., detailed description can be found in section 2.4). The states, where the electrons in d-orbitals of the metal ion are closer to the electrons in the p-orbital of the O2- ion have higher energy than the others. Therefore the e-orbitals will possess lower energy than t2-orbitals in the crystal environment of the Td-sites leading to the splitting of the five degenerated d-orbitals into two groups.
The situation will be opposite for the Oh-site as is depicted in Figure 2.2. The splitting of the d-orbitals in CF is characterized by the crystal field splitting parameter, Td (Oh) that is nearly two times smaller for the Td-sites than for the Oh- sites.
Depending on the strength of the CF with respect to the correlation energy (energy cost by putting two electrons into one orbital), two different cases can be distinguished: First, the strong field limit, where the CF dominates the correlation energy, this leads to the so-called low spin configuration. Second, the weak field limit in which the CF is weaker than the correlation energy that results in the high spin configuration. The resulting value of magnetic moment depends on the low/high spin configuration of the magnetic ion as is depicted in Figure 2.3.
The consequences of the presence of CF are the quenching of the orbital moment and the so-called Jahn-Teller effect. For more information see the appropriated references [26, 27, 30].
13
Figure 2.2: The splitting of the energy levels of the d-orbitals in the crystal field of different coordination and symmetry, the Td-site is depicted on the left, the Oh-site on the right. The values of the crystal field splitting parameters, Td and Oh respectively are shown [26, 30].
Figure 2.3: The CFT splitting of the d-orbitals for the Oh-sites on the left. The filling of the d-orbitals resulting in the so-called high spin and low spin configuration for the Fe3+ ions is shown on the right [26, 30].
14
2.3 Magnetic nanoparticles
This section gives an overview on magnetic properties of the single-domain NPs.
First, the theory of superparamagnetism (SPM) will be introduced, which is valid for an ideal system of well-crystalline NPs with no presence of particle size distribution and interparticle interactions. Thus, the key deviations from the basic SPM model in real systems of NPs will be described afterwards.
2.3.1 Theory of superparamagnetism
In order to decrease the magnetostatic (dipolar, demagnetizing) energy that is associated with the dipolar fields, the ferromagnetically (or ferrimagnetically) ordered crystal is divided into the magnetic domains. Within each domain the M reaches the saturation (see Figure 2.4). The creation of the domains depends on the competition between the reduction of the magnetostatic energy and the energy required to form the domain walls separating the adjacent domains. The size of the domain wall is a balance between the exchange energy that tries to unwind the domain wall and the magnetocrystalline anisotropy with the opposite effect.
In the magnetic NPs, the typical dimensions are comparable with the thickness of the domain wall, thus at some critical size, dcrit it is energetically favorable for the NP to become a single domain. These critical dimensions ranging from 10-7 to 10-8 m were firstly estimated by Ch. Kittel in 1946 [25, 31]:
2 S 0
u ex crit
9 M
K d A
, (2.14)
where Ms is the saturation magnetization, Aex and Ku is the exchange and uniaxial anisotropy constants, respectively. It is obvious, the dcrit is material specific, e.g. for the -Fe2O3 the dcrit reaches 40 nm [32].
In small magnetic NPs reaching the single domain limit (given by the equation (2.14)), the paramagnetic-like behavior can be observed even below the Tc. The phenomenon is therefore called the superparamagnetism (SPM) as the whole particle behave as one “giant” superspin consisting of the atomic magnetic moments, thus the magnetic moment of the whole NP is 104 - 105 times larger than the atomic magnetic moment. The theory of SPM and relaxation of the NPs were treated by L. Neel and C. P. Bean [33].
The SPM theory contains several assumptions. First, all moments within the NP rotate coherently, in other words the magnetic moment of the whole NP behaves as a one giant superspin with magnitude equal to summation of the individual atomic moments. Second, the magnetic energy required to magnetization reversal is given by the uniaxial magnetocrystalline anisotropy (MCA) that can be expressed in the form:
M 2 effVsin K
E , (2.15)
15 where is the angle between the magnetic easy axis and the direction of the magnetic moment, Keff effective anisotropy constant and V the volume of the NP (see Figure 2.5). Third, the interparticle interactions are neglected.
In the case of the zero external magnetic field, 0H the NP superspin will be trapped by the MCA in one of the two equilibrium state as is depicted in Figure 2.5. The relaxation time, of the NP into the equilibrium state can be described by:
T k
V K
B eff 0exp
, (2.16)
where kB is the Boltzmann constant and T temperature, the relaxation time 0 is of the order of 10-9 - 10-12 s.
Because of the small size, the energy barrier KeffV can be overcame by the thermal fluctuations at the certain temperature, the so-called blocking temperature, TB. Below the TB the NPs are in the blocked state analogous to the quasi-ordered state (such as ferromagnetism, ferrimagnetism). Reaching the TB the NP come in the SPM state.
However, the TB depends not only on the particle size, d and the Keff, but also on the time window of the measurements, m. If the m > the NPs have enough time to fluctuate and the SPM state would be observed, on the other hand if m < the blocked state of NPs will be observed. Thus, determination of TB is dependent on the used experimental technique (10-8 s for Mössbauer Spectroscopy, 1 s for dc magnetic measurement, 10 - 10-3 s for ac magnetic measurement).
Figure 2.4: Schematic illustration of the domain structure in bulk and submicron ferromagnetic material. a) The sample is uniformly magnetized with the high demagnetizing field associated with presence of the free poles at the surface. b) The sample is divided into two domains with opposite direction of magnetization, the demagnetizing field is twice lower; the inset shows the reorientation of the magnetic moments within the domain wall. c) The monodomain magnetic NP.
16
Figure 2.5: The angular dependence of the energy barrier, KeffV in zero external field with definition of the axis system on the right, the dashed line corresponds to the magnetic easy axis of the NP.
The application of an external magnetic field, 0H can be described by an additional term in the equation (2.15):
)
cos( M
0
H
E , (2.17)
where is the angle between the 0H and the easy axis of a NP. The 0H affects the shape of the angular dependence of the energy barrier (depicted in Figure 2.5). When the 0H is smaller than the coercivity,
S eff C
0
2 M H K
(detailed description will be given in section 2.5.3.2), the energy profile displays two minima, however, the 0H favors the one along the magnetization easy axis. With increasing 0H above the
0Hc the one minimum disappears and only the direction along the easy axis is favorable. More information can be found in [12, 34].
The magnetization of a single NP in applied field can be easily computed for the NPs in SPM state [12, 27, 35]:
T k L H M
M
B S
, (2.18)
where L is the well-known Langevin function:
x x x
L 1
coth )
( . (2.19)
The conditions for the ideal SPM model are not usually met in real systems of NPs due to presence of particle size distribution and interparticle interactions between the NPs in the sample. Furthermore, ratio of the surface to volume spins become important in small NPs and the presence of the surface disordered spins results in the non-uniform Keff in the whole volume of NP. These aspects will be described in next sections.
17
2.3.2 Particle size distribution
The size of NPs is usually not uniform through the whole system and the resulting particle size distribution is described by the log-normal distribution function [36]:
, exp 2
2 ln 2 exp
) 1 (
2 0
2 M 0 2
x x
x x
x x
f , (2.20)
where is the distribution width and x0 and xM are the mean a median property of the distribution. The particle size distribution results in the distribution of the anisotropy energy, TB and magnetic moment of NP, (will be described in section 2.5.3.2).
2.3.3 Effective anisotropy constant, Keff
In ideal system of NPs, the bulk Keff is considered. However, the Keff varies through the volume of a NP in real systems of NPs and is usually described by equation:
S V
eff
6 K K d
K , (2.21)
where Kv is the bulk value of the Keff and the Ks describes the contribution from the surface spins [37].
2.3.4 Surface effects in nanoparticles
Decreasing the NP size, the numbers of atoms located at the surface of a NP (surface atoms) increases with respect to the all atoms and at some defined size, surface atoms dominate the volume atoms. The ratio of the surface atoms to the atoms located in the whole NP is given by the relation:
V unit 3
S unit 2
1 2 3 4
1 4 2
V N d
S N d ratio
, (2.22)
where d is the particle diameter, Sunit and Vunit are the surface and volume of the unit cell, respectively and Ns, Nv are the number of atoms located at surface and in the whole volume of a NP. The dependence of the ratio on the d is depicted in Figure 2.6 for the spinel structure (details of the spinel structure will be discussed in section 2.4.2). First, only the atoms located at the surface were taken into account (depicted in left panel), in right panel the thickness of the surface layer was given by the thickness of the unit cell of the spinel structure (around 0.83 nm).
18
Figure 2.6: The ratio of the surface atoms vs. all atoms in the whole volume. Left panel: Only the surface atoms lying exactly on the surface were considered. Right panel: The thickness of the surface layer was equaled to the one unit cell of the spinel structure.
It is well seen, that for NPs with smaller d (the exact value depends on the thickness of the surface layer) the surface spins become dominant affecting the magnetic response of the whole NP.
The atoms at the surface exhibit lower coordination numbers originating from breaking of symmetry of the lattice at the surface. Moreover, the exchange bonds are broken resulting in the spin disorder and frustration at the surface leading to the undesirable effects such as low saturation magnetization of the NP and/or the unsaturation of the magnetization in the high magnetic applied field [16, 17]. The determination of the thickness of the shell is disputable, as not only the surface spins have to be considered, but a gradient of the orientation of individual spins should be expected. Many studies have been devoted to the surface effects in NPs, the most important results will be summarized further.
In the 60-70-ties of the last century, the lower Ms of the NPs was observed in comparison to the bulk values [38]. Several works tried to explain this feature including the presence of nonmagnetic layer [38] or presence of hydrogen in the lattice. In 1971, J. M. D. Coey [16. 17] suggested that the non-collinearity of the spin arrangement in NPs is responsible for the lower Ms and proposed the so-called magnetic core-shell model of the NP (Figure 2.7). In the model, the NP consists of a core with the bulk spin arrangement and the disordered shell, where the spins are inclined at random angles to the surface, the so-called spin canting angle. The spin canting angle depends on the number of the magnetic nearest neighbors connecting with the reduced symmetry and dangling bonds. This theoretical model was supported by the experimental evidence of the persistence of the 2nd and 5th absorption line in the In-Field Mössbauer Spectroscopy (IFMS) at low temperature performed on the 6 nm maghemite NPs pointing to the non-collinearity of some spins
19 with the applied magnetic field (details given in section 2.5.1.8). However, he also pointed out, that the possibility of some canting of the ions in the interior cannot be excluded.
In 1976, Morrish et al. [18] tried to verify the Coey’s model by examined the maghemite acicular NPs with 57Fe-enriched surface, however, the proof of enriching only the surface had not been given. Furthermore, it has to be noted, that the enriching (thus further chemical treatment) usually induces additional disorder in the surface layer. They observed the broadening of the lines and the difference of the peak position of the 1st/6th line and 2nd/5th line, attributed to the lower effective field that they connected with the surface spins.
There have been several studies concerning the IFMS with rather contrary results. On one side, they suggested increase of the spin canting angle with decreasing particle size [39, 40]. Moreover, both articles also investigated influence of the interparticle interactions (through the dilution of the samples into the polymer matrices) with no significant effect on the spin canting angle.
On the other hand, there are several studies showing that the internal structure of the NP is more important [23, 24, 41]. Morrish et al. [41] compared the maghemite NPs with different particle size and relative crystallinity – the well-crystalline 7 nm NPs and 95 nm NPs with high degree of disorder (the crystalline part had diameter around 30 nm). Their results (only small difference in the spin canting of both samples - 20 vs. 30°, respectively) suggested that the NP internal structure is important and must be considered in any analysis of the surface spin canting. Morales et al. [23] studied 100 nm maghemite NPs with different vacancy ordering and found out that the spin canting angle varied with the vacancy ordering, suggesting the spin canting is volume effect.
There has been also several theoretical studies trying to solve the presence of the surface effects [42, 43, 44]. For instance, Pankhurst et al. [42] proposed that the complete spin alignment cannot be achieved in the iron(III) oxide NPs due to the large magnetic anisotropy. This theoretical suggestion was elided by Hendriksen et al. in 1994 [45] who measured the IFMS of the frozen ferrofluids. They found out, that the spin canting angle is independent on the initial orientation of the magnetic moments in the NPs with respect to the external field, thus the origin of spin canting cannot be explained by enhanced magnetic anisotropy.
The surface effects in the NPs together with the origin of the spin canting angle are still discussed within the scientific community and lots of reviews or books have been devoted to this problem [11, 12, 43, 44, 46].
In most of these studies, the spin canting angle is determined by means of the IFMS.
The background of the IFMS and the approaches of determination of the spin canting angle will be given in 2.5.1.8.
20
Figure 2.7: The magnetic core-shell structure proposed by J. M. D. Coey [16] of the -Fe2O3 NP, the magnetically ordered core with disordered spins in the shell.
2.3.5 Interparticle interactions
The interparticle interactions play an important role in the NPs because the NPs are usually not well separated and the ideal SPM system can be achieved only in much diluted systems. There are two interactions that can be taken into account: First, the exchange interaction that affects the surface spins of the NPs in close proximity, thus can be neglected in most of the cases. Second, the long-range order dipolar interaction that is the dominant interaction due to the high magnetic moment of the NPs (at least 10^4 times the atomic moment in paramagnetic systems). For the randomly distributed NPs with average magnetic moment, and mean interparticle distance, r the dipolar interaction can be expressed as:
3 2 0
D 4 r
E
. (2.23)
The systems of NPs can be arbitrarily divided into the weakly interacting systems (the representatives are much diluted ferrofluids or NPs embedded in matrix in small concentration) and strongly interacting system with the powder samples as representatives. The strength of the interparticle interactions can be controlled by the concentration of ferrofluids or by the matrix-to-NP ratio.
The interparticle interactions affect the of the NP thus the Arrhenius law (equation (2.16)) is no longer valid. Furthermore, the strong interparticle interactions can result in the collective magnetic state at low temperature that resembles the typical physical properties of spin glasses [47].
2.3.5.1 Weakly interacting systems
In weakly interacting system the dipolar interaction is treated as a perturbation to the SPM state (known as interacting SPM model) by adding a phenomenological temperature, T0 that has no further physical meaning:
) exp (
0 B
eff
0 k T T
V
K
. (2.24)
21 The formula is known as the Vogel-Fulcher law [48].
Theoretically, interparticle interactions are described with two conflicting models giving contradictory results on the relaxation times – the Hansen-Morup model (HP model) [49] and the Dormann-Bessais-Fiorani model (DBF model) [50]. The decrease of the and thus TB is predicted by the HP model, on the other hand, the increase of the TB was obtained by the DBF model. So far, there have been no experimental evidences for a preference of one of these models [51, 52, 53, 54].
2.3.5.2 Strongly interacting systems
In the case of the strong interactions, the collective state of NPs occurs below a characteristic temperature – the so-called glass-transition temperature, Tg and the equation for the relaxation time is usually given by:
zv
T
T
1
g M
0
, (2.25)
where zv is the dynamical-critical exponent, for spin glass system is in the order of 5 - 12, TM is maxima on the ac susceptibility curve [55].
22
2.4 Iron(III) oxides
Iron(III) oxides have been widely used in everyday life for more than 3000 years for their very useful magnetic properties (for example, the mineral lodestone was utilized as a compass). In recent years they have found many applications in various fields such as the magnetic recording, catalysis or biomedicine [1, 2, 6, 7]. The iron(III) oxides can be found in five different polymorphs: the magnetite, Fe3O4
comprising both Fe2+/Fe3+ ions, that is oxidized to the maghemite, -Fe2O3. Maghemite is a low temperature stable phase of the iron(III) oxide that transforms to the high temperature stable phase, hematite, -Fe2O3 at temperatures higher than 750 K [30, 56]. There are two metastable phases occurring during this transition, the so-called epsilon and beta phases, -Fe2O3 and -Fe2O3, respectively. The two phases can be prepared only artificially as nanosized objects in defined range of the particle size due to the minimalization of the Gibbs free energy [57]. The scheme of the formation and transformation of the iron(III) oxides is depicted in Figure 2.8.
Generally, the crystal structure of all iron(III) oxides consists of the oxygen anions, O2- creating the tetrahedral, Td or octahedral, Oh environment for the Fe3+ or Fe2+
ions. The O2- ions form layers that are in the hcp (hexagonal close-packed, ABABAB) or ccp (cubic close-packed, ABCABC) structure. In the Oh-site the Fe3+
ion is surrounded by six O2- ions occupying the corners of an octahedron. In the Td- site the Fe3+ ion is located in the centre of tetrahedron where the three O2- ions lie in one plane and the remaining O2- ion is found in the plane above [58, 59]. The space group and lattice parameters of individual phases are summarized in Table 2.1, the hyperfine parameters determined from Mössbauer Spectroscopy (MS) in Table 2.2 and the atomic positions of individual phases can be found in Appendix (Table 7.1- 7.6).
The crystal structure, magnetic properties and the hyperfine parameters determined from MS of individual phases will be now discussed in more details.
Figure 2.8: The scheme of the formation and phase transformation between the individual phases of the iron(III) oxides.
23
Table 2.1: Lattice parameters together with the space group of the iron(II,III) oxide phases:
Phase Structure Space
group
Lattice parameters
a (Å) b (Å) c (Å) (°) (°) (°)
-Fe2O3 Trigonal1 R-3c 5.040 5.040 13.750 90.0 90.0 120.0
-Fe2O3 Cubic2 Ia3 9.398 9.398 9.398 90.0 90.0 90.0
-Fe2O3 Cubic3 Fd3m 8.350 8.350 8.350 90.0 90.0 90.0
Cubic4 P4332 8.345 8.345 8.345 90.0 90.0 90.0
Tetragonal5 P41212 8.346 8.346 25.034 90.0 90.0 90.0
-Fe2O3 Orthorhombic6 Pna21 5.072 8.736 9.418 90.0 90.0 90.0
Fe3O4 Cubic7 Fd3m 8.396 8.396 8.396 90.0 90.0 90.0
Monoclinic8 P21/c 5.944 5.924 19.775 90.0 90.2 90.0 Comments: The PDF2 database card no. 1 892810; 2 830112; 3 00-022-1086 (PDF4 database); 4 895892; 5 802186; 6 897047; 7 892355; 8 ref. [60].
Table 2.2: The Mössbauer parameters of the iron(II,III) oxide phases at selected temperatures in zero applied field: the isomer shift, ; the quadrupolar shift, 2 and the hyperfine field, Bhf:
phase Interpr. T (K) (mm.s-1) 2 (mm.s-1) Bhf (T)
-Fe2O3 [56] > TM 295 0.37 -0.20 51.8
< TM 4 0.49 0.40 54.1
-Fe2O3 [56] Td + Oh 300 0.32 0.02 50.0
Fe3O4 [56] Td 300 0.26 0.02 49.0
Oh 300 0.67 0.00 46.0
-Fe2O3 [60] Oh1 + Oh2 300 0.37 -0.25 44.5
Oh3 300 0.38 -0.03 38.7
Td4 300 0.23 -0.15 25.6
-Fe2O3 [62] d-sites 300 0.36 0.69 -
b-sites 300 0.37 0.90 -
2.4.1 Hematite, -Fe
2O
32.4.1.1 Crystal structure
The hematite phase is isostructural with the well-known mineral, corrundum (- Al2O3) that crystallizes in the rhombohedral structure with the space group R-3c. The O2- ions stacked along the [001] axes form the hcp planes and create the Oh-sites for the Fe3+ ions that occupy only 2/3 of these Oh-sites. The crystal structure is depicted in Figure 2.9, where the hcp structure of the oxygen anions is shown.
24
Figure 2.9: The crystal structure of the hematite phase, the blue circles denotes the Fe3+ ions, the small black dots represents the O2- ions, the hcp structure of the O2- ions is marked on the right.
2.4.1.2 Magnetic structure
The hematite is magnetically ordered below the Néel temperature, TN = 955 K at which the Fe3+ ions get ordered in the so-called weak ferromagnetic state (wf). In this state, the spins are oriented in the basal (001) plane perpendicular to the c-axis with minor spin canting from this direction resulting in very small magnetic moment, [63]. At the so-called Morin temperature, TM = 260 K the magnetocrystalline anisotropy changes its sign resulting in the spin flop transition of the towards the c- axis ([001] direction). Below the TM the
are oriented strictly antiferromagnetically with respect to each other, the hematite is in the antifferomagnetic (af) state. The TM
observed at 260 K for the monocrystal can be lowered by defects, lattice strain, substitution by paramagnetic ions (Al3+ ions) and decreasing the particle size. It can be even totally suppressed in the case of NPs with sizes smaller than 20 nm [56].
2.4.1.3 Mössbauer Spectroscopy
The MS of the hematite phase consists of one resolved sextet at room temperature.
The principal axis of electric field gradient (EFG) lies in the [001] direction, therefore the rotation of the spins into the direction of the EFG axis at TM causes a significant change of the quadrupolar shift, 2 (see Table 2.2) with the increase of Bhf. The typical MS are depicted in Figure 2.10.
Application of the sufficient high external magnetic field, app
B (higher than 6 T) below TM can induced the reorientation of the spins from the [001] direction to the energetically favorable wf state. In single crystals depending on the direction of the
25
app
B with respect to magnetic easy axis (c-axis), either the spin flop transition or screw rotation occurred [64, 65]. In the powder samples the orientation of the c-axis is random with respect to the app
B , therefore only part of the sample undergoes the spin flop or screw rotation. Contrary results have been published, observing the change of the line width [66] or occurrence of both the af and wf states [65].
Figure 2.10: The Mössbauer spectra above and below the Morin temperature, TM on the left with the simple image of the orientation of the spins on the right.
2.4.2 Maghemite, -Fe
2O
32.4.2.1 Crystal structure
The maghemite crystallizes in the spinel structure that is named after the mineral spinel (MgAl2O4). The general formula of the structure can be written as A2+B3+2O4, where A and B correspond to the metal ions. Generally, the spinel structure consists of the 32 oxygen anions with 1 Oh-site and 2 Td-site per one O2- ion. However, only 1/8 of Td-sites and ½ of Oh-sites are occupied by metal ions [30, 67].
Depending on the value of the crystal field stabilization energies of the divalent and trivalent ions in the individual sites, two boundary possibilities of the occupation of the sites are obtained: the so-called normal and inverse spinel structure, respectively [30].
In the normal spinel structure, all divalent ions are located in the Td-sites while all trivalent ions occupy the Oh-sites. In the inverse spinel structure, the divalent ions preferentially occupy the Oh-sites and the trivalent ions are accommodated in the rest
