Single-Particle Phenomena in Magnetic Nanostructures

Single-Particle Phenomena in Magnetic Nanostructures

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures David S. Schmool1, Hamid Kachkachi PROMES, CNRS-UPR 8521, Universite´ de Perpig...

9MB Sizes 143 Downloads 169 Views

ARTICLE IN PRESS

Single-Particle Phenomena in Magnetic Nanostructures David S. Schmool1, Hamid Kachkachi PROMES, CNRS-UPR 8521, Universite´ de Perpignan Via Domitia, Rambla de la Thermodynamique—Tecnosud, Perpignan, France 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Theoretical Aspects of Single Nanoparticles 2.1 Models and Approaches: A Representation of a Nanoparticle 2.2 Computing Methods 2.3 A Sample of Results: Finite-Size and Surface Effects 2.4 Magnetic Excitations and Reversal 3. Overview of Experimental Results on Single-Particle Measurements 3.1 Micro-SQUID and Nano-SQUID 3.2 Micro-Hall Magnetometry 3.3 Lorentz Transmission Electron Microscopy 3.4 Magnetic Force Microscopy 3.5 X-Ray Magnetic Circular Dichroism (XMCD) and X-Ray Photoemission Electron Microscopy (X-PEEM) 3.6 Ferromagnetic Resonance 3.7 Magnetic Resonance Force Microscopy 4. Summary References

301 309 310 328 350 355 363 364 369 373 377 382 391 399 406 409

1. INTRODUCTION Nanoscaled magnetic systems are the subject of much current research in magnetism. This includes research on the magnetic properties and behavior of both individual magnetic nanoelements and arrays thereof, with special emphasis on the interplay between their intrinsic and collective properties. The novel features of nanomagnets, as compared to bulk systems, can be categorized as being due to one, or a combination, of the following

Solid State Physics ISSN 0081-1947 http://dx.doi.org/10.1016/bs.ssp.2015.06.001

#

2015 Elsevier Inc. All rights reserved.

301

ARTICLE IN PRESS 302

David S. Schmool and Hamid Kachkachi

effects: (i) confinement or finite-size effects, (ii) surface effects, and (iii) interactions between nano-objects in assemblies of nanoparticles and their interactions with the hosting medium. In this chapter, we will mainly be interested in the magnetic properties related to nanostructures, with an emphasis on single-element magnetic nanostructures and particles, leaving the study of their collective behavior for a subsequent work. With regard to magnetic nanosystems, confinement effects are strongly related to surface effects, since the surface acts as a discontinuity of the magnetic object and this defines the boundary conditions of the system. These are of particular importance especially when considering magnetic excitations or magnons. Surface effects in magnetism are generally related to the disruption to the magnetic body at the discontinuity that constitutes a surface. This means that the local symmetry is broken and results in a number of effects, which are both structurally related and due to the alteration of the number of nearest neighbors between which exchange interactions occur on the atomic scale. In terms of the crystalline structure, the presence of a surface means that the bonding between neighboring atomic sites changes and frequently affects the nearest neighbor distances. Such a relaxation of the structure can also produce surface reconstructions, allowing atoms to take up new positions with respect to the normal bulk crystalline lattice. Alternatively, a disordered region can be formed on the outer surface and can act as a shell. This also depends on the nature of the surrounding medium in which the nanoparticle is located. Both the effect of changed interatomic separation as well as the difference in the nearest neighbor coordination at a surface can be expected to produce a modification of the magnetic structure of the body in the region of the surface and the ensuing magnetic disorder may propagate inside the system. Such changes are particularly important with respect to magnetic anisotropies. Such considerations will be discussed in detail in Section 2.1.2. Confinement effects are typically considered in physical systems with respect to charge carriers, such as electrons, in the system. An example would be the quantum confinement of electrons in quantum wells, where the limited electron momenta produce discrete energy states, whose allowed energies depend on the size of the structure and the boundary conditions, or electron penetration characteristics, of the system. In magnetic systems, the boundary conditions and size confinement act in a similar way to define the allowed standing wave or magnon states of the system. Such considerations are extremely important, for example, in ferromagnetic resonance (FMR) experiments, also more generally referred to as standing spin-wave

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

303

resonance. Due to the size relation of a magnetic object and the allowed spin-wave wave vectors, below a certain dimension standing spin-wave modes are not usually observed. For this reason, we will not explicitly discuss these issues here. In nanomagnetism, there are two main characteristic length scales that are of importance: the domain wall width and the exchange length. In the simplest case of a ferromagnetic material with uniaxial anisotropy, we can express the domain wall width (here we consider a 180° Bloch wall) as [1, 2] rffiffiffiffi A (1) δDW ¼ 2 K where A represents the exchange stiffness constant and K the uniaxial anisotropy constant. The exchange stiffness constant is related to the exchange energy as well as to a geometric factor accounting for the crystalline structure of the material and gives a measure of the strength of the exchange interaction between neighboring spins, which tries to maintain their magnetic moments parallel to one another. The anisotropy constant is also dependent on the samples’ crystalline structure and arises from spin–orbit interactions. It provides a measure of this interaction and the strength to which the magnetization is maintained along its “easy axis.” The domain wall width represents the distance over which the transition of spin orientation occurs from one magnetic domain to the next. The exchange length has a similar formula to the domain wall width and is expressed as rffiffiffiffiffiffiffiffiffiffiffi 2A Λexch ¼ : (2) μ0 Ms2 The denominator in the square root is related to the magnetostatic energy. An inspection of Eqs. (1) and (2) shows that these characteristic lengths are related to the (squareroots of the) ratios of the exchange energy and the magnetocrystalline anisotropy energy or the magnetostatic energy, respectively. They are obtained by the minimization of the energies involved with respect to distance. As long as a ferromagnet has dimensions greater than those given by Eqs. (1) and (2), the magnetization configuration can form multidomain patterns to reduce the energy of the ferromagnet. Once the dimensions of the magnetic body are reduced to those of the characteristic length scales (or below), only single-domain magnets can be produced. This means that the size and shape of the magnetic element or particle can be used to tailor their magnetic properties and behavior. A consideration of the magnetostatic and

ARTICLE IN PRESS 304

David S. Schmool and Hamid Kachkachi

anisotropy energies can be used to estimate the size of the magnetic particle for it to be a single magnetic domain. In the simplest case of a spherical particle with uniaxial anisotropy, this can be expressed as the critical diameter [2] pffiffiffiffiffiffiffiffi 9π AK Dcr ¼ μ0 Ms2

(3)

This predicts sizes of around 7 nm for Fe, 14 nm for Co, and 16 nm for Ni, but can be much larger in high anisotropy materials such as NdFeB and on the order of hundreds of nm. For nonspherical particles, shape anisotropy can also play a significant role, creating monodomain particles for diameters larger than Dcr. Dynamical properties of magnetization have received increased attention in recent years and in particular with regard to the switching times available for magnetic systems. Indeed, developments in dynamical methods, and especially with regard to ultrafast laser systems, have dominated recent research in ultrafast magnetic processes [3–8]. The dynamics of magnetization in confined magnetic systems, such as nanosystems, can be severely modified with respect to the bulk situation, and this is a major concern for many researchers in the field. Such changes can be expected due to the modification of the magnetic properties of the system, especially with regard to the magnetic anisotropies that govern the magnetic nano-object. We will discuss these effects in more detail in Section 2. In practical terms, most applications of magnetic nanosystems employ assemblies of nanoparticles or elements since the magnetic signature is extremely weak for isolated magnetic entities. In such respects, magnetic interactions are almost always of practical concern when we consider an assembly of nano-objects. In such cases, the measured properties depend on the magnetic interactions averaged over the whole assembly of nanoparticles. In general, when there are several magnetic objects, interactions are always present; however, for very dilute systems, these interactions can be negligible. The separation, spatial orientation, and the particle-size distribution are the main parameters which control the strength of the interactions, though the mechanism of interaction also depends on the intervening material between the magnetic objects. The nature of nanostructures depends intimately on the method used for their fabrication. We can broadly categorize the type of nanostructure as being either a random array of randomly oriented (in a crystallographic sense) nanoparticles or a regular array of identical nanostructures. For

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

305

randomly oriented NPs, the magnetization (or particle magnetic moment) will align along one of the easy (local) axes with respect to its crystalline structure. However, no orientational correlation is to be expected to exist between the NPs themselves. For nanostructured samples, shape anisotropies will be aligned and if they are epitaxial with a substrate, then the magnetization of the various nanoelements will also be aligned. In either case, there are many options on the fabrication techniques available. Nanoparticle assemblies can be produced via a number of processes. For example, gas aggregation sources work on the principle of cluster formation in temperature- and pressure-controlled cells via a sputtering process. The average size and size distribution can to a certain extent be controlled by these parameters. Typically, a chopper is used to narrow the range of sizes to a more controlled size distribution width [9–11]. The particles are usually deposited onto a substrate. Co-deposition of another atomic species allows nanoparticle arrays to be prepared in a solid matrix. Further control of the relative deposition rates determines the particle concentration. This is a relatively sophisticated method of producing random arrays of nanoparticles. Much cheaper and simpler methods are available using chemical routes, such as the Brust and related methods [12–15]. Such chemical synthesis of nanoparticles has shown the power of chemistry and astounded many physicists as to the level of sophistication in both the size and distribution control, as well as the wealth of shapes that are possible to routinely fabricate [14, 16, 17]. The nanoparticle formation process can be characterized, for example, with iron nanoparticles, where typically the synthesis occurs by the thermal decomposition of an iron stearate precursor in a high boiling point solvent in the presence of oleic acid molecules. Variation of the synthesis parameters, such as the nature of the solvent and the temperature profile, produces different size distributions of particles [17, 18]. This type of nanoparticle synthesis has the advantage of producing nonagglomerated nanoparticles since they have an oleic coating and form highly stable suspensions in organic solvents. It is further possible to make core-shell heterostructured nanoparticles using a multistep chemical process [19]. Selective coverage of a core nanoparticle, which is nonspherical in form, is also possible, as in the case of goldtipped Co nanorods [20]. A simple oxidation of the particle can also produce oxide coatings on many types of magnetic nanoparticle [21]. An alternative method of fabricating random arrays of nanostructures is via the growth of discontinuous films, i.e., before coalescence and percolation. The form of the particles depends on the wetting conditions (substrate/overlayer), as well as the growth temperature and deposition rates used. This produces

ARTICLE IN PRESS 306

David S. Schmool and Hamid Kachkachi

a two-dimensional array of particles. Three-dimensional arrays can be obtained via the alternate growth of a discontinuous layer with a thicker layer of a different material, e.g., [CoFe/Al2O3]n [22, 23]. Regular arrays of nanostructures typically require more complex sample fabrication methods. Commonly lithographic methods are employed for such purposes. Such two-dimensional arrays require a pattern transfer which contains the structure shape and periodicity of the final array. Photoresists are usually used since they offer a very reliable method of transferring the pattern structure from a mask or template onto a wafer. The resolution or minimum feature size depends strongly on the type of photoresist used and the nature of the radiation used for exposition to define the nanostructure through a mask. For nanostructures, this typically means X-rays or electrons. Depending on whether the photoresist is a positive or negative type, after pattern transfer into the photoresist, etching or lift-off processes are employed to further transfer the pattern into a more stable metallic form. Three-dimensional structures can be produced via the multistage process with several steps of lithography, though this requires accurate mask alignment and processing. With electron beam lithographies (EBLs), since the electron beam can be precisely controlled (via electric or magnetic fields), it is also possible to write directly into the photoresist. The best resolutions that can be attained with such lithographic methods are of the order of 10–20 nm [24–26]. Interference lithography is an adaptation of the normal lithographic method, in which the interference of a coherent (laser) source is used to expose the photoresist. This eliminates the need for a mask in the lithography. Such techniques can produce well-defined periodic structures, the periodicity of which can be expressed as: p ¼ λ=ð2 sinθÞ, where λ expresses the wavelength of the radiation used and θ depends on the angle of incidence of the radiation. More complex structures can be formed by multiple exposures, for example, by rotating the sample between expositions. Using similar interferometry techniques in conjunction with photoactive polymers, or azopolymers, it is possible to make surface relief structures, which are periodic and sinusoidal in profile. This can be used to dispense with more complex lithographic processing for certain types of structure [27, 28]. The optical, X-ray, and electron forms of lithography while producing well-defined structures can be limited in the areas of arrays they can produce. Nanoimprint lithography may overcome some of the shortfall. As a technique, the principle of nanoimprint lithography is very simple. Essentially, a nanostructured mold is used to stamp the nanostructure into the resist. The

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

307

master mold is usually manufactured from a hard material, usually via EBL or XBL. The mold can be reused for a simple a rapid throughput. Structures of around 25 nm have been prepared using this technique. At the other end of the spectrum, focused ion beam (FIB) provides a method for the fabrication of nanostructures with a unique or nonperiodic form. FIB uses a beam of ionized Ga atoms, which are accelerated with sufficient energy onto a surface as to eject atoms from that surface via momentum transfer. Gallium is used for a number of reasons, mainly because it is liquid near room temperature and is a convenient source of ions which are relatively heavy (i.e., good for momentum transfer). The ionized Ga beam can be focused much in the same way as an electron beam in a scanning electron microscope (SEM). Milling can then be controlled by focusing on rastering the Ga+ beam to form the desired structure. It is possible to program the motion of the Ga beam via computer so as to make well-defined structures. In addition to its milling capabilities, FIB can also be used for selective growth via assisted dissociation of metal precursors injected near to a surface. The Ga+ beam selectively dissociates a precursor molecule with a metallic component; the precursor is removed from the surface leaving behind the metal ion at the surface which forms the nanostructure in the region where the Ga+ beam impinges the surface. A further advantage of FIB technologies is that it can also be used to image a surface. By using a low-energy beam, the Ga+ ions can excite secondary electrons from a surface, which can be captured in exactly the same way as in the SEM, thus providing a direct image of the nanostructure fabricated by the FIB [29, 30]. The resolution of the FIB apparatus depends on the beam focus, and this is strongly influenced by the working conditions and beam energy. Best resolutions are in the tens of nanometers for very low beam currents. While having the advantage of direct writing abilities, FIB is a specialized tool, producing well-defined nanostructures, but is slow and expensive. Low-cost fabrication of quasi-regular arrays of nanostructures can be found using alumina membranes. A porous aluminum oxide membrane can be produced via anodic oxidation [31]. The size, shape, and separation of the regularly spaced circular pores can be controlled by setting the experimental conditions: electrolyte, temperature, anodizing potential, and time. Usually, a two-step anodization is used to obtain the best-quality template structures, which is a hexagonal arrangement of the pores. It is possible to make rather long pores, with diameters in the range 2–500 nm with densities of 109–1012 pores/cm. These can be produced over relatively large areas, up to around 10 cm2. Once a template has been produced, it is possible to

ARTICLE IN PRESS 308

David S. Schmool and Hamid Kachkachi

deposit a film on top, to produce an antidot array or to deposit material within the pores. In the latter case, this can be done in a variety of ways to deposit on the inner walls and then inside to produce a core-shell cylinder [31], or via alternately depositing different materials to form a multilayered nanowire [32]. In any experimental study of magnetic nanostructures, a chemical, structural, and morphological characterization is essential. A vast array of techniques are employed for such purposes, such as X-ray diffraction, electron microscopies, scanning probe microscopies (STM and AFM), and electron spectroscopies (EELS, AES, UPS, XPS, etc.). This is usually before any magnetic characterization is performed. It is not our aim to outline such techniques. In this chapter, we will consider the magnetic properties and behavior of (ferro)magnetic nanoparticles and nanostructures. Of particular interest are the control and variation of their magnetic behavior as a function of their physical dimensions. Indeed, as we discussed above, the simple control of physical size of an object can be used to alter, modify, and even tailor their subsequent physical properties. Intrinsic physical properties of materials typically depend on the chemical species of atom and their spatial arrangement. Size reduction changes the atomic environment of a significant portion of the atoms in a nano-object and hence changes their fundamental physical properties. This can also bring about temporal/thermal instabilities in particular with regard to magnetic properties. In this first part of our review, we are principally concerned with the manner in which such processes occur and ways in which to experimentally measure them and theoretically study them. Isolated magnetic nanostructures are considered to be nano-objects with no interactions with other magnetic bodies. However, the surrounding environment can also influence the physical behavior of the nanostructure, though for the most part we will not be concerned here with such effects. In a future work [33], we will consider the effects of interactions between magnetic objects, with particular emphasis on dipolar interactions. We will review the static and dynamic magnetic properties of such ensembles of nano-objects as well as providing a general overview of the theoretical approaches and the experimental methods most commonly used in their study. This will include a review of the important work on the theoretical modeling of nanoparticle assemblies as well as an overview of experimental studies using magnetometries, ac susceptibility, ferromagnetic resonance, small-angle neutron spectroscopy, and M€ ossbauer measurements.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

309

We are well aware of the enormous development in magnetic materials and research in magnetism over the past two to three decades. Much of the progress in magnetic materials has been made in areas related to nanostructures and nanomaterials, such as magnetic thin films and multilayers, magnetic nanoparticles, and nanostructured arrays of objects. Parallel with the development in materials and sample preparation techniques, there has been tremendous progress in the development of new techniques and the enhancement of existing methods, tailored specifically to the study of nanomagnets. Such a wealth of sophisticated scientific methods has allowed much to be learned of the nature of materials and magnetism in general. Furthermore, new theoretical methods and models have been elaborated, partially in reply to the progress in experimental work but also as a means to spur on experimental development. In this work, we aim to give a general overview of our current understanding of single-particle magnetism from both a theoretical background and an experimental overview. The sheer extent of research in these areas means that we cannot hope to provide an exhaustive account of the research in this field, but we hope it provides a general introduction for both the newcomer and the veteran researcher.

2. THEORETICAL ASPECTS OF SINGLE NANOPARTICLES The study of nanoscaled spin systems was conducted over the last few decades on polydisperse assemblies of nanoparticles with randomly distributed anisotropy easy axes [34]. The main experimental techniques used are M€ ossbauer spectroscopy, VSM magnetometry, and FMR measurements rendering the magnetization and susceptibility of the nanoparticle assembly as functions of temperature, applied ac/dc field, and time. Even though these observables are averages over the whole assembly, they did help investigate the global properties of the system both at equilibrium and out of equilibrium and constitute a relatively efficient means to characterize the system with respect to its physical parameters such as the effective anisotropy, coercive fields, relaxation time, “mean blocking temperature,” and so on. However, as the size of the nanoparticles is reduced, partly in order to meet the requirement of higher densities in various technological applications, these experiments started to reveal limits as to the interpretation of the observed phenomena. It was quickly realized that small particles show new features related with their enhanced surface effects and that the latter were rather difficult to resolve from the macroscopic observables, mostly because the (intraparticle or intrinsic) surface effects are smeared out by the assembly

ARTICLE IN PRESS 310

David S. Schmool and Hamid Kachkachi

collective effects and by the averaging procedure of the experimental probe. In order to access the particle’s intrinsic features, such as the local effective fields and spin configuration, some experimental groups have dedicated their research to single-particle physics and to the development of adequate measuring techniques with better space-time resolution. Despite the enormous progress that has been achieved in this regard, up to now, no experimental technique available today is able to probe a single nanoparticle and to measure its properties fully independently from the hosting medium. Nevertheless, theoretical activity has been conducted for single isolated particles as well as on nanoparticle assemblies. In the absence of unambiguous experimental evidence, single-particle calculations have been performed on the basis of models mostly borrowed from the physics of thin films. In particular, this applies to the anisotropy models and exchange energy parameters for spins on the particle’s surface. The main reason for this is that the spatial arrangement of the surface atoms is not known and thereby no atomic quantum calculations [35, 36] have been performed on clusters of reasonable sizes. More precisely, surface anisotropy in magnetic nanoparticles has been modeled with the help of Ne´el’s model or using a uniaxial anisotropy with an easy axis that is either normal or parallel to the surface. We will now try to take stock of what has been learned on the equilibrium and dynamic behavior of single particles. For simplicity, to illustrate the main ideas we consider a cluster of N spins cut from a cube of side N (i.e., N  1 atomic spacings) with simple cubic lattice structure. Of course, several shapes and lattice structures have been studied. Due to the underlying (discrete) lattice structure, the particle thus obtained is not a sphere with smooth boundaries because its outer shell presents apices, steps, and facets, resulting in many sites with different coordination numbers. This is assumed to mimic to some limit the real situation of a magnetic cluster grown by chemical methods or etched by lithographic means.

2.1 Models and Approaches: A Representation of a Nanoparticle Here we give a brief account of the models and approaches that have been in use within the community of nanomagnetism for computing the equilibrium and dynamic properties of magnetic nanoclusters. There are two main approaches: • the so-called macrospin approach, leading to a one-spin problem (OSP), which is a macroscopic approach that models the magnetic state of a

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

311

nanoparticle using its net magnetic moment of constant magnitude and with two continuous degrees of freedom (the polar and azimuthal angles). This approach is based on the assumption of coherent rotation of all atomic magnetic moments. It is exemplified by the Stoner– Wohlfarth [37, 38] model for equilibrium properties and the Ne´el– Brown model [39–43] for dynamics [44, 45]. • Many-spin particle (MSP), leading to a many-body problem. This is a microscopic approach that involves the atomic magnetic moment with continuous degrees of freedom as its building block. It takes account of the local environment inside the particle, including the microscopic interactions and single-site anisotropy [46]. This approach becomes necessary in small nanoparticle with a number of surface atoms that exceeds those in the core. There is also an intermediate approach regarding the scale of the building block. This is the coarse-grained approach of micromagnetism [47, 48] where the magnetic system is organized into a (3D) array of cells each containing a certain number of atomic spins that depends on the meshing procedure and computational power. Today, there are several ready-to-use packages such as μMAG (Micromagnetic Modeling Activity Group), OOMMF (Micromagnetic Modeling Tool), and μMax (GPU-accelerated Micromagnetic Modeling Tool). This technique is mostly suited to the study of domain structures and their dynamics in thin magnetic films. Indeed, as the computing power continues to increase, the size of the cell decreases, but at the same time the size of nanoparticles also decreases. This means that nanoparticle remains the building blocks of micromagnetic calculations, but stacked in rather dense arrays. Another limitation of micromagnetics is that the temperature dependence of the cell magnetization has to be introduced in some ad hoc manner, for instance, by using meanfield theory. 2.1.1 Macrospin Approach In small magnetic particles, with a set of appropriate physical parameters, it is believed that exchange interaction may be strong enough to hold all spins tightly parallel to each other and prevents spatial dependence of the particle’s magnetization. Since the seminal work of Stoner and Wohlfarth (SW) [37] and up to now, a simplified model is used which consists in representing the magnetic state of a nanocluster by its (macroscopic) net magnetic moment with of course redefined (effective) anisotropy contributions, including magnetocrystalline anisotropy and shape as well as surface anisotropies. This

ARTICLE IN PRESS 312

David S. Schmool and Hamid Kachkachi

macroscopic model is what is commonly known as the macrospin approach, or what we called above the OSP, in analogy with the MSP. Obviously, in this approach one cannot take account of the local atomic environment within the nanoparticle and is quite sensible for studying the overall behavior of the nanomagnet in an applied magnetic field. In addition, at finite temperature, one uses this model, now referred to as the Ne´el–Brown model [39, 41, 42, 49], to investigate the effect of thermal fluctuations and to compute the relaxation rate of the nanomagnet related with its switching between its various macrostates. These aspects will be discussed at length below. Ignoring spatial inhomogeneities, the exchange energy is a constant and plays no role in the energy minimization. In the latter, only the anisotropy and Zeeman energies are relevant. Consequently, the Hamiltonian reads H¼

KV ðM  eÞ2  ðgμB ÞH  M: M2

(4)

This is the macroscopic model with an effective uniaxial anisotropy constant K and axis e. V is the volume of the particle. Upon writing M ¼ Ms, H ¼ Heh, and introducing the dimensionless anisotropy and field parameters σ¼

KV ðgμ ÞHM , h¼ B , kB T 2KV

(5)

the energy in (4), measured with respect to thermal energy (β  1/kBT), becomes   (6) βH ¼ σ ðs  eÞ2 + 2hs  eh : Let us consider the case of a longitudinal field, i.e., eh ke kez, and denote by θ the angle between s and the field direction eh. Then, Eq. (6) simplifies to   (7) βH ¼ σ cos 2 θ + 2h cosθ : For simplicity, we only consider the case of an easy axis, i.e., σ < 0. For σ > 0 (easy plane), the results are the same only that the energy maximum and minimum are interchanged. The extrema of H and their nature are obtained by setting to zero its first derivative with respect to θ and evaluating the second derivative at these extrema. The results are presented in Fig. 1. Thus, for jhj < 1 the energy has minima at θ ¼ 0 and θ ¼ π, separated by a maximum at θs ¼ arccosðhÞ. For jhj > 1, the upper (shallower) energy minimum (θ ¼ π for h > 0) turns into a maximum as it merges with the intermediate maximum at θs, which disappears with increasing h (see Fig. 1).

ARTICLE IN PRESS 313

Single-Particle Phenomena in Magnetic Nanostructures

Field

Minima

Maxima

|h| < 1

q = 0,p

q = arccos(–h)

h>1

q=0

q=p

h < –1

q=p

q=0

b H(q ) 0.5 0.0 –0.5 h=0 –1.0 h = 0.25 –1.5 –2.0 h = 0.75 –2.5 0.00

0.25

0.50

0.75 q/p

1.00

Figure 1 Magnetic energy in the case of a longitudinal field for some values of the (reduced) field h in Eq. (5). We see that upon increasing the field, the number of potential wells changes from two to one.

From the values of the energy at θ ¼ 0, π and, when it exists, at the intermediate maximum θs, one obtains the energy-barrier heights (jhj < 1) ΔH ¼ β½Hðθs Þ  Hð0,πÞ ¼ σð1  hÞ2 :

(8)

The analysis presented above helps solve for the hysteresis curve, an important observable in magnetism, for small ferromagnetic particles. Such calculations are known as the SW model [37, 38]. In their original study, Stoner and Wohlfarth considered shape anisotropy instead of uniaxial anisotropy. However, their model is usually applied to the latter. In addition, they also considered the more general situation of a field applied at an arbitrary angle ψ to the easy axis. From the above analysis, we see that there is a unique minimum for jhj > 1, while there are two minima for jhj < 1. This is due to the multivaluedness of the trigonometric functions entering the energy (7) and its derivatives with respect to θ. In order to obtain a unique solution, one has to specify and follow the history of the

ARTICLE IN PRESS 314

A

David S. Schmool and Hamid Kachkachi

B hy

m

1.0

1.0 Stoner–Wohlfarth Our calculation

y = 0° y = 60°

0.5

y = 85°

0.5

y = 90°

0.0

0.0

–0.5

–0.5

–1.0 –1.0 –1.0

–0.5

0.0

0.5

h

1.0

–1.0

–0.5

0.0

0.5

hx

1.0

Figure 2 Left: (numerical) hysteresis loops for different values of ψ, the angle between the applied magnetic field and the easy axis, increasing inward: ψ ¼ 0, 60°, 85°, 90°, for a 33 particle with uniaxial anisotropy. Right: (numerical in squares and analytical in full line) Stoner–Wohlfarth astroid for the same particle; j ¼ 10. Source: Reprinted with permission from Ref. [50]. Copyright (2002), AIP Publishing LLC.

field h for each angle ψ. One usually starts at saturation and increases or decreases the field across zero until it reaches the value at which the energy barrier disappears. This field marks the limit of metastability and is called the critical field or the saturation field. In SW model, this is given by hSW ¼

1 ð cos 2=3 ψ + sin 2=3 ψ Þ

3=2

:

(9)

For example, for ψ ¼ 0 we have hSW ¼ 1. One can also define what is called the switching field that is the field at which the net magnetization, or the projection of net magnetic moment on the field direction, changes sign. Figure 2 (left) shows the hysteresis loop at different angles ψ, and on the right (in full line) the angular dependence of the switching field, the so-called SW astroid, as obtained from the SW macrospin model. 2.1.2 Many-Spin Approach For nanoparticles of a diameter less than 10 nm (e.g., of cobalt), more than 50% of its atoms are located on its surface (see Fig. 3). The surface of a nanoparticle may undergo lattice reconstructions and atomic rearrangement which in turn leads to a crystal-field symmetry breaking with strong local inhomogeneities (see Fig. 4).

ARTICLE IN PRESS 315

Single-Particle Phenomena in Magnetic Nanostructures

Number of ∝ 1/R Total number

1/Radius

Figure 3 Increasing number of surface spins and effects on the spin configuration.

Figure 4 Surface atoms for a quasi-spherical particle (left) and an icosahedral (right) nanoparticle.

In order to account for such features and study their effects on the magnetic properties of such systems, one has to use a microscopic approach that involves the atomic magnetic moments with continuous degrees of freedom as its building block. This approach has to be able to take account of the local environment inside the particle, including the microscopic interactions and single-site anisotropy [46]. Consequently, this amounts to adopting a manyspin approach where the nanoparticle is considered as a crystal of N atoms with associated intrinsic physical properties.

ARTICLE IN PRESS 316

David S. Schmool and Hamid Kachkachi

Therefore, in this approach one considers a nanoparticle as a crystallite of N atomic spins. The prototype model Hamiltonian is the (classical) anisotropic Dirac–Heisenberg model [46, 51, 52] X Jij si  sj + HZ + Han , H¼ (10) hi, ji where si is the atomic unit spin vector (ksik ¼ 1) on site i and Jij the nearestneighbor exchange coupling which may be ferromagnetic or antiferromagnetic and whose nominal value depends on the link i $ j. HZ ¼ ðgμB ÞH 

N X si

(11)

i¼1

is the Zeeman energy of interaction of the magnetic field H with all spins si. Finally, Han is the single-site anisotropy energy. Nanoscaled magnetic systems are finite-size systems which exhibit the additional surface effects. As such, one should distinguish, at least from a conceptual point of view, between finite-size, boundary, and surface effects. To illustrate the difference between the finite-size and boundary effects, let us consider the simplest, though somewhat unrealistic, cluster grown or cut out of a simple cubic (sc) lattice. For periodic boundary conditions (pbc), there is only one environment (crystal field) with coordination number z ¼ 6. In this case, the temperature behavior of the magpffiffiffiffiffi netization is marked by the well-known M  1= N tail in the critical region. With free boundary conditions (fbc), a cube with sc structure yields four different environments with z ¼ 3, 4, 5, 6. Since the system is of the same size, the contribution of finite-size effects to the magnetization is the same, but there are now the additional boundary effects due to the inhomogeneous local environment. In addition, boundary effects induce stronger fluctuations that suppress the magnetization of the system. To make this discussion more quantitative, let us summarize part of the work in Ref. [53] where it was shown that the magnetization can be written in a simple form. Indeed, at low temperature and zero-field magnetization M deviates from 1, its (normalized) saturation value, according to θ M ffi 1  WN , 2

(12)

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

317

where WN ¼

1 X0 1 , N k 1  λk

(13)

with, for a three-dimensional (d ¼ 3) sc lattice, λk ¼ ð cos kx + cos ky + coskz Þ=d.It is important to note that WN in (13) for pbc differs from that for fbc only by the definition of the discrete wave vectors. Indeed, we have [53, 54] 8 2πnα > < , pbc (14) kα ¼ πnN , nα ¼ 0,1, …, N  1 > : α , fbc N where α ¼ x, y, z. This subtle difference is responsible for much stronger thermal fluctuations in the fbc model due to boundary effects. The difference between (i) the sums WN (with finite N, which is the side of the cube) and (ii) the so-called Watson’s integral (for bulk systems) W¼

R d3 k 1 , ð2πÞ3 1  λk

(15)

with W ¼ 1.51639 for d ¼ 3, describes the finite-size effects for the pbc case [55, 56] and boundary effects in the fbc case [53]. Indeed, this difference is given by [53] 8 0:90 > , pbc, WN  W < N ffi (16) > W : 9 lnð1:17N Þ , fbc: 2πWN Therefore, this shows that the coefficient in the linear θ term in Eq. (12) is smaller than in the bulk for the pbc system and greater for the fbc system. This implies that the boundary effects suppress the magnetization, while finite-size effects enhance it. Moreover, boundary effects render a larger (in absolute value) contribution than finite-size effects, making the net magnetization smaller than that of the bulk. Now, in real systems finite-size and boundary effects cannot be disentangled as they both coexist in a nanoscaled magnetic system. From a purely formal point of view, we can separate, as shown above, these two kinds of effects for box-shaped systems. Still formally, we can attribute some energy

ARTICLE IN PRESS 318

David S. Schmool and Hamid Kachkachi

parameters (e.g., coupling and anisotropy constants) to the atoms at the boundary of the system that are different from those in its core. Then, we can investigate, for instance, the additional effects of surface anisotropy on the magnetic state. Therefore, we have finite-size effects because the system is not a bulk system, boundary effects because the boundary conditions are not periodic, and surface effects because the physical parameters on the boundary are different from those in the core. However, in reality boundary effects cannot be disentangled from surface effects because as soon as boundary effects are present, there is a breaking of crystal field symmetry that induces inhomogeneous local effective fields. Nevertheless, this somewhat formal distinction between the various effects allows us to assess the role of each contribution in the onset of a given magnetic state as the parameters of the system are varied. Let us now discuss the details of the various models that have been used for the anisotropy contribution to the energy of a nanomagnet. In Eq. (10), Han is usually taken to be the uniaxial single-site anisotropy energy X Han ¼  Ki ðsi  ei Þ2 , (17) i

with easy axis ei and constant Ki > 0. If site i is located in the core, the anisotropy axis ei is taken along some reference z axis and Ki ¼ KC. In fact, KC is the effective anisotropy constant and ei is the easy axis of the effective anisotropy that is usually assumed to include the particle’s shape anisotropy. For nanoparticles grown out of a magnetic material with cubic anisotropy, the term Han may also comprise a cubic contribution. Altogether, in the absence of experimental data, the anisotropy constant KC and easy direction are often assumed to be the same as those of the underlying bulk material (see discussion at the end of Section 3.6). For surface spins, the anisotropy is also considered as uniaxial with a constant Ki ¼ KS and an easy axis that is taken along the radial direction (i.e., transverse to the cluster surface) for the so-called Transverse Surface Anisotropy (TSA). Several works have also considered the same model with Ki < 0, i.e., with an easy axis that is tangential to the surface (see Fig. 5). The transverse e 2n e 1n

Figure 5 Transverse Surface Anisotropy model.

ARTICLE IN PRESS 319

Single-Particle Phenomena in Magnetic Nanostructures

direction is given by the gradient (the vector perpendicular to the isotimic surface ψ ¼ constant defining the shape of the particle, e.g., a sphere or an ellipsoid), because in the case of a spherical particle the transverse and radial directions coincide, whereas for another geometry such as an ellipsoid they do not. A more physically appealing model of surface anisotropy was introduced by Ne´el [57] with eel HN an ¼

zi KS X X ðsi  uij Þ2 , 2 i j¼1

(18)

where zi is the coordination number of site i and uij ¼rij/rij is the unit vector connecting the site i to its nearest neighbors (see Fig. 6). This model is more realistic since the anisotropy at a given site occurs only when the latter loses some of its neighbors, i.e., when it is located at the boundary. The model in Eq. (18) is referred to as the Ne´el Surface Anisotropy (NSA) model. Qualitatively, the NSA model is not very different from the TSA model. For example, consider a site i lying on a [100] facet, e.g., in the upper most plane normal to the z axis. It has four neighbors on that facet and one in the lower adjacent plane perpendicular to the z axis. From Eq. (18), we infer the corresponding energy (upon using ksik ¼ 1) ¼ 2KS  KS s2i, z : HNSA i This implies that if KS > 0 the easy direction is along ez, i.e., normal to the facet, and if KS < 0 the facet becomes an easy plane. Therefore, upon dropping the irrelevant constant, we rewrite the above energy as HNSA ¼ KS ðsi  ez Þ2 i which is the same as the TSA in Eq. (17) for the site considered. More generally, averaging the NSA over a surface perpendicular to the direction n leads to (see Eqs. (6, 7) in Ref. [58])

i uij

Figure 6 Néel Surface Anisotropy model.

j

ARTICLE IN PRESS 320

David S. Schmool and Hamid Kachkachi

  2 2 2 HNSA ¼ K jn js + jn js + jn js S x y z i x y z , thus favoring the direction closest to the surface normal. This explains the similarity between the results obtained with TSA and NSA. Using the components jnαj, α ¼ x, y, z, the atomic surface density reads [58]   (19) f ðnÞ ¼ max jnx j, jny j,jnz j , and thereby we have the specific cases 8 nz ¼ 1 KS s2z , > > > > pffiffiffi > 1 < KS ðs2x + s2y Þ= 2, nx ¼ ny ¼ pffiffiffi NSA Hi ¼ 2 > > p ffiffi ffi > 1 > > nx ¼ ny ¼ nz ¼ pffiffiffi : : KS = 3, 3 For comparison, taking account of the atomic surface density, the TSA model can be described by ¼ KS ðn  si Þ2 f ðnÞ HTSA i

(20)

and the whole effect comes from the atomic surface density. Quantitatively, in the NSA model the effect is bigger, since for the surface cut perpendicular to the body diagonal of the cubic lattice the anisotropy completely disappears, whereas in (20) the surface anisotropy is only reduced by the factor pffiffiffi 1= 3. A more detailed comparison between NSA and TSA and their effects on the magnetic state and hysteresis cycles of a magnetic nanocluster can be found in Ref. [59]. For magnetic particles of finite size, there are two types of magnetization, m and M. First, we define the vector magnetization (per spin) M¼

1X si N i

(21)

whose thermodynamic average yields what we call induced magnetization m ¼ hMi ¼

1X hsi i: N i

(22)

Obviously, this magnetization vanishes for finite-size systems in the absence of a magnetic field due to the Goldstone mode corresponding to

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

321

the global rotation of the magnetization (21). On the other hand, it is clear that at very low temperatures the spins in the system are aligned with respect to each other and thereby there exists an intrinsic magnetization. The latter is usually defined for finite-size systems as v* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 +ffi q ffiffiffiffiffiffiffiffiffiffiffi u u 1X (23) : M¼ M2 ¼ t si N i If the temperature is low, all spins in the particle are bound together by the exchange interaction and M behaves as a rigid “giant spin,” jMj ffi Mffi1. If a magnetic field H is applied, M exhibits an average in the direction of H which leads to a nonzero value of the induced magnetization m. The relation between the induced and intrinsic magnetizations is frequently written in the form of what is called the superparamagnetic relation, m ¼ MBD ðMxÞ,

x  N H=kB T,

(24)

where BD(x) is the Langevin function [BD ðxÞ ¼ B3 ðxÞ ¼ cothx  1=x] for the isotropic Heisenberg model and BD ðxÞ ¼ B1 ðxÞ ¼ tanhx for the Ising model, D being the number of spin components. A very important question concerns the field dependence of M at nonzero temperatures, which can be responsible for deviations from the simple superparamagnetic behavior of Eq. (24). This question was considered at length in Refs. [53, 54]. It was shown that this relation cannot be used to account for the field dependence of M in the range where the argument of B is of order 1 or less. This formula can only be correct in the case of large particles for which the change of M in this field range is very small, and M actually changes for much larger fields where we already have m ffi M. However, we have shown that this relation is a good approximation everywhere below the critical temperature in the limiting cases of isotropic (Heisenberg) and infinitely anisotropic (Ising) systems. Experimentally, the field dependence of M and, in particular, the nonsaturation of the magnetization in the region x ≳ 1 have been observed in nanoparticles by different groups [60–62]. Usually, this dependence is close to linear and is used to extract the value of M at zero field by extrapolation to H ¼ 0. For the isotropic Heisenberg model, the field dependence of M in the range x ≳ 1 is due to suppression of the fluctuations of individual spins, i.e., spin waves, and this dependence disappears for T ! 0. The dependence M(H) is much stronger and persists at zero temperatures if the spins in

ARTICLE IN PRESS 322

David S. Schmool and Hamid Kachkachi

the particle are not perfectly collinear due, for instance, to surface effects. There exists a more general relation between the induced and intrinsic magnetizations which is valid at any temperature, field, and anisotropy. This is given by M 2 ¼ m2 +

dm ðD  1Þm + : dx x

(25)

Indeed, by rewriting m2 and M2 as follows N N X D D E

1 X 1 X α α , ¼ s  s s s i j i j 2 2 N i, j¼1 N i, j¼1 α¼1 N N X D D E

1 X 1 X 2 m ¼ sαi sαj , hsi i  sj ¼ 2 2 N i, j¼1 N i, j¼1 α¼1

M2 ¼

we obtain M 2  m2 ¼

N X D hD 1 X 2

N i, j¼1 α¼1

N X D E D Ei 1 X 1 αα sαi sαj  sαi sαj ¼ 2 χ ij , β N i, j¼1 α¼1

(26)

where β ¼ 1/kBT and the susceptibility tensor χ ij reads in its most general form [63] αβ α β α β χ αβ ij ¼ χ k ni nj + χ ? ðδij δ  ni nj Þ:

(27)

nαi is the α-component of the unit vector n ¼ h/h on site i, and δij and δαβ are Kronecker symbols for the lattice sites and Cartesian coordinates, respectively. Then noting that χk ¼

dm m , χ? ¼ , dx x

we arrive at Eq. (25). 2.1.3 Crossover from a Many-Spin to a Macrospin Approach With the desire to avoid the difficulties inherent to the MSP approach, especially when dealing with dynamics, one may address the question as to whether there exist some cases in which the full-fledged theory that has been developed for the OSP approach (see Ref. [64] and references therein) can still be used to describe an MSP (see Fig. 7). In Ref. [58], it was shown that when the surface anisotropy is much smaller than the exchange coupling,

ARTICLE IN PRESS 323

Single-Particle Phenomena in Magnetic Nanostructures

Under what conditions OSP

MSP

m

KS

Figure 7 Crossover from MSP to OSP.

and in the absence of core anisotropy, the surface anisotropy contribution to the particle’s energy turns out to be of fourth order in the net magnetization components and second order in the surface anisotropy constant (see Eqs. 33, 34 below). This means that the energy of an MSP with relatively weak surface anisotropy can be modeled by that of an OSP whose effective energy contains an additional cubic-anisotropy potential. In Ref. [65], it was found that because of surface anisotropy the minimum defined by the core uniaxial anisotropy splits into four minima, reminiscent of cubic anisotropy. The energyscape can be modeled by an effective energy of the net particle’s magnetization containing a uniaxial- and cubic-anisotropy terms, and this result holds for the two models of surface anisotropy (TSA and NSA). In order to illustrate this, let us consider a spherical particle of radius R with uniaxial anisotropy in the core and TSA on the surface. Then, to compute the contribution of surface anisotropy in Eq. (10), we replace the number of nearest neighbors of a surface atom by its average value   ziα ) ziα ¼ 2  jnα j= max jnx j, jny j, jnz j , (28) where nα is the α-component of the normal to the surface n. The surfaceenergy density can then be obtained by multiplying the contribution in   Eq. (10) by the surface atomic density f ðnÞ ¼ max jnx j,jny j, jnz j , leading to Eq. (20). At equilibrium, in the continuous approximation the Landau– Lifshitz equation reduces to m Heff ¼ 0,

Heff ¼ HA + JΔm,

(29)

where Δ is the Laplace operator and the anisotropy field, HA, contains contributions from the core and the surface HA ¼ 

δEC δES  δðr  RÞ, δm δm

1 R  ðN  1Þ, 2

where N is the side of the cube inside of which the particle is cut.

(30)

ARTICLE IN PRESS 324

David S. Schmool and Hamid Kachkachi

For KS ≪ J, the deviations of m(r) from the homogeneous state m0 are small and one can linearize the problem as follows mðrÞ ffi m0 + ψðr, m0 Þ ¼ m0 + e1 ψ 1 + e2 ψ 2 , ψ  jψj ≪ 1: Then, the correction ψψ is the solution of the Helmholtz equation with boundary conditions [66]

1 Δ  k2α ψ α ¼  H? SA  eα , α ¼ 1, 2, J  2KC  2 2KC 2 2 k1 ¼ 2m0z  1 , k22 ¼ m0z , J J

  δES ðm0 Þ δES ðm0 Þ ? HSA ¼   m0 m0 δðr  RÞ, δm δm

(31)

where n  r/R. The solution ψ of Eq. (31) may be written in the form [58] Z 1 ψðr, mÞ ¼ d 2 r0 Gðr, r0 Þf ðm, n0 Þ (32) 4π S where Gðr, r0 Þ is the Green’s function of the problem. In the absence of core anisotropy (KC ¼ 0), where the Helmholtz equation reduces to a Laplace have recently found [66] an approximate expression for the Green’s function which contains a correction to Gðr, r0 Þ of the order k2α ∝KC =J ≪ 1. This correction turns out to be a convolution of two G(0)’s. Collecting all contributions from the core and surface, we showed that the effective energy of an MSP particle, in the absence of magnetic field, is written as E eff ¼ K2 m2z + K4 ðm4x + m4y + m4z Þ:

(33)

The coefficient K2 of the second-order contribution is in fact the result of two contributions, one stemming from the initial core uniaxial anisotropy and a new contribution that is induced by the surface anisotropy (see below). The latter contribution is much smaller than the former because its coefficient contains the product (KC/J)(K2S/J) ≪ 1. The fourth-order coefficient K4 was found to be [58] ð0Þ

ð0Þ

K4 ¼ κ 2

N KS2 , zJ

(34)

where KS, z, and J are, respectively, the surface anisotropy constant (transverse or Ne´el), the coordination number, and the exchange coupling of the

ARTICLE IN PRESS 325

Single-Particle Phenomena in Magnetic Nanostructures

ð0Þ

many-spin particle. κ2 is a surface integral that depends on the underlying lattice, the shape, and the size of the particle and also on the surfaceanisotropy model. For a spherical particle (of  1500 spins) cut from a simple ð0Þ

cubic lattice with Ne´el’s surface anisotropy, κ2 ’ 0:53465. In Ref. [65], we confirmed this result by numerical calculations of the field behavior of the net magnetization and effective energyscape. Since we are dealing with an MSP, the energyscape cannot be represented in terms of the coordinates of all spins. Instead, we may represent it in terms of the coordinates of the particle’s net magnetization. For this purpose, we fix the global or net magnetization, m, of the particle in a desired direction m0 (jm0j ¼ 1) by using the energy function with a Lagrange multiplier (see discussion in Ref. [58]). More generally, taking account of all contributions from the core anisotropy, surface anisotropy (see Fig. 8), and Zeeman energy, one obtains the extended effective model for a nanocluster of a given shape [67], whose energy can be written as ð0Þ

EEOSP ¼ K4

X α

m4α + C0 G0 ðmÞ + Ch Gh ðmÞ

where   4 1 2 2m2z 2 4 m  3=4 m + m mz 1  m2z + x y , 1  m2z z # 2"  m2 m2 m2 1 4 x y z Gh ðmÞ ¼ ðeh  mÞ, mx + m4y + m4z + 1  m2z 2

G0 ðmÞ ¼

and KS (NSA, TSA)

KC

MSP

Figure 8 A many-spin particle with both core and surface anisotropy.

(35)

ARTICLE IN PRESS 326

David S. Schmool and Hamid Kachkachi

! ! 3z2 Ns KC 3zhe S ð1Þ , Ch  K 4 C0  K4 ð4π Þ2 J0 2ð4π Þ2 ð1Þ

with Ns being the number of surface spins, S ¼ 4πR2 the surface area, J0 ¼ zJ the Fourier component of the exchange coupling, he ¼ μaH/J, and ð1Þ  K4 ¼ κ ð21Þ VKS2 =J0 with Z       R5 ð1Þ d 2 rd 2 r0 Gð1Þ ðr, r0 Þ jnx j  ny  n0x   n0y  : κ2 ¼ 4π S Therefore, these calculations show that if the spin noncollinearities are not too strong, one can model a many-spin nanoparticle with the help of an effective energy that comprises several contributions which are functions of the components mx, my, mz of the particle’s net magnetic moment: 1. Core contribution: uniaxial, cubic, etc. This contribution is, in fact, normalized (modified) by the other contributions due to spin noncollinearities. 2. Pure surface contribution:   ð0Þ ð0Þ (36) E 2 ¼ K4 m4x + m4y + m4z : 3. Shape contribution: For example, for an ellipsoid with axes a and b ¼ a(1 + E), E ≪ 1, we obtain ð0Þ

ð0Þ

E 1 ¼ K1 m2z ,

ð0Þ

K1  KS N

1=3

E

(37)

which scales with the volume. This shape contribution stems from surface effects and should not be confused with the magnetostatic contribution that is usually included in the effective uniaxial anisotropy in 1. 4. Core–surface mixing terms (in zero field): relevant only if KC ≳ Ks2 =J ð1Þ

ð1Þ

E 2 ¼ K2 G0 ðmÞ,

ð1Þ

ð0Þ

K2 ¼ κ 2 ðKC Ns Þ

KS2 : J2

(38)

which comprises both second- and fourth-order contributions. In Ref. [68], other shapes and crystal structures were considered and the numerical results were favorably compared to the analytical expressions given above based on an extended effective model. From the results of Refs. [65] and [68], we can assess the validity of the effective model, or equivalently the weakness of spin misalignment, in terms of the surface anisotropy constant KS ¼ KS/J. More precisely, one finds that

ARTICLE IN PRESS 327

Single-Particle Phenomena in Magnetic Nanostructures

the larger the coordination number of the underlying lattice the larger the critical value of KS below which the EOSP (effective OSP) is applicable. For example, for a simple cubic lattice with z ¼ 6, KS ’ 0.25 and for a facecentered cubic lattice with z ¼ 12, KS ’ 0.35. The MSP and OSP models have been used previously by the nanomagnetism community for studying the equilibrium and dynamic properties of nanomagnets. There have been several discussions in the literature as to which is the most appropriate. In fact, these two approaches are complementary and their relevance depends on whether one is interested in the overall behavior or in the fine details. Indeed, the MSP is mostly used for investigating fundamental issues related with finite-size and surface effects, when these become nonnegligible (see Fig. 9), while the OSP is widely used for interpreting most of the experimental results, especially on dynamics, because the Arrhenius exponential law (69) turns out to dominate the global behavior [48]. In fact, the subtle fine features predicted by the MSP cannot yet be observed in experiments because of the general lack of sufficient space-time resolution discussed earlier. Nevertheless, several experimental results on assemblies of nanoparticles [60, 62, 69–71], with better controlled properties, have revealed new features such as spin canting and noncoherent switching of the magnetization which are at variance with the hypotheses of the OSP. From the purely technical point of view, the OSP is obviously much easier to handle, especially for investigating the magnetization switching, induced by a dc magnetic field, and/or thermal fluctuations. This is one of the main reasons for which the majority of experimentalists consider almost exclusively the OSP when interpreting their results, even in situations where the validity of this approach starts to be questionable. The intermediate EOSP approach comes as a compromise, though it is still unclear how to disentangle the quartic anisotropy contribution induced by Example: cobalt particles Multidomain (MD) MD

45 nm

Single domain OSP Dominating core m

OSP: One-spin problem MSP: Many-spin problem

Figure 9 Size regime for MSP and OSP.

10 nm

Diameter

MSP Dominating surface

ARTICLE IN PRESS 328

David S. Schmool and Hamid Kachkachi

surface effects and the cubic magnetocrystalline anisotropy. In Ref. [72], the authors studied the limits of the macrospin model in Co nanodots, with an enhanced magnetocrystalline anisotropy at the edge. It was concluded that the most important deviation from the OSP is due to the fourth-order anisotropy constant, in agreement with the effective OSP.

2.2 Computing Methods As for bulk systems, one can either adopt a (semi-)analytical approach such as mean-field theory or spin-wave theory and accept to work in limited ranges of the physical parameters (temperature, field, couplings) or a numerical approach such as the Monte Carlo technique or numerical solution of the equation of motion of the magnetization, namely the Landau–Lifshitz equation augmented with the help of Langevin’s approach. These methods, initially built for bulk systems, have to be adapted with special care before they can be applied to nanoscaled spin systems. The major change that is induced by the nanoscale is that the energy barriers are drastically reduced leading to a nonnegligible probability for the magnetization to shuttle between the various energy minima, the so-called superparamagnetic effect. This means that similarly to the atomic spins si, the macroscopic moment M of the nanoparticle undergoes strong fluctuations in a magnetic field and/or under heating. Consequently, this effect has to be taken into account in any approach used for studying the behavior of the nanoparticle. 2.2.1 Spin-Wave Theory In order to compute the spin-wave spectrum and other spectral characteristics, one computes the spin fluctuations with respect to the equilibrium configuration. As such, one has to determine the latter beforehand. In practice, one assumes that it exists and then determines its direction afterward. For nanoscaled systems, for which the energy barriers are very low, thus allowing for a change in direction of the net magnetic moment on rather short time scales, one assumes that there exists a net direction of the particle’s magnetization denoted by n so that M¼

1X si  M n: N i

Then, spin-wave theory describes the spin fluctuations around this direction n. One may use the Holstein–Primakoff representation (HPR) [73] of the spin variables upon passing to the new coordinate system in which the z

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

329

reference direction is now the direction n. This amounts to performing a rotation of the original variables si to the new ones σi around a given axis and of a given angle depending on n. Then, one uses the HPR for the new variables σi. Alternatively, one may write the spin variable si as a sum of its component along n (the longitudinal component) and its transverse component. That is si ¼ ðsi  nÞn + ½si  ðsi  nÞn:

(39)

Next, we define the new basis vectors e1 ¼

ez n , kez nk

e2 ¼ n e1 ,

(40)

and we rewrite (39) as si ¼ s1i e1 + s2i e2 + s3i n:

(41)

It is clear that once the direction n has been determined, the vectors e1 and e2 can be immediately obtained from Eq. (40). Next, one uses the HPR for the new spin components sα, α ¼ 1, 2, 3. It is, however, usually more 1 2 convenient to employ the raising and lowering variables s i ¼ si  isi . Classical spin waves At low temperatures, all spins in the particle are strongly

correlated and they form a “giant spin” M which behaves superparamagnetically. In addition, there are internal spin-wave excitations in the particle which are responsible for the fact that M < 1 and for the field dependence of M at nonzero temperatures. In the case of three dimensions, these excitations can be described perturbatively by small deviations of the individual spins si from the “central” spin M defined in Eq. (21). It is convenient to insert an additional integration over dM ¼ MD1dMdn in the partition function (see Fig. 10) R

Z¼ M

D1

dM dn

Y i

! 1X dsi δ M  si eH=T , N i

(42)

and first integrate over the magnitude M of the central spin. To do this, one should reexpress the vector argument of the δ function in the coordinate system specified by the direction of the central spin M, which yields

ARTICLE IN PRESS 330

David S. Schmool and Hamid Kachkachi

Global rotation mode M

Spin-wave (SW) modes Si

Figure 10 Local rotation of atomic spins supplemented by a global rotation of the net magnetic moment.

! ! ! 1X 1X 1X δ M si ¼ δ M  ðn  si Þ δ ½si  nðn  si Þ : N i N i N i Then after integration over the amplitude M, one obtains [54] R Z ¼ dnZ n ,

(43)

with " #  N 1 ðD1Þ=2 E0 + N n  h D1 ðπT Þ Z n ffi exp , N detA0ij T

(44)

where the matrix A0ij is obtained from   1 ðD  1ÞT 1 + n  h + zi J δij  Jij Aij  2 N 2 by elimination of the uniform mode. The value of det A0ij in Eq. (44) depends on the form of the particle and the type of boundary conditions. For a cubic particle, N ¼ N 3 , with periodic boundary conditions (pbc), one can use the Fourier representation and write Y0 1 2N ¼ , 0 detAij ðD  1ÞT=N + n  h + J0  Jk k

(45)

where the mode with k ¼ 0 is dropped in the product and the wavevectors k are quantized according to Eq. (14).

ARTICLE IN PRESS 331

Single-Particle Phenomena in Magnetic Nanostructures

Then, using these developments, one obtains explicit forms of the field dependence of the intrinsic magnetization M in three different field regions [54] 8

2 2 > D1 hN > > , h ≪ hV , cN > > J0 2D > > > < D  1 NhT hV ≪ h ≪ hS , cN 2 , M ffi1t + (46) 2 J0 > > > >

 > > D  1 T h 1=2 > > : , hS ≪ h ≪ J0 , c0 2 J0 J0 where t ≪ 1 is defined by D  1 WN T , t 2 J0

ðD  1ÞcN α 4N 2

2 T : J0

(47)

Here, D ¼ 3 is the number of spin components. For the simple cubic lattice, c0 ¼ (2/π)(3/2)3/2. N is the linear size of the particle and cN is a lattice sum which evaluates to (for N ≫ 1)  0:384  1:05=N , pbc cN ffi 1:90  1:17=N, f bc and hV ¼

T π 2 J0 : ≪ hS ¼ 2d N 2=3 N

In the second and third field ranges of Eq. (46), the particle’s magnetic moment is fully oriented by the field, thus m ffi M, and the spin-wave gap in Eq. (45) has the value h, as in the bulk. The region h ≪ hV in Eq. (46) is less trivial. Here the gap in Eq. (45) is n  h and depends on the orientation of the particle’s magnetic moment which is not yet completely aligned by the field. Indeed, one has in this region n  h  h2, which leads to a quadratic field dependence of M. Quantum spin-wave theory for finite temperature and arbitrary spin In order to

investigate the spectral properties of a spin system with an arbitrary spin (S) and at finite temperature, one can use the general formalism of Green’s functions (GF) for the spin operators themselves and not for the magnon creation and annihilator operators a, a+. Indeed, in the GF approach the

ARTICLE IN PRESS 332

David S. Schmool and Hamid Kachkachi

Holstein–Primakoff representation is not necessary and one deals directly with the spin components s, s3 themselves, or the rotated ones. So, instead of using the transformation in the form (41), we may use a matrix form where the transformation is a rotation Rðey , ψÞ of an angle ψ around the axis ey. To illustrate this method, we consider the Hamiltonian (10), in a reduced form and with uniaxial anisotropy, E ¼

N N X X 1X 2 λij si  sj  h ðsi  eh Þ  k szi : 2 i, j i¼1 i¼1

(48)

Now we replace the spin variable si by the new one σ i related by si ¼ R1 ðey , ψÞσi ¼ Rðey ,  ψÞσi with the 3D representation for Rðey , ψÞ 0 1 cos ψ 0 sin ψ (49) Rðey , ψÞ ¼ @ 0 1 0 A: sin ψ 0 cosψ Then, using the components σ μ, μ ¼ , 3, and their commutation relations h i σ i+ , σ  ¼ 2δij σ 3i , j h i (50) σ 3i , σ μj ¼ μσ μi δij , μ ¼ , inferred from the SO(3) Lie algebra h i sμi , sνj ¼ iEμνλ sλi δij ,

(51)

we can recast the Hamiltonian in the following general quadratic form E¼ 

N N X X X 1X σ μi Qijμν σ νj  L μ σ μi 2 i, j¼1 μ, ν¼ + , , 3 i¼1 μ¼ + , , 3

(52)

where the linear coefficients are given by L+ ¼

h R , 2

L ¼

hR+ , 2

with hR ¼ Rðey , ψÞh,

L 3 ¼ h3R ,

(53)

ARTICLE IN PRESS 333

Single-Particle Phenomena in Magnetic Nanostructures

while the quadratic coefficients read  k 2 1 sin ψ δij ¼ Qij , Qij+  ¼ λij + k sin 2 ψδij ¼ Qij+ , 2 2 2 3+ 3 3 ¼ λij + 2k cos ψδij , Qij ¼ Q+3 ij ¼ k sin ψ cos ψ δij ¼ Qij ¼ Qij

Qij++ ¼ Qij33

(54) or in matrix form 0

1  1 2 k sinψ cos ψ δ + k sin ψδ λ ij ij ij C B 2 B C  C: 1 k 2 Qij ¼ B B λij + k sin 2 ψδij k sinψ cos ψ δij C sin ψ δij @2 A 2 2 k sin ψ cos ψ δij k sinψ cos ψ δij λij + 2k cos ψδij k sin 2 ψ δij 2

Note that the lattice inversion symmetry Qijμν ¼ Qjiμν applies here. Now we define the retarded Green’s function Gμν ðri  rj , tÞ 

DD

EE Dh iE Siμ ðtÞ;Sjν ð0Þ ¼ iθðtÞ Siμ ðtÞ, Sjν ð0Þ , μ, ν

¼ + ,  ,3:

r

Next, using the equations of motion of these Green’s functions [74], one obtains a system of coupled equations whose solution leads to the magnon dispersion relation 2 ðℏωðkÞÞ2 ¼ ½L 3 + kσ ð2 cos 2 ψ  sin 2 ψ Þ + zσ ð1  γ k Þ  k2 σ 2 sin 4 ψ (55)

P where γ k ¼ δ eik  δ =z. Then, one may compute the magnetization and ac susceptibility and various other observables written in terms of the Fourier transform of the retarded Green’s functions. In Ref. [75], the spin-wave spectrum of a nanoparticle was calculated using this technique with a detailed analysis of the decoupling procedure for anisotropy contributions, and several approaches were compared. The thermal effects on the spin-wave spectrum and its consequence for the magnetization, as a function of field and temperature, are studied. Moreover, the crossover from the quantum regime to the classical regime is rigorously established.

ARTICLE IN PRESS 334

David S. Schmool and Hamid Kachkachi

2.2.2 Equations of Motion In the case of weak coupling to the environment, the dynamics of a classical spin si can be described by the stochastic Landau–Lifshitz, or Landau– Lifshitz–Langevin (LLL), equation 1 dsi fl eff , ¼ si ðheff i + hi Þ + λsi si hi γ dt

(56)

with λ ≪ 1 being the (dimensionless) phenomenological damping parameter that measures the magnitude of the relaxation term relative to the gyromagnetic term. The deterministic effective field is defined by heff i ¼

δH , δsi

(57)

which in general comprises the applied magnetic field, the effective field of exchange and dipolar interactions, and the anisotropy field stemming from magnetocrystalline anisotropy in the core or surface anisotropy, depending on the locus of the atom i in the particle. In Eq. (56), heff i is augmented by a fluctuating or stochastic field hfli , accounting for the effects of interaction of si with the microscopic degrees of freedom of the surrounding medium (phonons, conducting electrons, nuclear spins, and so on), which causes fluctuations of the spin orientation. These environmental degrees of freedom are also responsible for the damped precession of si, since fluctuations and dissipation are related (through the fluctuation–dissipation theorem) manifestations of one and the same interaction of si with its environment. The heat bath contains a large number of such microscopic degrees of freedom with equivalent statistical properties, and by the central limit theorem, they give rise to fluctuations of the spin orientations which are Gaussian distributed. Consequently, the field hfl ðtÞ is a Gaussian stochastic process with the following properties [76, 77] D E fl

hα ðtÞ ¼ 0, hflα ðtÞhflβ ðt 0 Þ ¼ 2Cd δij δðt  t 0 Þ, (58) where α, β ¼ x, y, z refer to the Cartesian components of hfl and Cd is the diffusion coefficient and measures the strength of thermal fluctuations. It is related to temperature as follows Cd ¼ λ

kB T , γ

(59)

where kB is the Boltzmann constant. The brackets h…i in (58) denote the average over different realizations of the fluctuating field, with a realization

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

335

being the field hfl as a function of time. The Dirac function δðt  t 0 Þ expresses that above certain temperature, the autocorrelation time of hfl (of microscopic scale) is much shorter than the time of rotational response of the system (“white” noise or absence of memory effect), while the Kronecker symbol δij expresses the fact that the different components of hfl are assumed to be uncorrelated. Moreover, it is also assumed that the fluctuating fields acting on magnetic moments at different sites are statistically independent. Note that one may also add a fluctuating field in the relaxation term in (56), but since the thermodynamic consistency of this approach entails that jhflj  λ1/2, this term is negligible in the weak damping case. This means that the fluctuating and relaxation terms are decoupled. Equation (56) is a system of 3N coupled equations for the Cartesian coordinates sαi , α ¼ x, y, z (or 2N coupled equations for the spherical coordinates θi, φi) and can only be solved with the help of an efficient numerical algorithm. There are hundreds of works that employ this technique (see, for example, Refs. [76–81]). Instead of dealing with the equation of motion of the magnetic moments themselves, one can also consider the equation of motion of the probability distribution of spatial orientation of these magnetic moments. Therefore, instead of solving the stochastic Landau–Lifshitz equation (56), in some special cases one may prefer, for certain technical reasons, to solve the associated Fokker–Planck equation governing the time evolution of the nonequilibrium probability distribution of magnetic moment orientations that was originally derived for nanoparticles by Brown [41, 42]. The Fokker– Planck equation corresponding to Eq. (56) is formulated for the distribution function PðN,tÞ ¼ hδðN  SÞi on the sphere jNj ¼ 1 (see Refs. [76] and [82]; see also the textbook by Risken [83] for the general formalism and the one by Coffey et al. [64] for the Langevin approach to the theory of Brownian motion). It reads 

 γλkB T @P @ @ eff eff N N ¼ γN h  γλN N h + P: @t @N μ0 @N (60) The stationary solution of this equation is the Boltzmann distribution P0 ∝ exp ðβHÞ:

(61)

For small coupling to the bath, λ ≪ 1, Eq. (60) coincides with the Fokker–Planck equation derived by Brown [41, 42].

ARTICLE IN PRESS 336

David S. Schmool and Hamid Kachkachi

The Fokker–Planck equation (60) can be solved by various numerical methods, but the most efficient of all is the so-called Matrix Continued Fraction Method (MCF) [64, 83–85]. Solving the LLL equation is a very versatile technique that can deal with multivariate systems and is thus well suited for investigating equilibrium and dynamic behavior of many-spin systems. However, this method inherently includes rather time-consuming subroutines that are necessary for (i) generating sequences of arrays of stochastic numbers and (ii) computing averages over sufficiently large ensembles of time spin trajectories. On the other hand, solving the Fokker–Planck equation requires writing a hierarchy of equations that depend on the energy potential, which means that this procedure is somewhat model dependent. Moreover, the technique is limited in practice to a small number of degrees of freedom, since otherwise this hierarchy is too cumbersome to write and rather costly to solve numerically. As such, and as far as magnetic nanoparticles are concerned, this technique has been applied to a maximum of two particles in the OSP approach [86]. Now, the choice of the computing method depends on the observable of interest and on the approach considered. For equilibrium properties, the hysteresis loop and the switching field (see Fig. 2), for instance, are computed by numerically solving the deterministic Landau–Lifshitz equation (56) without the Gaussian field. Within the OSP approach, Eq. (56) is in fact a system of two (three) coupled equations in the system of spherical (Cartesian) coordinates, whereas within the MSP this leads to a system of 2 N (3N ) coupled equations. However, in all cases the numerical procedure is quite straightforward and uses standard routines such as the Euler, Heun, or Runge–Kutta methods [50, 52].

2.2.3 Ferromagnetic Resonance: A General Formalism Instead of studying the full spectrum of magnetic excitations, one may concentrate on the uniform modes and investigate the ferromagnetic resonance (FMR) characteristics (resonance frequency and field). Indeed, the FMR technique has proven to be one of the most precise methods for characterizing a magnetic material with regard to its physical parameters such as anisotropy and interactions. At very low temperatures, one can ignore damping effects related with thermal fluctuations and consider the time evolution of each spin si given by the Landau–Lifshitz equation (56) without the Gaussian field, i.e.,

ARTICLE IN PRESS 337

Single-Particle Phenomena in Magnetic Nanostructures

1 s i ¼ si heff , i + λ si si heff , i , γ

(62)

where heff,i is the effective field given by Eq. (57) which can be rewritten as the gradient of the system’s (total) energy E heff , i ¼ 

δE  ri E: δsi

We may investigate the nonequilibrium situation when the spin si devið0Þ

ð0Þ

ð0Þ

ates from its equilibrium state si ðθi ,φi Þ. To do so, we write ð0Þ

δsi ¼ si  si , for i ¼ 1,…,N , and linearize the LLE (62) about this equilibrium state. Accordingly, we expand the first derivative of E i (or the effective ð0Þ

field) around the equilibrium state si . To proceed, we introduce the notation fs1 , …, sN g  fsg: Then, we use the vector Taylor expansion X1 ½ða  rr Þn ψ ðrÞ, ψðr + aÞ ¼ n! n 0 to obtain the first-order expansion of the effective field around the equilibrium state " # N n o n o n o X ð0Þ ð0Þ (63) + ðδsk  rk Þheff , i sð0Þ : heff , i s + δs ¼ heff , i s k¼1

Inserting this into the LLE (62) and dropping vanishing (at equilibrium) and second-order terms, we obtain (in spherical coordinates) ðδsk  rk Þðri E Þ ¼ Hik δsk , where 0

1 1 2 @θk φi E B C sin θi Hik ðEÞ  @ 1 A 1 2 2 @φk θi E @φk φi E sinθk sin θk sin θi @θ2k θi E

is the (pseudo-)Hessian of E. Therefore, the linearized LLE becomes δs i ¼ γ

N X k¼1

Hik I λ δsk , i ¼ 1,…, N ,

(64)

ARTICLE IN PRESS 338

David S. Schmool and Hamid Kachkachi

where

Iλ ¼

 0 λ , I 2λ ¼ λ2 2 2 : λ 0

(65)

Then for a solution of the form δsk ¼ δsk(0) eiωst, we obtain the eigenvalue problem N X ½Hik I λ δsk ð0Þ ¼ Ωik ðλÞI λ δsk ð0Þ ¼ iωs δik δsk ð0Þ

(66)

k¼1

with Ωik ðλÞ ¼ iωs δik I 1 λ ¼ iωs

1 δik I λ : 1 + λ2

There are mainly two types of FMR experiments in which one either fixes the applied field and varies the frequency to find the resonance frequency, or fixes the frequency and varies the applied (dc) field to find the resonance field. These two situations are captured in Eq. (66), which is an eigenvalue problem whose characteristic polynomial is a function of the applied field. Accordingly, for a given set of physical parameters (applied field, anisotropy constants, and interaction couplings), there are two procedures: either (i) we compute the eigenvalues of the matrix ½Hik I λ αβ and thereby obtain the resonance spectrum by selecting the pure imaginary eigenvalues, or (ii) for a fixed frequency ωs we find the roots of the characteristic polynomial corresponding to (66) and obtain the resonance fields. From the technical point of view, the first case is much easier to tackle. In a work in progress, using this technique we also compute the eigenvectors and study the corresponding spin-wave modes. Projecting the wave vectors on the core and surface contributions, we investigate the weight of the core and the surface of a nanoparticle in a given mode. This helps understand how to tune the two contributions upon varying the physical parameters of the nanoparticle, such as size and microscopic energy parameters. 2.2.4 Monte Carlo Simulations The Monte Carlo technique is ubiquitous in all areas of science, since it can virtually be used to solve any problem. In physics, it has been applied in all

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

339

areas. The main reason for this universality is that it is first quite straightforward to implement and second it renders, up to statistical fluctuations, the exact solution of the problem once the model has been well defined. For spin systems, the classical Monte Carlo (MC) method based on the Metropolis algorithm is now a standard technique (see, e.g., Ref. [87] for details). The general idea is to simulate the statistics of a magnetic system by generating a Markov chain of spin configurations and taking an average over the latter. Each step of this chain (an MC step) is a stochastic transition of the system from one state to another, subjected to the condition of the detailed balance. Usually, an MC step consists in generating a new trial orientation of a spin vector on a lattice site i and calculating the ensuing energy change of the system ΔE. The trial configuration is accepted as a new configuration if exp ðΔE=T Þ Rð0,1Þ

(67)

where Rð0,1Þ is a random number in the interval (0, 1), otherwise the old configuration is maintained. As follows from Eq. (67), for ΔE 0 the trial orientation is accepted with a probability 1. The trial orientation can be a completely random orientation, or a random orientation in the vicinity of the initial orientation of the spin si, which is more appropriate at low temperatures. For the Ising model, the trial orientation is generated by a flip of si with a probability 1/2. The MC steps are performed sequentially or randomly for all lattice sites. This conventional version of the MC method is not efficient for systems of finite size at low temperatures and small fields, if one is interested in the induced magnetization m. The Boltzmann distribution over the directions of the particle’s magnetic moment M of Eq. (21) is achieved by rotations of M itself rather than by rotations of individual spins si. Indeed, P each spin si is acted upon by the strong exchange field HE, i ¼ j Jij sj  J0 , and in the typical case H ≪ J0, all trial configurations with the direction of si significantly differing from that of its neighbors are rejected with a probability close to 1. Thus, in the standard MC procedure directions of individual spins can only change little by little, and the resulting change of M is extremely slow. For the Ising model, the situation is even worse since the spin geometry is discrete and there are no small changes of spin directions, whereas a flip of a single spin against the exchange field has an exponentially small probability. Hence if one starts in zero field with the configuration of all spins up or all spins down, the magnetization m will practically never relax to zero. This drawback can be remedied by

ARTICLE IN PRESS 340

David S. Schmool and Hamid Kachkachi

augmenting the procedure by a global rotation (GL) of the particle’s spins to a new trial direction of M and calculating the energy change. That is, before turning single spins on all lattice sites, one computes M, generates its new orientation M0 , and obtains the energy difference ΔE ¼ N H  ðM0  MÞ:

(68)

If the new orientation is accepted according to Eq. (67), one turns all spins si by an appropriate angle and proceeds with the standard Metropolis method outlined above. In small fields (x  N H=T ≲ 1), relaxation of the induced magnetization m becomes much slower than that of the intrinsic magnetization M, and one needs much more MC steps to find the former than the latter with the same precision. If in the procedure each global rotation is coupled with subsequent rotation of single spins on all lattice sites i, making enough global rotations to achieve a required precision for m costs much more computer time for larger particle sizes. Thus, it is more convenient to make many global rotations and gather the data for m after each GL before proceeding to the conventional (single-spin) part of the Metropolis algorithm. This improved method is especially fast for the isotropic Heisenberg or Ising models where the energy change is given by Eq. (68) since, after M has been initially computed, each subsequent rotation and calculation of ΔE requires O(1) operations. In contrast, for systems with anisotropy one has to perform a sum over all lattice sites for each orientation of M, i.e., to make OðN Þ operations. 2.2.5 Dynamics and Calculation of the Relaxation Rate In Fig. 11, we consider two blocks of a given material, one of “bulk” dimensions and the other nanoscaled. 1 cm

1 μm 1 cm 1 nm

1 nm 1 nm

Figure 11 Energy barrier and relaxation rate.

ARTICLE IN PRESS 341

Single-Particle Phenomena in Magnetic Nanostructures

Considering, for instance, cobalt at a temperature T ¼ 300 K, in the absence of a dc magnetic field, we find for the (effective) energy barrier σ ¼ kKV  1015 , leading to the switching time between the two minima BT which is given by τ ∝ eσ  exp ð1015 Þ. On the other hand, for the cluster on the right we find σ  102 and τ  1010 s. This means that as the particle’s size is reduced to the nanometer scale, the switching of the net magnetic moment between the various energy minima becomes possible at room temperature. In fact, even at much lower temperatures, this switching is observable experimentally. This fundamental new effect, called superparamagnetism, and which corresponds to a fast shuttling of the macroscopic magnetic moment between its macrostates, induces a shift in the temperature relevant to magnetism. Indeed, as depicted in Fig. 12, in nanoscaled systems the most relevant temperature is that which corresponds to the thermal energy that is sufficient for overcoming the energy barrier, rather than the Curie temperature as is relevant for bulk systems. This temperature is known as the blocking temperature and denoted by TB. In fact, it should be called the unblocking temperature. In the previous sections, we saw that the Stoner–Wohlfarth model accounts for the hysteretic rotation of the particle’s magnetization over the potential barrier under the influence of a field applied in an arbitrary direction and at zero temperature. We also considered the effect of temperature on the Stoner–Wohlfarth model but at quasi-equilibrium [88]. Now, we will discuss the thermoactivated switching of the particle’s magnetic moment, a process that occurs at short time scales. At nonzero temperatures, the magnetization vector of the particle can surmount the energy barrier due to thermal fluctuations as argued by Ne´el [39, 49]. This effect is particularly pronounced for small particles with lower values of the potential barrier (8), which is proportional to the particle volume. Indeed, the magnetization For 3 nm (Cobalt) 1014 s

t Ⰷt m

Stable magnetism Hysteresis

t Ⰶt m

10–6 s

TB ~ 14 K ∝ KV

Superparamagnetism 〈m〉T = O at H = 0

t KV

Figure 12 Temperature axis.

PM T Tc = 1400 K

ARTICLE IN PRESS 342

David S. Schmool and Hamid Kachkachi

vector of the particle shuttles between the two energy minima and the characteristic time for this thermoactivated rotation of the spin over the anisotropy barrier ΔH is approximately given by the celebrated Van’t Hoff–Arrhenius law [89, 90] τ ¼ τ0 eΔH=kB T ,

(69)

where τ0 is usually taken as temperature and field independent and of the order of 1010–1012 s. τ0 is not necessarily the same for different ferromagnetic materials. It can indeed be assumed as constant only if the magnetization vector is always in one of the energy minima, which happens only if the minima have zero width, or equivalently if the barrier is infinitely high. However, in any realistic case, there is a finite probability that the magnetization vector spends some time in the vicinity of either minimum, in which case the prefactor need not be constant, and certainly depends on temperature and field. Furthermore, in the case of cubic anisotropy the assumption of a constant prefactor τ0 turns out to be a poor approximation [48]. For a given measuring time, τm, the system is at thermal equilibrium when τm ≫ τ, and in this case the particle is said to be superparamagnetic, which corresponds to the temperature range ln ðτm =τ0 Þ > ΔH=kB T 0:

(70)

In fact, it has been argued [77] that due to the smallness of τ0, the range of thermal equilibrium can extend down to temperatures at which the energy-barrier height is much larger than thermal energy. More precisely, for magnetic measurements with τm  100 s (such as is the case typical for magnetometry measurements), this range is 0 ΔH=kB T < 25, which means that it is too restrictive to argue that superparamagnetism occurs only when the thermal energy is on the order of the energy-barrier height. For the case τ ≫ τm, the particle’s magnetization does not change during the time of observation, and the particle exhibits stable ferromagnetism and is said to be in a blocked state. The temperature at which such a transition occurs, namely the temperature at which the relaxation time τ is equal to the observation time τm, is called the blocking temperature and is denoted by TB. For nanoparticle assemblies with size distribution, the same temperature may sometimes be above TB for some particles, and below it for the others. Such systems may thus behave as superparamagnetic for some high values of the temperature T, ferromagnetic at low values of T, and as a mixture of both at intermediate T. For such systems, a different characteristic

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

343

temperature is used, this is denoted by Tmax , and is roughly given, at least in the dilute case, by an average of all TB’s. One should not forget, however, that the time scale of 100 s depends on the experimental apparatus. Indeed, in M€ ossbauer effect measurements, the experimental time is the time of Larmor precession, which is of the order of 108 s, while in neutron scattering experiments it is of the order of 1012 s. The time scale can be completely different in different areas of applications. For example, in magnetic recording, in order to keep the data stored on a magnetic tape for years, one should have τ ≫ 108 s, and in rock magnetism the magnetization decays (or relaxes) within geological times which may be millions of years. Owing to these important practical applications and many others, the relaxation time of the particle’s magnetization is a very important and fundamental physical quantity that deserves extensive and rigorous investigation. This actually started, in this context, with the work of Kramers on transition-state method [91]. Kramers showed, by using the theory of the Brownian motion, how the prefactor of the reaction rate (inverse of relaxation time), as a function of the damping parameter, and the shape of the potential well could be calculated from the underlying probability-density diffusion equation in phase space, which for Brownian motion is the Fokker–Planck equation (FPE). He obtained, by linearizing the FPE about the potential barrier, explicit results for the escape rate for intermediate-to-high (IHD) values of the damping parameter and also for very small values of that parameter. Subsequently, a number of authors [92, 93] showed how this approach could be extended to give formulas for the reaction rate which are valid for all values of the damping parameter. These calculations have been reviewed by Ha¨nggi et al. [94]. The relaxation rate is given by the smallest nonvanishing eigenvalue λ1 of the Sturm–Liouville equation associated with the Fokker–Planck equation. Brown [41] first derived a formula for λ1, for an arbitrary axially symmetric bistable potential of Fig. 1 with minima at θ ¼ (0, π) separated by a maximum at θm ¼ arccos ðhÞ, which when applied to Eq. (6) for hke, i.e., a magnetic field parallel to the easy axis, leads to the form found by Aharoni [95], θm, h i 2 2 2 λ1 ’ pffiffiffi λ3=2 ð1  h2 Þ ð1 + hÞ eαð1 + hÞ + ð1  hÞ eαð1hÞ π

(71)

where 0 h 1, h ¼ 1 being the critical value at which the bistable nature of the potential disappears. In order to describe the nonaxially symmetric asymptotic behavior, let us denote by βΔH the smaller reduced barrier

ARTICLE IN PRESS 344

David S. Schmool and Hamid Kachkachi

height of the two constituting escape from the left or the right of a bistable potential (see Eq. 8 for the axially symmetric case). Then for very low damping, i.e., for λβΔH ≪ 1 (with of course the reduced barrier height βΔH ≫ 1, depending on the size of the nanoparticle studied), we have the following asymptotic expression for the Ne´el relaxation time [41, 96] τ1 VLD ’

o λ1 λn ’ ω1 βðH0  H1 ÞeβðH0 H1 Þ + ω2 βðH0  H2 ÞeβðH0 H2 Þ 2τN 2π (72)

where τN ¼

1 m , λ 2γkB T

(73)

is the so-called Ne´el time, that is the free-diffusion time, i.e., the characteristic time of diffusion in zero potential. For the IHD limit, where λβΔH > 1 (again with the reduced barrier height βΔH much greater than unity), we have [96] the asymptotic expression o Ω0 n βðH0 H1 Þ βðH0 H2 Þ τ1 , (74) ’ ω e + ω e 1 2 IHD 2πω0 where γ 2 ð1Þ ð1Þ γ 2 ð2Þ ð2Þ 2 c c , ω ¼ c c 2 m2 1 2 m2 1 2 γ 2 ð0Þ ð0Þ ω20 ¼  2 c1 c2 , m" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# λγ 4 ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ Ω0 ¼ c1  c2 + ðc2  c1 Þ2  2 c1 c2 : 2m λ ω21 ¼

Here ω1, ω2 and ω0 are, respectively, the well and saddle angular frequencies associated with the bistable potential, Ω0 is the damped saddle angular freðiÞ

quency, and the cj are the coefficients of the truncated (at the second order in the direction cosines) Taylor series expansion of the crystalline anisotropy and external field potential at the wells of the bistable potential denoted by 1 and 2 and at the saddle point denoted by 0. A full discussion of the application of these general formulas to the particular potential, which involves the ðiÞ

numerical solution of a quartic equation in order to determine the cj with the exception of the particular field angle ψ ¼ π4 or π2, in Eq. (4) is given in Refs. [97, 98].

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

345

In Ref. [99], we used either Eqs. (71) and (72) or (71) and (74) and solved the equation τ ¼ τm for the blocking temperature TB, as a function of the applied field, for an arbitrary angle ψ between the easy axis and the applied magnetic field. In particular, for very small values of ψ, we used Eq. (71), as the problem then becomes almost axially symmetric and the arguments leading to Eqs. (72) and (74) fail [41, 42, 97], and appropriate connection formulas had to be used so that they may attain the axially symmetric limit. Application of the above asymptotic formulas to the calculation of the blocking temperature TB (or Tmax ) as a function of the applied field appeared to recover the experimental observation, but this result turned out to be spurious. An explanation of this behavior follows (see also Refs. [99–101]): in the nonaxially symmetric IHD asymptote (74) which is formulated in terms of the Kramers escape rate, as the field tends to zero, for high damping, the saddle angular frequency ω0 tends to zero. Thus, the saddle becomes infinitely wide and so the escape rate predicted by Eq. (74) diverges leading to an apparent rise in the blocking temperature until the field reaches a value which is sufficiently high to allow the exponential in the Van’t Hoff–Arrhenius terms to take over. When this occurs, the blocking temperature decreases again in accordance with the expected behavior. This is the field range where one would expect the nonaxially asymptote to work well. In reality, as demonstrated by the exact numerical calculations of the smallest nonvanishing eigenvalue of the Fokker–Planck matrix, the low-field behavior is not as predicted by the asymptote in Eq. (74) (it is rather given by the axially symmetric asymptote) because the saddle is limited in size to ω0. Thus, the true escape rate cannot diverge, and the apparent discontinuity between the axially symmetric and nonaxially symmetric results is spurious, leading to an apparent rise in TB. In reality, the prefactor in Eq. (74) can never overcome the exponential decrease embodied in the Van’t Hoff–Arrhenius factor. Garanin et al. [100] have discovered bridging formulas which provide continuity between the axially symmetric equation (71) and nonaxially symmetric asymptotes, leading to a monotonic decrease of the blocking temperature with the field in accordance with the numerical calculations of the lowest eigenvalue of the Fokker–Planck equation. An illustration of this was given in Ref. [100] (see also Ref. [101] for more details) for the particular case of ψ ¼ π/2, that is, a transverse applied field. If the escape rate is written in the form κ τ1 ¼ A exp ðβΔHÞ π

ARTICLE IN PRESS 346

David S. Schmool and Hamid Kachkachi

where κ is the attempt frequency given by κ¼

2Kγ pffiffiffiffiffiffiffiffiffiffiffi2ffi 1h , m

pffiffiffi then the factor A, as predicted by the IHD formula, behaves as λ= h for pffiffiffiffiffiffiffiffi h ≪ 1, λ2, while for h ¼ 0, A varies as 2πλ σ=π , which is obviously discontinuous so that a suitable interpolation formula is required. Such a formula (analogous to that used in the WKBJ method [102]) is obtained by multiplying the factor A of the axially symmetric result by eξI0(ξ), where I0(ξ) is the modified Bessel function of the first kind, and ξ ¼ 2σh [100]. This interpolation formula, as is obvious from the large and small ξ limits, pffiffiffi automatically removes the undesirable 1= h divergence of the IHD formula and establishes continuity between the axially symmetric and nonaxially symmetric asymptotes for ψ ¼ π/2, as dictated by the exact solution. Alongside the well-known work of Brown [41, 42] and also that of Aharoni [95], there is a fundamental approach developed by Langer [104, 105] for multivariate systems. The only limitation of Langer’s approach is that it applies to situations where the extrema of the energy potential are well defined in the sense that they are not flat. Indeed, this approach is based on the quadratic expansion of the energy potential at the various extrema (minima, maxima, and saddle points). In practice, this turns out to be applicable to intermediate-to-high damping regimes. Nevertheless, a general bridging formula has been obtained (see, e.g., Ref. [103]), thus allowing interpolation between very low and intermediate-to-high damping regimes. Let us now discuss Langer’s approach with a typical situation as depicted in Fig. 13. (2)

(0)

(1)

Figure 13 Elementary process of barrier crossing: the system starts in the metastable state (1) and crosses the saddle point to end up in state (2) of greater stability.

ARTICLE IN PRESS 347

Single-Particle Phenomena in Magnetic Nanostructures

Within this approach, the problem of calculating the relaxation time for a multidimensional process is reduced to solving a steady-state equation in the immediate neighborhood of the saddle point (0) that the system crosses as it goes from a metastable state (1) to another state (2) of greater stability. The basic idea [104, 105] is that “... one imagines setting up a steady-state situation by continuously replenishing the metastable state at a rate equal to the rate at which it is leaking across the activation-energy barrier. By identifying the current flowing over the barrier with the desired decay rate, one avoids having to solve the complete time-dependent problem...,” especially in the multidimensional case where this problem is too difficult to tackle. However, this approach is only valid in the limit of intermediate-to-high damping because of the assumption, inherent to this approach, that the potential energy in the vicinity of the saddle point may be approximated by its second-order Taylor expansion. All in all, since the escape rate is simply given by the ratio of the total current through the saddle point to the number of particles in the metastable state, it turns out that Langer’s result for the escape rate can be achieved by only computing the energy-Hessian eigenvalues near the saddle points and 

metastable states, from which one then infers the partition function Zs of the system, restricted to the region around the saddle point where the energyHessian negative eigenvalue is (formally) (see Refs. [104, 105] for a rigorous derivation) taken with absolute value, and the partition function Zm of the region around the metastable state. When computing these partition functions, one has to identify and take care of the Goldstone modes associated with the unbroken global symmetries of the different states. Finally, one computes the unique negative eigenvalue κ of the SSFPE corresponding to the unstable mode at the saddle point. More precisely, κ is given by e mn ¼ λn ðDMDT Þmn , the negative eigenvalue of the dynamic matrix M where the λn’s are the eigenvalues of the Hessian at the saddle point and D is the transformation matrix from ηi to ψ n. Consequently, Langer’s final expression for the escape rate, corresponding to the elementary process in Fig. 13, is rewritten in the following somewhat more practical form [106, 107] e ð0Þ jκj Z s Γð1Þ!ð2Þ ¼ ð1Þ , 2π Zm

(75)

where jκ j is the attempt frequency which contains the damping parameter α.

ARTICLE IN PRESS 348

David S. Schmool and Hamid Kachkachi

Note that at the level of an elementary process of barrier crossing, as shown in Fig. 13, there is full equivalence between the Brown and Langer approaches. Using one or the other formulation for the relaxation rate, one can investigate the effect of temperature on the Stoner–Wohlfarth model (for equilibrium), and Ne´el–Brown (for dynamic) models has been confirmed by these experiments on individual cobalt particles [44]. It is obvious that the OSP approach can account for the longitudinal fluctuations of the particle’s magnetic moment, either induced by thermal effects or by spin noncollinearities. At finite temperature, but at quasi-equilibrium, the magnetization switches at very low temperature via a coherent rotation of all spins, as in the Stoner–Wohlfarth model, whereas at higher temperature, it switches by changing its magnitude. This results in a shrinking of the Stoner–Wohlfarth astroid as described by the modified Landau theory [88] and (qualitatively) confirmed by experiments [44]. However, it is clear that the change of magnetization magnitude cannot be explained in the framework of the OSP approach. Indeed, it can only be understood as the result of a successive switching of individual (or clusters of ) spins inside the particle, which is necessarily a multi-spin system. Indeed, deviations from the singlespin approximation, and thereby from both the Stoner–Wohlfarth and Ne´el–Brown models, have been observed in metallic particles [60, 62] and ferrite particles [108, 109]. These deviations have materialized in terms of the absence of magnetization saturation at high fields, shifted hysteresis loops after cooling in field, and enhancement of the magnetization at low temperature as a function of applied field. The latter effect has been clearly identified in dilute assemblies of maghemite particles [71] of 4 nm in diameter. In addition, aging effects have been observed in cobalt single particles and have been attributed to the oxidation of the sample surface into antiferromagnetic CoO or NiO (see Ref. [44] and references therein for a discussion of this issue). It has been argued that the magnetization reversal of a ferromagnetic particle with antiferromagnetic shell is governed by two mechanisms that are supposed to result from the spin frustration at the core–shell interface of the particle. On the other hand, according to M€ ossbauer spectroscopy some of the above-mentioned novel features are most likely due to magnetic disorder at the surface, which induces a canting of spins inside the particle, or in other words, an inhomogeneous magnetic state. As argued earlier, understanding these effects requires the use of the MSP approach capable of distinguishing and accounting for the various crystallographic local environments that develop inside a nanoparticle and on its surface. Here one is faced with complex many-body aspects with the

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

349

inherent difficulties related with analyzing the energyscape (location of the minima, maxima, and saddle points of the energy potential). This analysis is unavoidable, though, since it is a crucial step in the calculation of the relaxation time and thereby in the study of the magnetization stability against thermally activated reversal. In principle, as discussed earlier the stochastic Landau–Lifshitz equation can be used to compute the relaxation rate for a many-spin particle, but this requires a rather time-consuming procedure for the calculation of averages over large ensembles of spin trajectories. In addition, it is not so obvious how to establish an efficient criterion for defining the switching time. A compromise is provided by the EOSP approach which deals with a macrospin while capturing some of the intrinsic features of the nanoparticle. It is possible then to investigate the effects of surface anisotropy on the switching field. Accordingly, in Ref. [110] it was found that the relaxation rate is a nonmonotonic function of the surface anisotropy constant KS. More precisely, owing to the variation of the energy barrier as a function of the surface anisotropy (see Fig. 14), the relaxation rate increases for (small) increasing KS since the (surface) quartic contribution to anisotropy induces saddle points at the equator. As KS further increases, the quartic anisotropy

Figure 14 Energyscape with increasing surface anisotropy and the corresponding energy barrier.

ARTICLE IN PRESS 350

David S. Schmool and Hamid Kachkachi

starts to dominate, inducing much deeper energy minima and thereby much high energy barriers, which finally makes the switching less likely. Compared to the effect of thermal fluctuations on the switching field, which is a simple scaling law, as can be seen in Fig. 28, the effect of surface anisotropy depends on the direction of the applied field. This leads to a flattening of the switching field curve (see Fig. 3 of Ref. [110]). In the case of an assembly of magnetic nanoparticles, this model is used to compute the ac susceptibility and to investigate the competition between (intrinsic) surface effects and dipolar interactions within an assembly of magnetic nanoparticles [111–114].

2.3 A Sample of Results: Finite-Size and Surface Effects The effects of finite size and surface anisotropy on the equilibrium properties of magnetic nanoparticles, such as the spin configuration, the magnetization, and susceptibility as functions of the applied magnetic field and temperature, have been studied for over two decades by many groups, in various kinds of nanoparticles. Since no experimental results pertaining to the intrinsic features of the particles have been available, these studies have been carried out on assemblies which are sufficiently dilute so as to consider them as weakly interacting and not too dilute for the magnetic signal to be detected. The main consensus is that in such well-separated nanoparticle systems, the collective effects due to interparticle interactions are minimized. In reality, such interactions are long ranged and their effect can never be completely suppressed. This implies that all fitting procedures that are used to extract intrinsic properties of the particles from experimental results obtained for assemblies have to be used with care. This situation has made it rather difficult to compare the available experimental results with the theoretical developments conducted in parallel by the community. Nevertheless, experimentalists and theorists have produced various results and a valuable knowledge database that have shed light on our understanding of the equilibrium properties of magnetic nanoparticles and have led to a set of useful procedures for estimating the various relevant physical parameters. Magnetic state and spin configuration It is now well established that surface effects are responsible for the spin canting, or in general spin noncollinearities, in magnetic nanoparticle, as was evidenced a few decades ago by M€ ossbauer measurements [34, 70, 115–118], by neutron scattering [69], or simply by magnetometry measurements [71]. From the theoretical point of view, this is obvious considering

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

351

the fact that atoms on the particle’s boundary have less neighbors than their counterparts in the core of the particle. For very small surface anisotropy, the cubic contribution becomes negligible and the Stoner–Wohlfarth (or OSP) model provides a good approximation to the many-spin particle. This means that one can simply assume that all spins within the particle are tightly held parallel to each other. Some experimental macroscopic estimates of the surface anisotropy constant yield, e.g., for cobalt KS/J ’ 0.1 [119], for iron KS/J ’ 0.06 [120], and for maghemite particles KS/J ’ 0.04 [121]. However, one should not forget that this effective constant depends on the particle’s size, among other parameters such as the material composition. Moreover, for a nanoparticle of diameter 2 nm, we may expect higher anisotropies. Altogether, the spin configuration can be considered as quasi-collinear (see Fig. 15) Hysteresis and switching field The hysteretic properties of magnetic nanoparticles have been extensively studied by many groups [46, 51, 52, 59, 65, 122–126]. Most of the calculations have been devoted to the effects on the coercive field of finite size, boundary conditions, and surface anisotropy. It is now well established that for small surface anisotropy, i.e., in the regime of quasi-collinear spin

Figure 15 Spin configuration of the middle plane of a spherical particle.

ARTICLE IN PRESS 352

David S. Schmool and Hamid Kachkachi

states, the hysteresis loop is rectangular for all sizes, and the critical field decreases when the particle’s size decreases, as can be seen in Fig. 16. In this case, the hysteresis loop can be scaled with that rendered by the macrospin SW model, even in the general case of a field applied at an arbitrary angle with respect to the core easy axis. The scaling constant is the ratio of the number of core spins to that of the particle’s total number of spins. Moreover, the limit-of-metastability curves for all particle sizes fall inside the SW astroid (see Fig. 17).

Figure 16 Switching field of a spherical nanoparticle with Ks/J ’ 0.01 as a function of the particle's diameter D ¼ Na (a is the interatomic distance). Source: Reprinted with permission from Ref. [50]. Copyright (2002), AIP Publishing LLC.

Figure 17 Switching field of a spherical nanoparticle with Ks/J ’ 0.01, for different values of the surface-to-volume ratio Nst ¼ Ns =N . Source: Reprinted with permission from Ref. [50]. Copyright (2002), AIP Publishing LLC.

ARTICLE IN PRESS 353

Single-Particle Phenomena in Magnetic Nanostructures

hc 20 1.0

15

M.eh

Non-SW

0.5 0.0

10

1.0 ν = 60°

–0.5

0.5

–1.0 –1.0 –0.5 0.0

5

0.0

0.5 h 1.0

–0.5

SW scaling = const.

–1.0

SW scaling (y )

0 0.0

M.eh

–20 –10

0

10 h 20

T=0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

kS

Figure 18 Effects of surface anisotropy on the hysteresis loop and validity of the macrospin approach.

As the surface anisotropy increases, the switching mechanism becomes less coherent and mostly clusterwise leading to steps in the hysteresis loop and the limit-of-metastability curve can no longer be scaled with the Stoner–Wohlfarth astroid. The situation can be summarized in Fig. 18. Magnetization as a function of the dc applied field and temperature Likewise, the behavior of the magnetization as a function of temperature and applied magnetic field has revealed special features in nanoscaled systems. Finite-size and boundary effects are discussed at length in Section 2.1.2, where approximate analytical expressions are given for the magnetization that allows for some quantitative estimates. The main effect of finite size on the thermal behavior of the magnetization is exemplified by pffiffiffiffiffi the well-known tail (M  1= N ) in the critical region. As a consequence, the critical temperature is drastically reduced in magnetic nanoparticles. Obviously, this is not restricted to nanoparticles; it applies to any finite-size magnetic system. At low temperatures, free boundary conditions lead to a smaller magnetization than in the bulk because, as was shown in Section 2.1.2, there are more low-lying spin excitations within the spectrum. As the surface anisotropy and the ensuing spin noncollinearities are taken into account, one finds that the magnetization saturation requires much higher magnetic fields (see Fig. 19). It is worth mentioning that cobalt particles, for instance, were found [60, 62] to exhibit a magnetization that is larger than in the bulk of the same

ARTICLE IN PRESS 354

David S. Schmool and Hamid Kachkachi

1.0

Magnetization

0.8

t = kBT/J

0.6

t = 0.005 t = 0.01 t = 0.05 t = 0.1

0.4 Sphere of 515 spins Anisotropy: uniaxial kC = 0.0024 transverse kS = 0.4

0.2

† = 0.01 → T ~ 1 K 16 Tesla

0.0 0.0

0.1

0.2

h = maH/J

0.3

Figure 19 Magnetization as a function of temperature and dc field obtained by MC calculations.

substance. This effect was attributed by some groups [127–129] to the unquenched orbital contribution that would stem from the breaking of symmetry at the surface of metallic particles. In addition, XMCD and FMR measurements have been performed to evaluate the spin and orbital contributions to the magnetic moment (see end of Sections 3.5 and 3.6 for the latest experimental results). However, the study of the effect of the orbital contribution on the magnetic properties of nanoparticles has not been met with the same enthusiasm as for the Heisenberg spin-only Hamiltonians. Magnetization profile The magnetization profile has been obtained by Monte Carlo simulations [130] and from the spherical model [53]. The main outcome of these calculations is that the local magnetization decreases as one goes from the center of the particle toward the boundary. This is at variance with the simplistic idea that was in vogue for some time during the early 2000s, according to which a magnetic nanoparticle could be pictured as having a magnetically ordered core surrounded by a magnetically dead shell. In fact, the magnetic disorder indeed starts at the boundary but penetrates into the core of the particle, but in a progressive manner (see Fig. 20).

ARTICLE IN PRESS 355

Single-Particle Phenomena in Magnetic Nanostructures

Local magnetization

1.0

t core << 1

0.8 t core = 0.5

0.6

0.4 t core = 1–

0.2

0.0 0.0

0.2

0.4 0.6 Radial distance

0.8

1.0

Figure 20 Magnetization profile of a magnetic nanoparticle with surface anisotropy. Left: as obtained by Monte Carlo (numerical) calculations for a spherical particle. τcore  T/Tcore where Tcore is the critical temperature of the core. Right: as obtained (semic c analytically) by the spherical model for a cube-shaped particle. The reduced temperature here is defined as θ ¼ T/TMFA . Source: From Ref. [130]. With kind permission from c Springer Science and Business Media.

2.4 Magnetic Excitations and Reversal As discussed earlier, since the switching of the net magnetic moment of the nanoparticle becomes possible on a time scale that is observable in laboratory and, for some given sizes, at a temperature that is not too low, many potential applications can be envisaged. There are several ways in which the

ARTICLE IN PRESS 356

David S. Schmool and Hamid Kachkachi

magnetization switching can be triggered. The simplest one consists in applying a dc magnetic field that is sufficient to suppress the energy barrier induced by anisotropy. This is the Stoner–Wohlfarth switching field defined in Eq. (9). However, for practical applications higher energy barriers are sought in order to reach a longer temporal stability with respect to thermal fluctuations. This can be achieved by using nanoparticles with a high anisotropy material, if one wants to maintain the particle’s volume to be nanosized in view of high storage densities. The consequence of this is that the magnetic field that is required for suppressing the energy barrier is rather large for such small systems, in that it deteriorates the signal-to-noise ratio, apart from the fact that it is not so easy to achieve these fields in a stable manner. This issue is what we call the superparamagnetic trilemma, as it involves three physical parameters that have to be optimized at the same time, namely the size, temperature, and the magnetic field. 2.4.1 Microwave-Assisted Reversal With this problem in mind, the nanomagnetism community turned toward the investigation of other means for achieving magnetization switching with more practical parameters. Most of the routes explored mainly relate to the injection of other forms of energy into the system. Two of these routes that have attracted much attention are the all-optical technique [3, 131–134] and the application of a microwave (MW) magnetic field on top of the dc field [135–137]. It is also worth mentioning the alternative provided by the technology of Heat-Assisted Magnetic Recording (HAMR) which consists in heating the part of the disk that is being written to. The basic idea underlying this technology is that by heating locally the magnetic system, the effective anisotropy and thereby the energy barrier can be reduced. One of these routes consists in applying a radiofrequency field in addition to a dc magnetic field. For nanoclusters, it has been shown in previous works how a monochromatic microwave (MW) pulse can, by means of a nonlinear resonance, substantially reduce the static field required to reverse the magnetization of an individual nanoparticle. The switching curves obtained in these measurements present some irregular features that depend on the MW and dc field characteristics, the potential energy of the nanomagnet, and the damping parameter. In particular, the strong dependence of these features on the damping parameter might be used to estimate the latter in such clusters. Several theoretical works have been devoted to the understanding of the magnetization dynamics, and in particular its reversal, under the effect of a time-dependent magnetic field. The theoretical work may be divided into

ARTICLE IN PRESS 357

Single-Particle Phenomena in Magnetic Nanostructures

two categories. The first deals with the effect of a given MW field with a given polarization [138, 139], while the second seeks optimal strategies for achieving the magnetization switching [140]. For example, Mayergoyz et al. [141] assume a given dynamic response for the magnetization and attempt to determine the MW field that realizes it. In Refs. [142, 143], a numerical method was proposed based on the theory of optimal control that is capable of rendering an exact solution for the MW field vector necessary for the switching of a nanomagnet with a given potential energy (comprising anisotropy and an oblique static field). The standard formulation of this method consists of minimizing a cost functional using the conjugate gradient technique. We will briefly summarize the main steps of this method. The main problem (see Fig. 21) consists in finding the (vector) MW field bðt Þ to be applied to the nanomagnet, with its given energy potential comprising a dc magnetic field and an effective anisotropy, in order to switch its magnetization from an initial state sðiÞ ðtÞ to a final (target) state sð f Þ ðtÞ, with a minimal injected electromagnetic energy. To be more precise, we consider the Landau–Lifshitz equation for a magnetic moment s with uniaxial anisotropy and oblique field ds ¼ s heff + λ s ½s heff , dτ where the effective time-independent (constant) field reads heff  

δE ¼ ξ eh + kðs  nÞ n, δs

λ is the damping parameter and

b(t)?

s(i ) = s(ti) s(f ) = s(tf)

Figure 21 Double-well potential of a magnetic nanoparticle in presence of a microwave field.

ARTICLE IN PRESS 358

David S. Schmool and Hamid Kachkachi

t τ , τs

τs 

1 γHK

is the characteristic time (2 1011 s for a 3-nm cobalt cluster). Next, we apply the time-dependent field b(τ) leading to the total effective field ζðτÞ ¼ heff + bðτÞ: Then, we obtain what we call the “driven” Landau–Lifshitz equation ds ¼ s ζðτÞ + λ s ½s ζðτÞ: dτ The field b(τ) is then obtained by minimizing the cost functional 2 α 1 F ½sðτÞ, bðτÞ ¼ sðτf Þ  sð f Þ  + 2 2

Zτf dτ b2 ðτÞ 0

along the “driven” spin trajectory s(τ). Obtaining the field b(τ) amounts to solving the following boundary problem 8 8 9 Zτf > < = >   1 α 2 > > dτ b2 ðτÞ , min F ½sðτÞ, bðτÞ ¼ sðτf Þ  sð f Þ  + > > < : ; 2 2 ds > > > ¼ s ζðτÞ + λ s ½s ζðτÞ, > > >   : dτ sð0Þ ¼ sðiÞ , τ 2 0, τf :

0

Equivalently, one can define the Hamiltonian α H½s, η,b ¼ b2 ðτÞ  ηðτÞ  ½s ζðτÞ  λ s ðs ζÞ 2 and transform the problem into solving the Hamilton–Jacobi equations with boundary conditions 8   δH > > s_ ¼ , sð0Þ ¼ sðiÞ , τ 2 0, τf , > > δη > < δH η_ ¼  , ηðτf Þ ¼ sðτf Þ  sð f Þ , > δs > > > > : δH ¼ 0: δb

ARTICLE IN PRESS 359

Single-Particle Phenomena in Magnetic Nanostructures

In general, this problem can be solved only numerically using, for instance, the conjugate gradient method. As this is a technique of local convergence, it can be supplemented by a Metropolis algorithm for a global search of the minima [142, 143]. We have shown that the optimal MW field is modulated both in frequency and magnitude (see Fig. 22).

0.04

b(t)

0.02

0

–0.02 1 mx(t ) my(t ) mz(t ) 0.5

0

–0.5

Time (s) –1 4 × 10–9

6 × 10–9

8 × 10–9

Driven precession

1 × 10–8 Free relaxation

Figure 22 Time-dependent rf field and time trajectory of the components of the particle's magnetic moment. Source: Reprinted (figure) with permission from Ref. [142]. Copyright (2011) by the American Physical Society.

ARTICLE IN PRESS 360

David S. Schmool and Hamid Kachkachi

The role of this MW field is to drive the magnetization toward the saddle point, then damping leads the magnetic moment to the stable equilibrium position. For the pumping to be efficient, the MW field frequency must match, at the first stage of reversal, the proper precession frequency of the magnetization, which depends on the magnitude and the direction of the static field. Moreover, the intensity depends on the damping parameter. This result could be used to probe the damping parameter in experimental studies of nanoparticles. This study also shows that the Stoner–Wohlfarth field that is required for the magnetization switching can be significantly reduced in the presence of a small MW field (15 mT). In fact, the present problem and the results obtained for a magnetic nanocluster are reminiscent of the general and fundamental issue, namely the problem of taking a system out of an energy minimum by nonlinear resonance. This has previously been addressed in many areas of physics and chemistry, especially in the context of atomic physics. For example, the dissociation of diatomic molecules by a chirped infrared laser pulse requires a much lower threshold laser intensity than with a monochromatic field [144–147]. According to the classical theory of autoresonance or the quantum theory of ladder climbing [146–150], exciting an oscillatory nonlinear system to high energies is possible by a weak chirped frequency excitation. Moreover, trapping into resonance followed by a (continuing and stable) phase locking with the drive is possible if the driving frequency chirp rate is small enough. It has also been shown that a slow passage through and capture into resonance yields efficient control of the energy of the driven system. The results for a magnetic nanoparticle [142, 143] do confirm these general features.

2.4.2 Switching via Internal Spin-Wave Processes With the same objective on mind, i.e., optimizing the magnetization switching in a nanoparticle, one can investigate the role of internal spinwave excitations. As was shown earlier, in order to account for the magnetization reversal within spin-wave theory, one has to include the global rotation of the net magnetic moment. This means that the spin-wave excitations have to be described in a moving frame bound to the net magnetic moment. In Refs. [151, 152], a general formalism was developed for the spin-wave theory in the frame related with the particle’s global magnetization for an arbitrary direction of the applied field. Two anisotropy models were studied, namely the uniform uniaxial anisotropy and random

ARTICLE IN PRESS 361

Single-Particle Phenomena in Magnetic Nanostructures

anisotropy. Then, we investigated the ensuing spin-wave instabilities, which are found to be exponential and linear, respectively. We start from the precession equation for atomic spins obtained from Eq. (56) upon dropping the damping term and the Langevin field. Then, the microscopic effective field can be written as heff i ¼h

X δHan $ Jij sj , + 2g isi + δsi j

(76)

where Han contains only nonrandom anisotropy, whereas the random anisotropy is in the third term. In particular, for uniaxial anisotropy Han is given by Eq. (17) and thereby one has (for uniform anisotropy) $

$

$

δHan /δsi ¼ K si  K  si and the components of the tensor K read $

ðK Þαβ ¼ Kδαz δβz : Similarly, the components of the random anisotropy ten$

sor ðg i Þαβ ¼ gi, αβ are given by

 1 gi, αβ ¼ KR uiα uiβ  δαβ , 3

KR > 0:

(77)

Spin-wave excitations can then be described by writing the atomic spin si in the form X ψ i ¼ 0, si ¼ M + ψ i , (78) i

where M is the average spin defined by Eq. (21) and ψ i contains the Fourier components with k6¼0 and describes spin waves in the particle. As discussed above, whereas in the standard spin-wave theory M is a constant corresponding to the ground-state orientation, here it is treated as a timedependent variable. Since the atomic spins are subject to the constraint pffiffiffiffiffiffiffiffiffiffiffiffiffi s2i ¼ 1, one can use [153] M ¼ n 1  ψ 2i with n ψ i ¼ 0, where n is a unit vector. Although this reduces to two the number of the ψ i components, the formalism becomes much more cumbersome, but the final results are not affected. Thus, one can avoid using the constraint explicitly, with the understanding that the properly written equations should satisfy this constraint that can be used to check them. The equation of motion for si then leads to a set of equations for M and ψ i which can be solved analytically in the case of a random anisotropy and only numerically in the case of uniaxial anisotropy (see details in Refs. [151, 152]).

ARTICLE IN PRESS 362

David S. Schmool and Hamid Kachkachi

A

A

1.0

1.0 Box-fbc particle N = 28 × 31 × 34 = 29,512

m

0.5

0.5

0.0

z

–0.5

ez Uniaxial anisotropy Si (0) y

m⊥

0.0 Prepared with all spins aligned antiparallel to h = hez

–0.5

–mx

x

mx

Box-fbc particle N = 28 × 31 × 34 = 29,512

m

D/J = 0.03, h/J = 0.1 m⊥

DR/J = 0.1,

mz

h

–mz

–1.0

h/J = 0.1

Thermal equilibrium

–1.0

0

5000

10,000

15,000

(J/h)t

B

0

20,000

40,000

60,000

80,000

(J/h)t

B 1.0

1.0 m

m

0.8

0.8

0.6

0.6

Prepared with all spins aligned perpendicular to h = hez

m⊥ - Perpendicular to h

m⊥

0.4

0.4 Prepared with all spins aligned along z with h = hex

0.2 mx

0.2 mz

0.0

0.0 0

10,000

20,000

30,000

40,000

(J/h)t

0

10,000

20,000

30,000

(J/h)t

Figure 23 Magnetization switching via exponential spin-wave instability in a boxshaped particle with uniaxial anisotropy and transverse field. (a) The particle is prepared with all spins opposite to the magnetic field (the maximal-energy state). (b) The particle is prepared with all spins perpendicular to the magnetic field. Source: Reprinted (figures) with permission from Ref. [152]. Copyright (2009) by the American Physical Society.

In Fig. 23, we show a sample of the results obtained for a box-shaped nanoparticle showing the exponential spin-wave instability in the case of a uniaxial anisotropy and a linear spin-wave instability for a random anisotropy. Comparing the results on the left to those on the right, we see that the exponential instability (EI) leads to a faster relaxation than the linear instability (LI). Indeed, the instability increment for EI contains pffiffiffiffiffiffiffi the factor K while for LI there are the factors K/J and h=J . However, for long times, the relaxation rate scales with K2 in both cases and thereby the difference between the two kinds of instability is small. In Ref. [151], an estimate was given of the magnetization reversal rate, due to internal spin waves, for cobalt with K/J ’ 0.0024 and for h/J ¼ 0.001 (H ’ 0.07 T), Γ  105–106 s1. For this field, the precession frequency is ωH ’ 1010 s1, which is much larger than the rate Γ.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

363

3. OVERVIEW OF EXPERIMENTAL RESULTS ON SINGLE-PARTICLE MEASUREMENTS In this section, we will summarize the state of the art with regards to the main experimental work that has been performed on single magnetic particle systems or studies which are relevant to single magnetic particles. There are many experimental techniques available for the study of magnetic materials, which can be employed for the characterization of their intrinsic (and extrinsic) magnetic properties. However, when we consider the magnetic properties of isolated magnetic elements or nanoparticles, the number of techniques available rapidly declines. The stray field emanating from a magnetic nano-object can be of the order of just a few mT at a distance of tens of nanometers from its surface. This means that only a few techniques will have a sufficient sensitivity to magnetically characterize a single nano-object. We can therefore characterize the difficulties encountered in performing single nanoparticle measurements: (i) making a device with reduced dimensions which is sufficiently sensitive to detect the reduced number of spins in the nano-object, and (ii) placing the magnetic nanoobject (sample) in the vicinity of the device for it to couple with it and hence allow its measurement. In this section, we will review some of the principal results of those techniques which have been successfully employed for the purposes of studying the magnetic behavior. As we will discuss later in this chapter, the magnetic properties of nanoparticle assemblies and multielement nanomagnetic arrays have modified magnetic properties with respect to isolated nanomagnetic entities due to interactions between these elements. The nature of such interactions is related to the intervening material. This remains true even for very magnetically dilute systems, which exhibit only weak magnetic interactions. Therefore, we only consider here those measurements which are truly applicable for single-particle studies. Of the magnetic measurement techniques available, there appear to be three main classes of measurement that really seem to be able to offer anything approaching single nanoparticle sensitivity. These are mainly adaptations of previous techniques: magnetometry, resonance-type measurements, and imaging/scanning probe techniques. Here we will outline the basic principles of these methods and give some representative results of the following techniques: the micro-SQUID, micro-Hall magnetometry, Lorentz transmission electron microscopy (LTEM), magnetic force microscopy (MFM), ferromagnetic resonance (FMR), and magnetic resonance force

ARTICLE IN PRESS 364

David S. Schmool and Hamid Kachkachi

microscopy (MRFM). While in some cases the single-particle nature of the measurements can be debated, these techniques either have been shown to be sensitive to single nanoparticle or demonstrate the potential to be.

3.1 Micro-SQUID and Nano-SQUID The first experimental measurement of a single-domain magnetic nanoparticle dates to around two decades ago by Wernsdorfer et al. [154], using a micron-sized superconducting quantum interference device (SQUID) bridge (see also review by Barbara [155]). The SQUID is constructed of two Josephson junctions, which are planar tunnel junctions formed in a loop structure and typically enclosed in a magnetically shielded environment to protect it from stray magnetic signals from other sources. Early devices were of around 2 μm linear dimension. Further development of the technique has managed a roughly tens times reduction of the SQUID structure, with nano-SQUIDs fabricated with carbon nanotube Josephson junctions [156]. The measurement of the magnetic flux change, ΔΦ, from a nanoparticle is directly related to the variation of the magnetization, ΔM, which is associated with the reversal of the magnetic moment, where ΔΦ ¼ αΔM, here α is a flux coupling factor dependent on geometry of the SQUID and the nanoparticle. Ultimately, the total magnetic flux can be expressed as: Φ¼

1 m μ 2 0R

(79)

where m is the magnetic moment of the nanoparticle and R is its radius. A reversal of magnetization of the particle will lead to a flux change of 2Φ.1 The SQUID bridge is fabricated using the electron beam lithography method, which provides good structure definition as can be seen in Fig. 24. The SQUID bridge is constructed with planar geometry such that the magnetic flux through the loop produced by a magnetic nanoparticle in its immediate proximity is the maximum. The sensitivity to the magnetic flux from the measurement of the critical current was estimated to be around 2 1015 Wb [158]. For the tracing of the hysteresis loop, the external field is applied in the plane of the SQUID, making the device sensitive only to the flux produced by the stray field of the nanoparticle. An estimate of the senpffiffiffiffiffiffiffi sitivity of this method was evaluated to be of the order of 103 μB = Hz, which roughly translates to the magnetic moment of a single Co nanoparticle of 2–3 nm diameter [158]. 1

For technical reasons, the nano-SQUID has a limited sensitivity.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

365

Figure 24 Scanning electron micrograph of Nb micro- and nanobridge dc SQUIDs produced by electron beam lithography. Source: From Ref. [157]. ©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.

An important aspect of the single-particle measurement using this technique is the location of a nanoparticle (NP) on the SQUID detector. The simplest method for achieving this is by dispersing nanoparticles in an ethanol solution and placing a droplet of the liquid on a chip which has an array of SQUIDs fabricated on it [159]. As the droplet dries, the nanoparticles will adhere to the chip via van der Waals forces. The location of the NP on the SQUID bridge will be by chance; hence, a large number of SQUID structures are usually produced in order to obtain a working device with a NP sufficiently coupled to the SQUID loop. Alternatively, the positioning of the NP can be performed using an atomic force microscope [160, 161]. Improved flux coupling can be achieved by embedding the nanoparticles within the nanobridge junction which has the added advantage of protecting the nanoparticles against oxidation [162, 163]. With regard to the nanoSQUID structures using carbon nanotube (CNT) junctions, the NPs can be introduced into the CNT using an electrospray technique or by a chemical functionalization of the CNT [164]. There are a number of ways in which magnetic information is processed using the micro- and nano-SQUID technique. These are more fully reviewed in Refs. [157, 165, 166]. Here we will outline the main principles along with some representative results on single magnetic nanoparticles. One of the main difficulties in the SQUID detection method is the transition from the superconducting to the normal state, which occurs if the current in the device is ramped up. This critical value of the current at this transition is referred to as the switching current, ISW. Once in the normal state, Joule heating in the device creates a hysteretic response so that currents

ARTICLE IN PRESS 366

David S. Schmool and Hamid Kachkachi

below ISW are required for the device to return to the superconducting state. The problem can be resolved by using the SQUID as a trigger [154]. Since the switching current is a periodic function of flux traversing the SQUID, see Fig. 25, the measurement of the magnetization reversal can be made by biasing the SQUID close to the switching current using an applied magnetic field in the direction perpendicular to the SQUID loop such that it is in the state A or B for the measurement of positive or negative jumps induced by the reversal of the samples’ magnetization, respectively. This reversal will trigger the SQUID to shift from a superconducting to a normal state. This is referred to as the “cold mode” method. The cold mode method can be used in the study of the macroscopic quantum tunneling (MQT) of the magnetization. The process can be understood by the shift of the magnetization from an initial metastable state (before switching), over a saddle point or tunneling through the energy barrier, to a more stable configuration of lower energy. This can be schematically represented as shown in Fig. 26 (left). As the particle overcomes the energy barrier, which is accompanied by a shift of the magnetization through a few degrees, at the initial stages of the reversal process, the coupling of the particle with the SQUID loop means that the change of magnetic flux can drive the SQUID from its metastable state to a new equilibrium, see Fig. 26 (right), and in the process pushing it from the superconducting to the normal state. This process explains why the cold mode method is only used for measuring the switching field associated with the magnetization reversal.

Figure 25 Switching current of a micro-SQUID as a function of the applied magnetic field perpendicular to the SQUID plane. Source: From Ref. [157]. ©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.

ARTICLE IN PRESS 367

Single-Particle Phenomena in Magnetic Nanostructures

E(q )

Magnetic particle

Micro-SQUID U(f )

kBT ΔE MQT

Figure 26 Schematic illustration of the metastable states for a magnetic nanoparticle close to a switching field due to an applied magnetic field and the SQUID which is close to the switching current. Source: From Ref. [157]. ©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.

The principal disadvantages of this mode of operation is its limitation to low applied magnetic fields and low temperatures (below 7 K for the Nb-based SQUID bridges). This initially limited the measurement of magnetic nanoparticles to below the critical temperature, TC, of the superconductor and to two dimensions [167–169]. Later improvements lead to 3D measurements and the possibility to perform studies above TC using the so-called blind method [170, 171]. This can be done using a three-step technique characterized by 1. Saturation: The magnetization of the nanoparticle is at saturation in a certain orientation. 2. Testing: A test field is applied at a temperature between 35 mK and 30 K, which could possibly cause the magnetization to switch. 3. Probing: After cooling to 35 mK, the SQUID is turned on and a field is swept in the SQUID plane to probe the state of the magnetization as in the cold mode method. If a jump in the magnetization is detected (3), this implies that the previously applied test field was below that of the switching field for the particular probing direction used in step (2). The subsequent iteration must then be performed with a larger test field. However, if there is no detected switch in the SQUID, then the implication is that the magnetization has already switched in step (2). Therefore, the following iteration will be made with a lower applied field. An algorithm is used to zero in on the exact switching field in this way. This method allows measurements to be made in various orientations of the applied field and hence can be used to scan in three dimensions.

ARTICLE IN PRESS 368

David S. Schmool and Hamid Kachkachi

Measurements of a single cobalt nanoparticle of 3 nm diameter are illustrated in Fig. 27a and c using the blind method. A good agreement for the switching fields can be obtained using the Stoner–Wohlfarth model with an anisotropy energy given by [172]: B

0.4

0.4

m0Hx (T)

m0Hx (T)

A

0

−0.4

0

−0.4 −0.2

0 m0Hy (T)

m0Hz (T) 0

C

−0.2

0.2

m0Hz (T) 0

0.3 m0Hz (T) 0.3

0 m0Hy (T)

0.2 0.3

m0Hy (T) 0.2

0 0 −0.4

D

m0Hz (T) 0.3

0 m0Hx (T)

0.4

0 m0Hx (T)

0.4

m0Hy (T) 0.2

0 0 −0.4

Figure 27 (a,c) Top and side views of the experimental 3D angular dependence of the switching field of a 3 nm diameter Co nanoparticle. (b,d) Corresponding theoretical switching fields based on a Stoner–Wohlfarth model of the particle energy (see text). Source: Reprinted (figure) with permission from Ref. [170]. Copyright (2001) by the American Physical Society.

ARTICLE IN PRESS 369

Single-Particle Phenomena in Magnetic Nanostructures

0.3

m0Hz(T)

0.04 K 1K 0.2

TB ≈ 14 K

2K 4K 8K

0.1 12 K 0 –0.3

–0.2

–0.1

0 m0Hy(T )

0.1

0.2

0.3

Figure 28 Temperature dependence of the switching field distribution in the y–z plane. Source: Reprinted (figure) with permission from Ref. [170]. Copyright (2001) by the American Physical Society.

  EðmÞ ¼ K1 m2z + K2 m2y  K4 m2x0 m2y0 + m2x0 m2z0 + m2y0 m2z0 v

(80)

Here K1 and K2 are uniaxial anisotropy constants along the z and y directions, corresponding to the easy and hard magnetization axes, respectively. K4 is a fourth-order anisotropy constant in which the (x0 , y0 , z0 ) coordinates are rotated by 45° with respect to (x, y, z) around the z-axis (i.e., z0 ¼ z). Fitting the experimental data with the model expressed by Eq. (80) allows the evaluation of the anisotropy constants: K1 ¼ 2.2 105 Jm3, K2 ¼ 0.9 105 Jm3, and K4 ¼ 0.1 105 Jm3 [170]. The temperature dependence of the switching field was also studied, see Fig. 28, from which the blocking temperature for the particle was evaluated as TB ¼ 14 K. This is obtained from an interpolation of the Stoner–Wohlfarth astroids as a function of temperature. It will be noted that the form of the switching field distribution in the y–z plane is in excellent agreement with that predicted by the Ne´el–Brown model or the Stoner–Wohlfarth model at very low temperature, particularly for the lower temperature measurements, i.e., well below the blocking temperature (see Fig. 2). One should note, however, that the μ-SQUID technique only allows us to measure the field that marks the magnetization reversal without access to the magnetization itself. In particular, this does not allow for a detailed study of the switching mechanisms. Nevertheless, the histogram of the switching probability has made it possible to verify the validity of the Ne´el–Brown model.

3.2 Micro-Hall Magnetometry Micro-Hall sensors can be used to detect the magnetization in submicronsized elements. As with the micro-SQUID technique, the micro-Hall

ARTICLE IN PRESS 370

David S. Schmool and Hamid Kachkachi

magnetometer is a lithographically fabricated device whose reduced dimensions make it sensitive to stray magnetic field in low-dimensional magnetic structures [173]. One of the principal advantages of the micro-Hall over the micro-SQUID measurements is that the former allows the measurement of the magnetic stray field from individual magnetic objects to be made over a much larger range of temperatures (measurements from say 4 K to room temperature are in principle possible) and in the presence of a strong applied magnetic field. The central principle of this technique is clearly based on the Hall effect, i.e., current-carrying charge carriers, in the cross-shaped Hall bar, are deflected via the Lorentz force F ¼ qðv BÞ. The deflected charge is measured as a Hall voltage in the direction perpendicular to the original current path. In the case of the Hall magnetometer, the magnetic field B is generated by the stray field of the magnetic sample under study. The Hall voltage being proportional to the stray field then allows the sample moment and magnetization to be then evaluated. The Hall voltage can be expressed as: UH ¼

IB , nc e

(81)

I represents the applied current in the device, nc is the carrier density, and e the elementary charge of the charge carriers. The factor, 1/nce, represents the Hall coefficient, RH, which is specific to the sensor, i.e., material and doping. The best materials for Hall probes and Hall devices are semiconductors, which have a relatively low carrier density to produce a large Hall coefficient, but not too small as to make measurement difficult. Typically, such devices are made from GaAs/AlGaAs heterostructures which form a twodimensional electron gas (2DEG). For the micro-Hall sensor, a micrometric Hall bar is fabricated, usually via optical or electron beam lithography, from appropriate heterostructures to form the 2DEG [174]. It has been demonstrated that in the ballistic regime, the Hall voltage is proportional to the magnetic stray field averaged over the active region of the Hall sensor, which is determined by the intersection of the current and voltage paths [175]; this corresponds to the lighter blue (gray in the print version) region illustrated in Fig. 29. One important consideration for the micro-Hall sensor is that the sample must be smaller than the geometrical area of the active region of the device. One method of ensuring a good sample/sensor geometry is using the implantation of oxygen ions into the GaAs/AlGaAs heterojunction, which helps to define the

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

371

Figure 29 Schematic illustration of a Hall sensor with a magnetic disk/particle on its surface. Source: Reprinted from Ref. [174]. Copyright (2002), with permission from Elsevier.

effective geometric shape of the Hall sensor [176].2 The magnetic nanostructure is then deposited on top of the Hall sensor, as illustrated in Fig. 29 [174]. As an example, we can consider the study, by Rahm et al. [176], of a nanoring deposited on the top of a double Hall device. The Ni ring has inner and outer dimensions of 0.3 and 1.2 μm, respectively. The Hall measurement shown in Fig. 30a were made at a temperature of 110 K and show good agreement with micromagnetic simulations and other measurements made on magnetic nanoring structures [177–179]. The distinct jumps in the magnetization are attributed to the switching between different stable states as the magnetization reverses. The first jump occurs when the magnetization changes from a bidomain state, where the magnetization circulates in opposition in each half of the ring, to a vortex state. A second jump takes place when the vortex state is annihilated at saturation, as illustrated in Fig. 30a. (It should be noted that the jumps in this hysteresis loop are not of the same origin as those obtained in the multispin model, in Fig. 18.) Another study by Rahm et al. [174] on a Ni pillar-shaped element (170 370 850 nm) is shown in Fig. 31. Due to the shape anisotropy and the orientation of the applied magnetic field, the magnetic dot acts like a single-domain particle for magnetization reversal along its long axis. The temperature dependence of the switching field is clearly seen as the sample temperature increases. The reversal process is governed by a surface oxide layer which is capable of pinning the magnetization. A curious feature of the measurements shown in Fig. 31 is that while the noise is relatively small for low-temperature and the high-temperature loops, the MH loop at 130 K shows a significant increase in the noise level. This is associated with the 2

It should be noted that regions where there is sufficient oxygen implantation, the semiconductor becomes insulating. This can therefore be used to define the active region of the device.

ARTICLE IN PRESS 372

David S. Schmool and Hamid Kachkachi

Figure 30 (a) Hysteresis loop, which was measured at 110 K, is typical of magnetization reversal via a magnetic vortex state. The insets sketch different magnetic configurations during the reversal process. (b) SEM image of the Hall sensor with the Ni nanoring deposited on its surface. The areas within the broken white lines are ion implanted and depleted of free electrons. The SEM image does not reveal any visible change of the surface originating from ion implantation. Source: Reprinted with permission from Ref. [176]. Copyright (2002), AIP Publishing LLC.

Figure 31 Micro-Hall measured hysteresis loops of a Ni pillar taken at sample temperatures. The inset shows an SEM image of the nanopillar deposited on top of the microHall sensor. Source: Reprinted from Ref. [174]. Copyright (2002), with permission from Elsevier.

mean free path of the charge carriers in the micro-Hall device. The maximum of the noise level occurs when the mean free path of the carriers is of the order of the lateral dimensions of the Hall sensor, that is, when the transport regime changes from the ballistic to the diffusive regime.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

373

A slightly different approach to the electrical measurement of the magnetization of single particles was adopted by Li et al. [180], who used spinvalve sensors fashioned into rectangular strips of roughly 1 μm in width. The spin valve has a free and pinned layer, and when a magnetic field aligns the free layer with the pinned layer a significant change in electrical resistance is measured in the structure. This magnetoresistance (MR) is the basis of the measurement of the magnetic particle, whose stray field acts as the aligning magnetic field in the device. In this work, the MR was about 10% and the sensitivity is enhanced using a Wheatstone bridge circuit in which two spin valves are used, one with the magnetic sample to be measured, the other as a reference in the bridge. The sensitivity of the device will also clearly depend on its size. Using a 1 μm wide device, the authors were able to detect 11 nm diameter Co nanoparticles. The micro-Hall technique represents a novel method for measuring the hysteresis loop of relatively large nanostructures. It shows good sensitivity, particularly at low temperatures. While no specific studies have been performed on single nanoparticles, it would be a candidate technique, though it is not sure if the sensitivity would be sufficient. It would also suffer from the problems encountered in the micro-SQUID with regard to placing the nanoparticle in the active device region. At higher temperatures, it would probably not measure any signal from a superparamagnetic particle, so low temperatures would probably be best suited for single nanoparticle studies.

3.3 Lorentz Transmission Electron Microscopy Of the imaging techniques available, transmission electron microscopy was probably one of the earliest to be applied to the study of magnetic domains in ferromagnetic materials. Electrons have an intrinsic charge and are therefore subject to the Lorentz force as they traverse a magnetic sample and are deflected. This force can be expressed as: F ¼ jejðv BÞ

(82)

e being the electric charge, v its velocity, and B is the effective magnetic field averaged along the trajectory of the electrons. When using Eq. (82), it should be noted that only components of the magnetic field normal to the electron trajectory will give rise to a deflection of the electron path. Since the image formed in Lorentz TEM (or LTEM) arises from the transmission of the electrons through the sample, it will be dependent on the specific magnetic domains of the ferromagnetic material under examination. It is a simple

ARTICLE IN PRESS 374

David S. Schmool and Hamid Kachkachi

matter to deduce the deflection angle of the electrons as they emerge from a magnetized sample and can be expressed as: γL ¼

eλL ðv BÞ, h

(83)

where h is the Planck constant, L is the sample thickness, and λ the electron wavelength, which is evaluated from the accelerating potential of the electron gun. The deflection angle will result from the average magnetic induction in the sample and over its thickness. Such deflections are typically less than 100 μrad [181]. One of the main attractions of LTEM is the excellent spatial resolution it offers and hence applicability for the study of magnetic nanostructures. While TEM as a structural measurement has a spatial resolution of around 0.5–2 nm, this increases to tens of nanometers for magnetic information in LTEM [181]. The adaptation of the LTEM technique means that ferromagnetic samples, and in particular their domain state, can be observed in both the remanent state as well as under an applied magnetic field to observe the magnetic reversal process and as a function of temperature. Furthermore, electrical currents can also be applied in situ for studies of spin currents on nanostructures. Another important application of Lorentz microscopy is the study of domain wall motion and dynamics in nanostructures. There are various imaging modes for the electron microscope and for Lorentz microscopy in general; these will not be discussed in detail here, see, for example, Refs. [181, 182] and references therein for more detailed information. The deflection of electrons as they pass through a sample with magnetic domains can give rise to regions of light and dark contrast, which for a stripe domain pattern, with 180° domain walls, can be depicted as shown in Fig. 32. Here the electron beam is slightly defocused at the sample and leads to an intensity enhancement or an intensity reduction at the domain boundaries. In analogy to the variation of light deflected in a specimen of varying optical thickness in a defocused image, this mode of operation of Lorentz microscopy is called the “Fresnel mode.” The contrast here is dependent on the relative orientation of the magnetization in the magnetic domains. For head-to-head (180°) domain walls, the electrons will not experience a net deflection since contributions from within the sample will cancel out with the stray magnetic field. Other imaging modes are possible and result from the manipulation of the focusing of the electron beam and partially blocking certain orders of the diffracted beam, as in the Foucault method [181, 182]. Higher image resolution and domain orientation

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

375

Figure 32 Schematic explanation of Lorentz microscopy. Electrons that traverse the sample are deflected by the Lorentz force from the B field inside and outside the sample. 180° domains are separated by domain walls running along the magnetization axis. The B field is confined to the interior of the sample. The deflection of the electrons after transmission through the sample leads to a modulation of the electron beam intensity.

analysis are possible using the differential phase contrast (DPC) mode of operation and require scanning capabilities (STEM). The beam is focused on the sample and deflection is monitored using a quadrant detector. In Fig. 33, an example of a DPC image is shown where the domain structure of a rectangular permalloy microelement is illustrated. The DPC mode can be used under applied magnetic fields and can thus image domains throughout the hysteresis cycle. This technique has been very successfully exploited for the study of domain wall motion in magnetic nanostructures, such as the effects of pinning via notches and other traps in nanowires [183, 184]. While the domain imaging of magnetic elements is reasonably common in LTEM, where typically the samples are multidomain, those for magnetic nanoparticles are not so frequent. Both Foucault and Fresnel modes have been applied to the imaging of magnetic particles. In the Fresnel mode, the sample can be imaged with the focal plane either above or below the sample [185]. In this case, the electron trajectories either converge or diverge and the particle appears as bright or dark features. Moving through underfocus to overfocus, Co particles of 50–100 nm in precipitates of AuCo were shown to change from light to dark regions [186]. Needle-shaped particles of γ-Fe2O3 with lengths between 65 and 300 nm have been studied using the Foucault imaging mode by Salling et al. [187]. These were magnetized along their long axes. Shifting the objective aperture, dark lobes would appear on either side of

ARTICLE IN PRESS 376

David S. Schmool and Hamid Kachkachi

Figure 33 Lorentz microscopy DPC images of a 40-nm-thick permalloy element illustrating the magnetic domain pattern. Source: From Ref. [181]. With kind permission from Springer Science and Business Media.

the particle. This was attributed to the stray field around the particles. Reversing the magnetization of the particles would cause the lobes to disappear, with the particles exhibiting the same appearance as for the bright-field image. Nanoparticles of SmCo5 with sizes 5–50 nm and dispersed in a thermoplastic matrix were also studied using the Foucault mode by Majetich and Jin [188]. Using bright-field conditions, the objective aperture was shifted to cut off part of the transmitted electron beam. Images were taken for various aperture shift directions. The aperture shift angle, θA, was determined from electron diffraction images. With all aperture shifts, the nanoparticles show the two dark lobes observed by Salling et al. [187]. The orientation of the lobes was the same for all particles in an image, with a size scaling to the particle diameter. A sequence of images for the SmCo5 particles was studied as a function of the aperture angle. For certain shift angles, the center appears dark (this can be seen in fig. 1 of Ref. [188]). The dark lobes are evidence of the fringing field of the nanoparticles. The angle at which the dark center is maximized reveals the orientation of the particles’ magnetization.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

377

Majetich and Jin also studied Fe3O4 nanoparticles of about 10 nm diameter using the same technique [188]. The images LTEM (given in figure 4 of Ref. [188]) show a switching event, where a change of the magnetization direction between two equivalent h111i directions has occurred. The fact that the image shows a stable magnetization is taken as evidence of the strongly enhanced anisotropy, due to surface effects, being about 20–25 times larger than the bulk magnetocrystalline anisotropy for Fe3O4. Yamamoto et al. [189] also studied Fe3O4 nanoparticles, though in closely packed arrays where strong dipolar forces act to produce some coherent behavior. It is possible to perform other types of measurements simultaneously with LTEM, such as magnetoresistance, where electrical contacts are made to ferromagnetic nickel strips where a direct correlation between the appearance of magnetic domains and anisotropic magnetoresistance was observed. Using high-frequency feedthroughs, it should also be possible to perform correlated measurements of dynamic properties with magnetic imaging. Of the LTEM measurements made on nanoparticles, the study of relaxation properties and the relation to surface effects is probably the most significant.

3.4 Magnetic Force Microscopy Magnetic force microscopy or MFM is an example of a scanning probe microscopy (SPM) technique, such as scanning tunneling microscopy (STM) and atomic force microscopy (AFM). The STM was the first of the SPM techniques to be developed, which was in the early 1980s by Binnig, Rohrer, and coworkers [190–193]. This method relies on a small tunneling current that passes from an atomically flat metallic or semiconducting surface into a fine tip placed in close proximity. This method has shown an extraordinary spatial resolution and is commonly used to map surface crystallographies in real space with atomic resolution [194]. The STM technique is sensitive to the density of electronic states in both surface and tip and has been developed into scanning tunneling spectroscopy (STS) [195, 196]. In the year, Binnig and Rohrer were awarded the Nobel prize (1986) for their development of the STM; Binnig was also instrumental in the development of the AFM [197]. This adaptation of SPM measures the atomic forces between a tip placed on a soft cantilever and the surface of a sample, where there is no limitation to the type of sample studied. Forces as small as pN can be detected. The deflection of the tip is produced by the electrostatic forces between the tip and the sample surface. In the following year, the first development of the MFM was made by Martin and Wickramasinghe [198], by changing the AFM tip to a magnetic one. The

ARTICLE IN PRESS 378

David S. Schmool and Hamid Kachkachi

Figure 34 MFM images of cobalt cylinders patterned by interference lithography. Diameters are 100 nm (a) and 70 nm (b), while heights are 40 nm (a) and 100 nm (b). Shape anisotropy results in an in-plane (a) or out-of-plane magnetization (b), with very different contrasts matching those expected from in-plane and out-of-plane oriented dipoles, respectively. Source: From Ref. [201]. With kind permission from Springer Science and Business Media.

magnetic dipolar forces (or the force gradients) between a magnetic tip and a magnetic sample, such as a thin film, allow the deflection of the cantilever to be mapped into the domain structure of the sample.3 This allows both the detection of normal deflection of the cantilever as well as torsional motion. Tip–surface interactions can be crucial, since the tip can induce changes in the magnetic film under investigation [199]. Since the magnetic forces are of longer range than the atomic forces in AFM, the spatial resolution of MFM is inferior, typically being no better than tens of nanometers, while AFM technically has atomic resolution. Spatial resolution is also dependent on the tip size. To these ends, Winkler et al. [200] used multiwalled carbon nanotubes filled with Fe to produce tip sizes of the order of 100 nm. The MFM technique has been mostly applied to the study of magnetic domains in thin magnetic films and thin film microstructures. However, there have been some MFM studies on other systems. In Fig. 34, we show the MFM image for lithographically patterned Co cylinders with diameters of 70 and 100 nm [202]. The two sets of data show the in-plane nature of the magnetization in the larger diameter shorter length structures (D ¼ 100 nm, h ¼ 40 nm), while for the longer narrower cylinders (D ¼ 70 nm, h ¼ 100 nm) the magnetization is out of plane.

3

Typically using the reflection of a laser beam on the rear side of the cantilever onto a quadrant detector.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

379

The MFM can operate in dc or ac mode. In the former, the cantilever reacts to the static force between tip and sample, as given by Hooke’s law: F ¼ kΔz, where k is the spring constant and Δz the tip displacement. In ac mode, the tip/cantilever is driven near to its natural resonance frequency, which for an approximation as a classical harmonic oscillator can be expressed as [201]: rffiffiffiffiffiffiffi 1 keff (84) f¼ 2π m where m is the effective mass of the tip/cantilever and keff is the effective spring constant, which is comprised of two components: keff ¼ k 

@F , @z

(85)

the first term being the actual spring constant and the correction term depends on the force gradient with respect to the sample–tip separation. This can effectively stiffen the cantilever due to their mutual interaction. The force gradient will produce a shift in the resonance frequency of the cantilever, which can be expressed in the form: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 @F (86) , f ¼ f0 1  k @z with f0 being the free natural resonance of the cantilever. As the tip is scanned across a surface, any tip–sample interaction changes will result in a shift of the resonance frequency of the cantilever. The measurement for the magnetic force is made by either detecting changes in the amplitude of the resonance or in the resonance frequency itself. If we assume that the sample produces a magnetic stray field Hs to which the tip magnetization Mt is subjected, the tip– sample magnetic potential can be written: R Es  t ¼ μ0 V Mt  Hs dVt : (87) The force acting on the tip can then be written in the form: R F ¼ rEs  t ¼ μ0 V rðMt  Hs ÞdVt :

(88)

Integration is taken over the volume of the tip Vt. Considering the dipolar interaction between the tip and a nanoparticle, it can be shown that the force derivative takes the form [203]:

ARTICLE IN PRESS 380

David S. Schmool and Hamid Kachkachi

@F 6μ0 mp mt : ¼ @z πðs + z0 Þ5

(89)

Here mp,t are the magnetic moments of the particle and the tip, respectively, s is the tip–surface distance, and z0 ¼ dm/2 + dt/2 + d0, where dm and dt are the diameters of the NP and tip, while d0 is the thickness of a passive layer on the NP surface, which can account for any coating of the NP surface (see Fig. 35). To remove surface topography features from the magnetic (domain) image, a two-pass (tapping-lift) mode is frequently used. In the first pass, the tapping (semi-contact) mode is used to map surface topographical features. The second pass is made by lifting the cantilever a set distance and follows the surface contours from the first pass at a constant height above the sample. At the new set height, only the longer range magnetic forces are, in principle, detected. Magnetic force microscopy is primarily used as a technique to elucidate the magnetic domain structure in magnetic thin films and micron-sized elements, see, for example, Okuno et al. [205]. It has also been successfully applied to the study of reversal dynamics in arrays of micron-sized elements [206]. However, MFM has also been used to probe smaller structures of the 100 nm range. Mironov et al. [207] have used MFM to study magnetic vortex chirality in elliptical (400 600 27 nm) Co nanoparticles. Much smaller iron oxide nanoparticles (11 nm diameter) formed the study of Torre et al. [204]. The difference between topological features

dt/2 d0

s dm/2

Figure 35 Schematic diagram of the MFM tip and a magnetic nanoparticle indicating the diameters of the tip and particle as well as a passive layer at the NP surface. The dipole separation is therefore obtained from s + d0. Source: Reprinted by permission from Macmillan Publishers Ltd: ScientificReports [204]. Copyright (2011).

ARTICLE IN PRESS 381

Single-Particle Phenomena in Magnetic Nanostructures

(hills and valleys) and magnetic signals are shown and agglomerates of particles occur in elongated aggregates. Each chain of NPs, of a few hundred nm, acts as a magnetic domain. The results indicate that single-particle detection should be attainable. So while the MFM resolution is not high enough to see features of the nanoparticle dimensions, single-particle measurement can in principle be possible if the magnetic stray field generated by the particle is sufficiently strong. Indeed, Dietz et al. [208] claim single-particle detection for 5-nm superparamagnetic particles of ferritin and apoferritin in liquid and air. For these measurements, the authors use a bimodal mode of operation, modulating the tip at two frequencies. This involves the mechanical excitation of two cantilever eigenmodes, which allows for the simultaneous imaging of mechanical and magnetic information. In Fig. 36, we show the bimodal phase shift image indicating two different structures: ring-like and full nanoparticles. The ring-like structure is given by ferritin, while the flat disk is given by apoferritin molecules. The bimodal phase shift is sensitive to the presence of both short-range mechanical (repulsive) and long-range magnetic (attractive) forces. As with other magnetic imaging techniques, MFM has a relatively long measurement time and may not be the best method for studying superparamagnetic systems. However, there does seem to be sufficient sensitivity measuring weak stray magnetic fields from nanoparticles.

Iron oxide core

Mechanical and magnetic properties

Peptide shell f2

X

y

Figure 36 Bimodal phase shift image of the iron oxide core, the peptide shell, and an image reconstruction of ferritin revealing the mechanical and the magnetic properties. Source: From Ref. [208]. ©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.

ARTICLE IN PRESS 382

David S. Schmool and Hamid Kachkachi

3.5 X-Ray Magnetic Circular Dichroism (XMCD) and X-Ray Photoemission Electron Microscopy (X-PEEM) Photoemission is a well-known physical process whereby an electron is emitted from a solid surface upon the absorption of an incident photon. The photoelectron can be energy analyzed to obtain information about the electron states in the solid from which it emerged. This is the basic process for photoelectron spectroscopies such as ultraviolet photoelectron spectroscopy (UPS) and X-ray photoelectron spectroscopy (XPS). Photoemission electron microscopy (PEEM) combines photoelectron spectroscopies with microscopic techniques. PEEM was initially developed with UV sources, but later synchrotron-based X-ray sources were exploited due to the greater element specificity that it offers with soft X-rays [209]. Synchrotron sources offer a multitude of techniques among which excellent chemical and magnetic contrast is available using X-ray magnetic circular dichroism (XMCD). This method permits microscopy options with a spatial resolution in the range of tens of nanometers [210, 211], making it an excellent technique for the study of nanomagnetism. Further developments were made possible by exploiting the pulsed nature of synchrotron radiation, conferring the potential for dynamic studies of magnetization processes using pump–probe methodologies [212]. The pulse duration using X-rays defines the ultimate limit of temporal resolution available, as with all pump–probe measurements, which in this case is of the order of 50–100 ps, which is sufficient to observe various magnetic processes, such as the precessional dynamics of ferromagnetic resonance (see Section 2.2.3), relaxation mechanisms, and vortex gyrotropic motion in magnetic nanoelements [213, 214]. The basic structure of the experimental setup for PEEM is illustrated in Fig. 37. An image of the position from which the photoelectrons emerge from the sample is formed in the back focal plane by an area detector, which is usually a multichannel plate and scintillator screen. Since the photoelectrons (most of which are secondary electrons) are excited by a soft X-ray source, the photoelectrons are generated within the mean free electron path of the surface, which is 2–5 nm according to the universal curve [215]. The spatial resolution of PEEM is limited by spherical and chromatic aberrations in the electron optics as well as the electron mean free path. For an electrostatic system, this usually corresponds to about 20 nm. However, the soft X-ray excitation means that the kinetic energy distribution can be broadened and increases chromatic aberration. This can compromise the resolution to limit it to around 50–100 nm [209]. Since the photoelectrons are

ARTICLE IN PRESS 383

Single-Particle Phenomena in Magnetic Nanostructures

Screen

Projector lenses Aperture Incident photon beam

Extraction lens

Figure 37 Schematic illustration of the PEEM instrument using an X-ray source. Photoemission is stimulated by the excitation from the incident X-ray photons. These are then accelerated into the column by an extraction lens. A contrast aperture is used to limit the angular acceptance angle of the photoelectrons; this will also reduce spherical aberration in the system. The electrons then pass through a system of projection lenses and an image is formed on an area detector. Source: From Ref. [209]. ©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.

mainly secondary in nature, chemical information is available since core level states can be probed and chemical mapping is possible. Magnetic sensitivity is derived by polarizing the incoming radiation, making it sensitive to the spin and orbital occupancy of the electrons in the solid. For the range of energies of soft X-rays (200–2000 eV), absorption is dominated by core state resonances and transitions to unfilled bands where the magnetism originates. Such resonances can be particularly strong for 3d and 4f final states, making the photoelectrons very sensitive to the magnetism of the sample. If circularly polarized radiation is used, there is an additional contribution to the absorption, giving rise to the XMCD effect. This strong dependence arises in ferromagnetic materials due to the difference in the spin density of states for spin-up and spin-down electrons for the final state band. For transition metal ferromagnets, the relevant transitions correspond to 2p–3d levels for the L2,3 X-ray absorption edges. For rare-earth metals, it is the M4,5 edges which allow XMCD to directly probe the f states [211]. In Fig. 38, we show a schematic illustration of the absorption transitions for left and right circular polarized light for Fe. Early work on XMCD was performed by Schu¨tz and coworkers in 1987 for iron [216]. In the transition scheme, it is assumed that the d shell spin moment is given by the difference in number of spin-up and spin-down electrons for states below the Fermi

ARTICLE IN PRESS 384

David S. Schmool and Hamid Kachkachi

Fe metal 10 8

L3

L2

6

3d

Left circular polarization

Right circular polarization →

E

2p3/2 2p1/2

Linear absorption coefficient

4

EF

2 0 10 8 6 4 2 0 10 8 6 4 2 0 690 700

710 720 730 740

Photon energy (eV)

Figure 38 Principles of X-ray magnetic circular dichroism spectroscopy, illustrated for the case of L-edge absorption in a d-band transition metal. In a magnetic metal, the d valence band is split into spin-up and spin-down states with different occupation. Absorption of right (left) circularly polarized light mainly excites spin-up (spin-down) photoelectrons. Since spin flips are forbidden in X-ray absorption, the measured resonance intensity directly reflects the number of empty d-band states of a given spin. In XMCD spectroscopy, it is equivalent whether the photon polarization is changed and the magnetization direction is kept fixed, or whether the magnetization direction is changed and the photon helicity is fixed. The corresponding XMCD spectra for Fe metal [217] are shown on the right for three different orientations of the magnetization directions relative to the fixed photon spin (right circular polarization). Source: Figure reproduced with permission from Ref. [211].

energy, EF. For states above EF, the moment is given by the imbalance in the spin-up and spin-down holes. Since left circular polarized (LCP) and right circular polarized (RCP) light transfers angular momenta of amplitude + ℏ and ℏ, respectively, the absorption of each (LCP and RCP) will depend on the occupancy of initial and final states for the spin-up and spin-down electrons. Spin flips are forbidden in electric dipole transitions governing X-ray absorption; therefore, spin-up (spin-down) photoelectrons from the p core shell can only be excited into spin-up (spin-down) d hole states. Thus, the spin splitting of the valence shell will perform the function of spin detector, where the transition probability will be reflected in the intensity of the absorption. The dichroic effect varies with the angle between the

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

385

magnetization Ms and the angular momentum of the photon σ being a maximum when they are either parallel or antiparallel. Therefore, we can write XMCD ∝Ms  σ. The L3 and L2 resonances arise from the transitions between spin–orbit split 2p3/2 and 2p1/2 core levels and the d bands, respectively. Absorption, while strongest at the L-edges, is generally not sufficient for imaging purposes and for PEEM; the image is formed by taking the difference, divided by the sum of the LCP and RCP signals, which is the same as that used in XMCD spectroscopy. A linear dichroic effect also exists due to spin–orbit coupling (XMLD), though its use is more limited than that of XMCD. It is generally more used for the study of antiferromagnetic samples [211]. The construction of the XMCD spectra is relatively simple and can be performed as follows. The normalized transmission X-ray absorption spectra (XAS) are taken of the thin magnetic film with the projection of the spin of the incident photon parallel (I+) and antiparallel (I) to the magnetization (spin of majority 3d electrons) of the magnetic layer. The magnetizationindependent measurement of the substrate (IS) is measured to evaluate the relative absorption cross section using the formula: μ ðωÞ∝  ln ½I ðωÞ=IS ðωÞ. It is important to take into account the angle of incidence θ and degree of circular polarization P of the incident X-rays. This is done by multiplying [μ+(ω)  μ(ω)] by ½1= cosθ=P and maintaining μ+(ω) + μ(ω) [217]. In Fig. 39, we show these parameters of the XAS and XMCD for Fe and Co grown on parylene substrates. For the sum rules of XMCD, the orbital and spin magnetic moments can be determined from the XAS and XMCD spectra as [218, 219]: R

L + L ðμ +  μ Þdω morb ¼  R 3 2 ð10  n3d Þ, 3 L3 + L2 ðμ + + μ Þdω

4

mspin ¼ 

6

R

L3 ðμ +

(90)

R    μ Þdω  4 L3 + L2 ðμ +  μ Þdω 7hTz i 1 R ð10  n3d Þ 1 + : 2hSz i L3 + L2 ðμ + + μ Þdω (91)

These are given in units of μB/atom. n3d is the 3d electron occupation number for the particular transition metal in question, while hTz i is the expectation value of the magnetic dipole operator and hSz i is equal to half of mspin in Hartree atomic units [217]. The L2 and L3 denote the ranges of integration. Integrated quantities are also shown in Fig. 39(c) and (d). The

ARTICLE IN PRESS 386

David S. Schmool and Hamid Kachkachi

B 1.0

I+ I– IS

0.8

(b)

Iron

0.2

–0.1

–0.2

p m+ –m–

–0.2

0.0

–0.4

⌠ m+ –m– ⌡

I+ I– IS

0.8

(b)

Cobalt

m+ m–

0.4 0.2

0.2

(c)

0.0

0.0 q –0.2

–0.1

p

–0.2

m+ –m–

–0.4

⌠ m+ –m– ⌡

–0.6

–0.6

(d)

0.4

4.0 r

2.0

0.0

0.0 700

740 720 Photon energy (eV)

760

m+ +m–

0.8 XAS

⌠ m+ +m– ⌡

XAS integration

m+ +m–

0.8

4.0

⌠ m+ +m– ⌡

0.4

r

2.0

XAS integration

(d) XAS

L2

L3

MCD integration

q

MCD integration

0.0

(a)

1.0

0.0 0.1

0.2

(c)

1.2

0.6

m+ m–

0.4

0.0 0.1

MCD

Transmission

L2

L3

0.6 Absorption

(a)

Absorption

1.2

MCD

Transmission

A

0.0

0.0 760

780 800 820 Photon energy (eV)

Figure 39 XAS for the L2, 3 edges and XMCD for (A) Fe: (a) transmission spectra of Fe/parylene thin films, and of the parylene substrates alone, taken at two opposite saturation magnetizations; (b) the XAS absorption spectra calculated from the transmission data shown in (a); (c) and (d) are the XMCD and summed XAS spectra and their integrations calculated from the spectra shown in (b). (B) The corresponding quantities for a Co thin film. Source: Reprinted (figure) with permission from Ref. [217]. Copyright (1995) by the American Physical Society.

value of hTz i=hSz i can be determined from a first-principles band structure calculation, yielding values of  0.38% for bcc Fe and  0.26% for hcp Co [220, 221]. If hTz i=hSz i is neglected, the ratio of orbital to spin momenta (mspin/morb) can be evaluated from the values determined for p and q from Fig. 39(c) as 2q/(9p  6q). The values of the orbital to spin moments have been assessed as 0.043 for Fe and 0.095 for Co, and are in good agreement with other works (see Ref.[217] and references therein). The vortex magnetic structure of micrometric ferromagnetic elements has formed the subject of numerous studies by X-PEEM. The gyrotropic motion of the vortex core has also been performed using time-domain measurements. For example, Choe et al. [213] studied 20-nm-thick CoFe elements by imaging the Co L-edges as a function of delay time. The structures were deposited on a coplanar waveguide which formed part of a photoconductive switch in their pump–probe experiment. The core center was traced as a function of time,

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

387

Figure 40 (a): (A) The static domain configuration of two ferromagnetic rectangles of 1.5 μm 1 μm size. Both samples show an identical Landau flux closure structure. In (B), the domain walls have been made visible by calculating the derivative of M. (C,D) The trajectory of the vortex center. Surprisingly, the vortex centers of the two structures move in opposite direction, although they show the same Landau domain structure. (b): (A) Spin structure of a left- and right-handed vortex structure. The blue (dark gray in the print version) arrows represent the precessional motion of the magnetization caused by the magnetic field pulse. The magnetization direction of the vortex center determines the initial direction of vortex motion parallel or antiparallel to the magnetic field pulse. (B,C) The simulated trajectory for two magnetization directions of the vortex core. Source: Reproduced from Ref. [225]. Copyright ©2008 Elsevier Masson SAS. All rights reserved.

where results show the circular and elliptic gyrotropic motion for different shaped elements. This study is significant as it was the first to demonstrate the direct observation of gyrotropic motion of the vortex core in the time domain. Other authors have shown similar time-resolved measurements in elongated rectangular magnetic elements with cross-tie domains [214], in an asymmetric magnetic disk using current-induced vortex dynamics [222] and nonlinear nongyrotropic motion in permalloy disks as a function of an applied magnetic field [223, 224]. In Fig. 40, we show an example of a study of the gyrotropic motion in a magnetic element. In other measurements, Wegelin et al. have used X-PEEM to study standing spin-wave modes in permalloy thin film structures [226]. This is important since it shows that X-PEEM can be used to provide a direct observation of a spin-wave mode that cannot be attained by the measurement of the

ARTICLE IN PRESS 388

David S. Schmool and Hamid Kachkachi

resonance frequency. It can only be inferred with modeling and simulations. These authors noted that traveling spin waves can also be observed with this method. Single nanoparticle detection using X-PEEM has been demonstrated by Fraile Rodrı´guez et al. [227] for Fe NPs deposited on a Co surface. Of particular importance in this study was the fact that the element specificity of PEEM means that the Fe signal can be measured without “interference” from the Co signal, both are in fact measured separately by tuning the measured energies (708.1 and 778.1 eV, respectively, for the L3 edges of Fe and Co). In this way, the effects of the exchange coupling between the Fe particles and the Co substrate can be investigated. Since the magnetic coupling between the nanoparticle and the Co surface will be strong, by varying the size of Fe particle studied the magnetic anisotropy energy (MAE) relative to the exchange interaction will alter; thus, above a certain size, the MAE of the particle will become dominant over the exchange, resulting in an independent orientation of its magnetization with respect to that of the substrate. The critical size of 6 nm was found where the Fe nanoparticle has a noncollinear spin structure due to the competition between its local anisotropy and that exchange coupling to the Co substrate.4 Particles of a smaller diameter will be single domain and collinear with the Co magnetization. Since the PEEM resolution is significantly lower than the NP diameter, the NPs appear as bright spots, whose intensity is related to the particle size. Also, the projection of the local magnetization will vary on a grayscale depending on the orientation with respect to the photon propagation vector σ. Figure 41 shows the superposition of the Fe and Co XMCD images of the same area. Analysis of the image reveals a variation of the Fe magnetization orientation with respect to that of the Co underlayer. It is noted that due to the exchange coupling between the Fe NPs and the Co substrate, the magnetization of the Fe is stable during the measurement. A quantitative assessment of the NP magnetization was performed by taking measurements as a function of the azimuthal angle of the sample with respect the propagation direction. The variation of the asymmetry ratio for XMCD allows the local direction of the magnetization to be determined. A combination of XMCD and AFM on the same particles allowed the variation of angle between the magnetizations, Δϕ, of the Fe and Co to be

4

This is reminiscent of the variation of spin structures in exchange-spring systems where a consideration of the local spin energies can reveal the specific spin structure expected [228, 229].

ARTICLE IN PRESS 389

Single-Particle Phenomena in Magnetic Nanostructures

4 μm

Figure 41 Superposition of Fe and Co XMCD images. Different shades of gray correspond to different magnetization orientations. It will be noted that the Fe NPs appear as black dots in the areas of the Co surface for which there is a “dark” magnetic domain, while they appear as white dots for which the Co magnetic domain is bright. This reflects the fact that these NPs have a magnetization which is collinear with that of the Co substrate. Source: Reprinted (figure) with permission from Ref. [227]. Copyright (2010) by the American Physical Society.

found as a function of particle size.5 For particles larger than 6 nm, a broad distribution of canting angles was observed, while for those smaller than this size the magnetization of the Fe NPs is practically collinear with that of the Co substrate, where exchange coupling dominates. For larger particles, the variation of canting angle is indicative of a transition to an anisotropydominated regime. To understand the experimental results, the authors use a model of the magnetic energies [230] (see also model presented in Ref. [228]) in which the magnetization of the substrate is placed perpendicular to the easy axis of the nanoparticle magnetic anisotropy, nMAE. All magnetizations are considered to be in the plane of the sample. The exchange coupling between the Co substrate and Fe NPs is considered sufficient to pin the Fe spins at the interface, i.e., taken parallel to the Co magnetization. Spins above the interface atomic layer are free to rotate in the plane, ϕ(z). Optimization of the spin structure is taken by minimizing the total energy: exchange and MAE, EMAE ¼ Ku cos 2 ϕ. The relevant parameters used are the exchange stiffness constant for Fe, A ¼ 62 meV/m, and the anisotropy constant of Ku ¼ 50 μeV/atom. A spiral spin structure is obtained for a 10-nm particle, as illustrated schematically in Fig. 42. In the experiment, the PEEM 5

Note that for these experiments, a reference structure was integrated onto the substrate to permit the localization of the NPs for correlating the PEEM and AFM measurements.

ARTICLE IN PRESS 390

David S. Schmool and Hamid Kachkachi

z

Δf

D

Figure 42 Schematic illustration of the spiral spin structure deduced for a Wulff-shaped particle of 10 nm. Source: Reprinted (figure) with permission from Ref. [227]. Copyright (2010) by the American Physical Society.

measures the angle of the upper most spins of the nanoparticle, so only the difference between the Co magnetization orientation and that of the upper Fe spin can be observed. Fe nanoparticles with bulk-like properties (magnetization and bcc lattice), with sizes in the range from 8 to 20 nm, were studied by Balan et al. [231]. The authors measured the X-PEEM images at room temperature with no applied external magnetic field and found a significant quantity ( 40%) of particles which appear magnetically stable over a period of hours. This indicates a blocking of the particles in a ferromagnetic (FM) state for periods much greater than would be expected from the relaxation time determined by the usual Arrhenius law. The remainder of the particles show the usual superparamagnetic (SPM) behavior, where there seems to be no correlation between the particle size and FM/SPM behavior. Heating of the sample shows that the magnetic contrast is lost for the FM particles (between 360 and 375 K), indicating a decrease of the relaxation time to below the measurement time. Upon cooling the sample back to room temperature, only about a third of the FM particles recover their magnetic contrast; some show contrast reversal, showing that thermally activated magnetic switching has occurred. Heating to 420 K is sufficient for all FM particles to lose their magnetic contrast. Furthermore, transitions from FM to SPM are also encountered at room temperature, where there is a suggestion of a transition via transient states. A consideration of the magnetic anisotropy energy (MAE) contributions (due to magnetocrystalline, shape, and surface contributions, though the latter can be excluded from size considerations here) leads to an estimation of the relevant energies of MAE for FM/SPM behavior. For cubic Fe with bulk-like properties, a total MAE of 1.35 eV is expected for the magnetocrystalline anisotropy, leading to an energy barrier (EB ¼ EMAE/4) of 0.34 eV, for a 20-nm NP. From this, an SPM behavior is expected at room temperature. A further uniaxial contribution would be required to increase the energy barrier for FM to occur. The conclusion is that some form of strain-induced uniaxial anisotropy is introduced into the particles, possibly during the preparation of the NPs and their impact on the

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

391

surface during deposition. This can cause the introduction of dislocations or other defects, giving rise to a uniaxial magnetic anisotropy.

3.6 Ferromagnetic Resonance Ferromagnetic resonance is a well-established technique whose origins date back to the development of microwave technologies and the invention of radar. Microwave spectroscopy is a general method for the measurement of various types of resonances, such as FMR, ESR, and cyclotron resonance. The technique uses a microwave circuit constructed of waveguides and cavities and other components. The microwaves are typically generated by a source such as a klystron or more commonly now with a solid-state diode. The microwave radiation is then directed via the waveguide system to a cavity, via a coupling hole or iris, which is so designed to support a standing wave mode of the incident electromagnetic radiation. Its size and geometry are matched to the wavelength of the radiation and the specific standing wave mode of the cavity. There are many possible designs of microwave spectrometer and we will not go into the specifics here; the interested reader can find more information in books on the subject, see, for example, Poole [232] and Ingram [233]. Of critical importance to the sensitivity of the method is the coupling from waveguide to cavity. The sample to be studied is placed inside the microwave cavity in a position of maximum oscillating magnetic field. It is this field which is the driving force of the resonance, as discussed in Section 2.2.3. In the conventional FMR experiment, the cavity mode will determine the frequency of excitation, i.e., that of the microwave source, and the resonance condition is found by sweeping an externally applied magnetic field. This field can typically be rotated with respect to the sample, allowing angular studies of the resonance field to be plotted, thus permitting studies of magnetic anisotropies in sample under study. In the off-resonance state, the spectrometer will usually be set up such that the detector gives a null signal, typically using a bridge system. The resonance state corresponds to the maximum deviation of the magnetization precession angle with respect to its equilibrium orientation. As the magnetic field is varied, the magnetization of the sample under study will begin to precess at the microwave frequency, the precessional angle increases to a maximum and then decreases again as the field is further increased. The applied field corresponding to the maximum of the precession angle is defined as the resonance field. This is the root of the secular equation that derives from the eigenvalue problem in Eq. (66). In terms of the experiment, at this point the sample absorbs the maximum energy from the microwave field, thus changing the cavity conditions and as a consequence the reflected signal from the cavity itself. This is what is

ARTICLE IN PRESS 392

David S. Schmool and Hamid Kachkachi

measured in the typical FMR experiment. It is common to use field or frequency modulation techniques in conjunction with phase-sensitive detection. The results are then displayed as a derivative of the microwave power absorption as a function of the applied magnetic field. In the preceding discussion, we have described the traditional method for measuring FMR. In recent years, a number of alternative methods have been developed which have adapted the basic principles of the FMR experiment, making it more suitable for the measurement of nanostructured materials and nanoparticles [234, 235]. Of the methods available, the use of microresonators and stripline technologies in tandem with the vector network analyzer (VNA) is extremely promising and has now developed into a well-established method of performing ferromagnetic resonance (VNAFMR) on thin films and low-dimensional structures [236–240]. In this technique, the VNA acts as both source and detector, in which the two-port VNA device is connected, via high-frequency cables, to a coplanar waveguide system consisting typically of a coplanar waveguide (CW) or stripline. The use of a planar microresonator (PMR) can also increase sensitivity of the measurement [241], though limits measurement to a fixed frequency, as we will discuss shortly. For the coplanar stripline, there is no resonant cavity, which means that measurements can be made over a broad range of frequencies (commonly referred to as a broadband FMR measurement). In this case, measurement can be made continuously up to tens of gigahertz. The twoport VNA is connected via high-frequency cables to the CW through which a high-frequency electrical signal is passed from the VNA. The detection is made by measuring the four scattering or S parameters; these consist of the two transmitted signals (port 1–port 2, S12, port 2–port 1, S21) and the two reflected signals (port 1–port 1, S11, port 2–port 2, S22). These four parameters make up the elements of the S matrix. Since the CW is impedance matched (50 Ω) to the VNA output, this will maximize the transmitted signal, which makes the technique very sensitive to changes in the line impedance. The method requires a full two-port calibration to be implemented to remove background reflections from the cable/waveguide system. It should be noted that the sensitivity of this method can be limited by the quality of the cables and connectors. Often poor-quality components will introduce further reflections, thus limiting the transmission characteristics of the high-frequency signals. This is particularly true of measurements made at the high-frequency end and above around 40 GHz in general. The magnetic sample, usually in thin film form, is placed (face-down) on top of the waveguide and located inside the poles of an electromagnet whose field direction should be ideally parallel to the stripline. Placing the sample on the stripline

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

393

Figure 43 Schematic diagram of the vector network analyzer ferromagnetic resonance spectrometer. The sample is placed on the coplanar waveguide (CW) structure, as indicated. The mutually perpendicular static applied field Hext and the microwave field h are in the plane of the film. Source: Reprinted with permission from Ref. [238]. Copyright (2006), AIP Publishing LLC.

changes the characteristic impedance of the waveguide.6 The measurement of the FMR spectrum can then proceed in one of two methods: (i) field sweep at fixed frequency or (ii) frequency sweep with a fixed static magnetic field (see Section 2.2.3). The VNA provides a measurement of the line impedance via transmission and reflection coefficients, which are related to the various S parameters. An overview of the theory of broadband measurements can be found in Refs. [237, 240, 242, 243]. It should noted that the electrical signal which passes through the CW will produce a small oscillating magnetic field around the CW. It is this high-frequency magnetic field which is the driving field for the resonance measurement. As the field or frequency is swept through the resonance of the ferromagnetic sample placed on the CW, the line impedance will change, hence altering the S parameters, providing the measurement of the resonance itself. Figure 43 shows a schematic representation of the VNA setup. A limitation of the traditional FMR experiment resides in the fact that it must be, by its very nature, a fixed frequency measurement. The VNAFMR technique, however, overcomes this problems since it does not require a cavity and broadband measurements are possible. This therefore allows for a direct measurement of the frequency–field dispersion relation for a magnetic sample. Indeed, excellent agreement with theory is found

6

Signal to noise is improved by covering as much of the stripline as possible. This can be important for broadband measurements where there is no signal amplification due to Q-factors.

ARTICLE IN PRESS 394

David S. Schmool and Hamid Kachkachi

Resonance fequency f (GHz)

30

GaAs(001) 14Fe 14Au 40Fe 30Au

25 f vs. H cucp

20 15 10 5 0 0

1

2 3 Magnetic field H (kOe)

4

5

Figure 44 Experimental variation of the resonance field as a function of the applied field for Fe films of 14 and 40 monolayers thickness, both taken with the magnetic field applied along the direction of the hard axis. Source: Reprinted from Ref. [239]. Copyright (2006), with permission from Elsevier.

using this technique [244]. An example of the dispersion relation for a thin Fe film is illustrated in Fig. 44. Other derivatives of the coplanar waveguide method are also available using fixed frequency and variable frequency microwave generators, see, for example, Refs. [245, 246]. For the case of a planar microresonator (PMR), a one-port setup can be used, where the measurement is analogous to that of the traditional FMR method. Here the PMR acts as a cavity and the VNA or microwave generator is set to its resonance frequency. The advantage of this technique is the improvement of the sensitivity due to the quality factor of the resonator, though this is typically around 50 and much lower than that of the normal microwave cavity, typically of the order of 104 [247]. Another important parameter is the filling factor of the sample. In a conventional FMR measurement, the filling factor is quite small ( 108) since the cavity is quite large with respect to the sample dimensions. The combination of filling factor and Q-factor plays important roles in the practical sensitivity of resonance-type measurements. The increased effective filling factors are of great importance in VNA-type measurements, being one of the main reasons why its sensitivity makes it a viable alternative to conventional FMR techniques. The microresonator improves significantly the sensitivity in this respect, where filling factors can be several orders of magnitude larger. Since the resonators can be relatively small, smaller samples can be measured. Banholzer et al. [248] and Schoeppner

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

395

et al. [249] have recently reported on the measurement of micron-sized ferromagnetic elements using this method. In Fig. 45, the FMR spectra, measured and simulated, are shown for a single Co stripe (5 1μm2) in the easy and hard axes configurations. The angular dependence of the FMR for this sample is illustrated in Fig. 46.

Figure 45 Measured and simulated FMR spectra of the single Co stripe. (a) Measured spectrum for the easy direction. The stripes’ long axis parallel to the x-axis and external field. (b) Simulated spectrum for the same orientation as in (a). The inset shows resonance mode (1) at 36.5 mT and a magnification by a factor of 100. (c) Measured spectrum for the hard direction. The stripe's long axis is parallel to the y-axis and perpendicular to the external field. (d) Simulated spectrum for the same orientation as in (c). The inset shows the resonance mode (8) at 509 mT and a magnification by a factor 100. Source: Reprinted with permission from Ref. [249]. Copyright (2014), AIP Publishing LLC.

ARTICLE IN PRESS 396

David S. Schmool and Hamid Kachkachi

Figure 46 Examples of spectra and results of angular-dependent FMR. (a) Spectra for the hard direction in the measurement (top) and simulation (bottom). The vertical axis represents the external field, while the horizontal axis represents the normalized FMR signal. The angular dependence is shown in a grayscale plot for the measurement (b) and simulation (c). Horizontal axis: in-plane angle of the external field. 0° : the stripes’ long axis is parallel to the x-axis. Vertical axis: external magnetic field. Red (light gray in the print version) and blue (dark gray in the print version) lines indicate the single spectra at 0° and 90° orientation, respectively. Color (different gray shades in the print version) symbols show resonance positions taken from single spectra. Source: Reprinted with permission from Ref. [249]. Copyright (2014), AIP Publishing LLC.

The planar microring resonator structure lends itself to size reduction and hence should allow its sensitivity to increase due to improvements in the filling factor. However, decreasing the diameter of the resonator will mean that its natural resonance frequency for standing-wave modes will subsequently increase. This has the knock-on effect of pushing up the applied magnetic field required to observe the FMR of the nanoelement or particle, as illustrated in the dispersion relation (55) (see, for example, Fig. 44). This means that in practice, the measurement of the FMR for nanosystems, a

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

397

compromise must be made between what magnetic fields are available and practical with the size of the resonator. This method has been shown to be sufficiently sensitive to be able to measure FMR spectra for a single nanoparticle for reduced-size cavities. Indeed, it has been used to measure small assemblies of NPs, though difficulties in placing a single NP into the resonator have hindered progress [250]. Ferromagnetic resonance measurements have indicated the presence of enhanced surface anisotropy in assemblies of maghemite nanoparticles, where a strong resonance field enhancement was observed for lowtemperature measurements. The effects were observed to scale with the inverse of particle size, which is an indication of surface effects [251]. In a further development, de Loubens and coworkers combined the use of electron paramagnetic resonance (EPR) and micro-SQUID techniques.7 The integration of magnetic sensors then allows for the simultaneous measurement of magnetization and absorbed microwave power, which was applied to the study of Ni4 single-molecule magnets (SMM) as a function of sample temperature [252, 253]. This study permitted the evaluation of relaxation rates in the SMM. This work is further developed by Quddusi et al. [254], where they have used improved coupling between the microwave source and resonator to deliver higher power rf fields at the sample position.8 The Ni4 crystal was located on the sensor such that the easy axis is collinear with the rf magnetic field, which is perpendicular to the applied static magnetic field. Results of the simultaneous micro-Hall and VNA-FMR are given in Fig. 47. The magnetization is observed to change due to the absorption of the microwave radiation, as seen in the difference between the red (light gray in the print version) and black curves in Fig. 47a; this occurs due to photoninduced transitions between quantum superposition states of the spin of the molecule [255]. The microwave absorption is assessed from the changes in the signal, S21, transmitted through the resonator. The simultaneous measurement of EPR and magnetization permits the direct determination of the energy relaxation time associated with the spin–phonon

7

8

EPR is experimentally identical to the FMR measurement, with a paramagnetic sample instead of a ferromagnetic one. Increasing the driving power of the microwave source is another way in which to improve sensitivity. The larger the rf field at the sample, the harder it will drive the spin system, increasing the precessional angle, for example, and thus enhancing the absorption. Care should, however, be taken since nonlinear effects can result from excessive driving fields.

ARTICLE IN PRESS 398

David S. Schmool and Hamid Kachkachi

A

B

0.0

T = 200 mK HT = 2.2 T f = 15 GHz

C ON OFF

–0.5

Magnetometry –1.0 –0.2

–0.1

0.0 H (T)

0.1

6.0

EPR

4.0 2.0 0.0 0.1

ΔM/Ms

M/Ms

0.5

Pabs (nW)

1.0

0.2

0.0 –0.2

–0.1

0.0 H (T)

0.1

0.2

Figure 47 (a) Magnetization with (red (light gray in the print version)) and without (black) microwave radiation (15 GHz) applied to the sample of Ni4 at 200 mK under a transverse field of 2.2 T. (b) Absorbed microwave power obtained from the S21 transmitted signal. (c) Microwave-induced change of the magnetization in the sample, as extracted from the difference of the red (light gray in the print version) and black curves in (a). Source: Reprinted with permission from Ref. [254]. Copyright (2008), AIP Publishing LLC.

(spin–lattice) coupling, provided that only the lowest lying states are significantly populated. This is expressed as: τ¼

hfN0 ΔM 2Meq Pabs

(92)

where h is the microwave magnetic field, f the frequency, and N0 the population difference between the two spin levels at equilibrium and the total number of molecules in the sample, NT. This is then evaluated as 30 ms from the measurements taken in Fig. 47. Twig et al. [256, 257] report on the enhanced sensitivity for EPR measurements using a surface loop-gap microresonator, which reduces the effective volume of the resonator, though has a low Q factor of around 15. The system operates in the frequency range 6–18 GHz at room temperature and has the capability for measurements down to 5 K. Furthermore, it offers a pffiffiffiffiffiffiffi sensitivity of 1 106 spins/ Hz, corresponding to around  2.5 104 spins for a 1-h measurement. In a study of Co nanoparticles, Wiedwald et al. [258] were able to use a combination of FMR and XMCD to determine the existence of an oxide shell around the NP. Furthermore, they were able to determine the orbitalto-spin magnetic moment ratio for both magnetic phases, confirming a bulk-like Co core. This was deduced from measurements of the resonance

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

399

frequency at various frequencies, which allowed the determination of the g-factor: g ¼ 2.150  0.015. This is in good agreement with the bulk Co g-factor which is 2.16. From this, it is possible to evaluate the orbital-to-spin magnetic moment ratio with morb/mspin ¼ (g  2)/2, which yielded a value of 0.075  0.010. The XMCD measurements performed on the same samples yielded a significantly higher value, with morb/mspin ¼ 0.24  0.06. The apparent disagreement between the two measurements is reconciled by a consideration of the probing depths of the two experimental techniques. FMR at the measuring frequencies used has a penetration depth much greater than the NP diameter and thus measures the entire entity. On the other hand, XMCD is a surface-sensitive technique, probing just to a depth equal to the inelastic mean free path of the electrons, which for Co is about 2 nm (this happens to be roughly the same value for CoO). In order to explain the difference between the two results, it is necessary to assume the presence of Co2+ ions, which has a large uncompensated magnetic moment and a large moment ratio [259]: morb/mspin 0.6. It is common for metallic Co to form a passivating CoO layer of a few nm. It can then be assumed that the XMCD value of the orbital-to-spin magnetic moment ratio arises from a combination of the metallic Co and the CoO oxide contributions. High-resolution TEM studies with EELS (electron energy loss spectroscopy) were able to confirm that indeed a Co oxide layer was present on the Co surface. The absence of the CoO signature in the FMR is due to the fact that the antiferromagnetic resonance frequency is at much higher values and well above those used in the measurement; as such, no contribution was observed.

3.7 Magnetic Resonance Force Microscopy Another very sensitive technique to measure the dynamic magnetic properties of individual nanostructures is based on the scanning probe technologies, referred to as magnetic resonance force microscopy (MRFM). This is another form of scanning probe microscopy, as discussed in Section 3.4. MRFM uses a mechanical detection of the magnetic resonance and significantly improves on the sensitivity of conventional FMR experiments while also providing spatial resolution for magnetic resonance imaging [260–263]. This is a general technique which utilizes the mechanical force between a sample and a cantilever to detect the magnetic forces exerted by the electron or nuclear magnetic moments of a sample on the magnetically modified probe tip. The magnetization of the sample is modulated at the mechanical

ARTICLE IN PRESS 400

David S. Schmool and Hamid Kachkachi

Figure 48 Principal components of the MRFM experiment. Source: Reprinted by permission from Macmillan Publishers Ltd: Nature [267]. Copyright (2004).

resonance frequency of the cantilever. The technique was first developed to obtain three-dimensional images of biological molecules [261, 264]. The technique was first applied to ferromagnetic resonance for the study of microscopic elements [265, 266]. In fact, so sensitive is this method that it has been used for the detection of an individual electron spin [267]. The principal components of the experimental apparatus are illustrated in Fig. 48. A soft cantilever is loaded with a magnetic tip which acts as a sensor to the spin system under study. These interact via dipolar coupling, creating a deflection of the cantilever, which can be readily observed using the deflection of an incident laser beam onto a four-quadrant photodiode detector. The force on the cantilever is generated by the field gradient @B/@z from the tip and the sample magnetization M; F ¼ Mz(@B/@z)(Bjjz) [265]. The minimum detectable force on the cantilever, which behaves as a linear harmonic oscillator with a single degree of freedom, can be expressed as [268] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kB TkB Fmin ¼ πfc Q

(93)

where k is the spring constant of the cantilever (of the order of mN/m), fc is its natural resonance frequency, Q the quality factor, and B is the detection bandwidth. This gives a sensitivity of about 1 fN.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

401

There is a second purpose to the field gradient, which is very important for its spatial resolution and sensitivity; it defines a thin shell or “resonant slice” at which the field is such that in the sample, the conditions are met for resonance. This corresponds to a local resonant excitation of the equivalent Larmor frequency, ωL ¼ 2πγH, where H is the effective field in the sample and γ is the gyromagnetic ratio. The resonance is a forced resonance with a frequency given by that of the applied rf field (from the microwave coil) [267], as shown in Fig. 48. The distance of the resonant slice from the tip is estimated to be roughly 250 nm. Alternatively, the sample can be placed on a waveguide, which will deliver the rf driving field [266, 268]. Spin selectivity is achieved since it will only be the spins within the resonant slice that contribute to the signal driving the cantilever at its natural frequency, fc. Modulation and lock-in techniques are typically used to improve detection sensitivity. By modulating the mechanical resonance of the cantilever, it is possible to exploit the Q-factor of the cantilever (Q ¼ 4500) [266]. Imaging capabilities are derived from the fact that the experimental apparatus is a scanning probe. The spatial resolution in a ferromagnetic sample will be related to the magnetic correlation length, which lies in the range 10–100 nm for most ferromagnetic materials [265]. As with most FMR-type experiments, the setup allows various measurement schemes. In the work of Zhang et al. [265], anharmonic modulation was used in which both the bias field and the rf field were modulated, with a difference frequency matched to the mechanical resonance frequency of the cantilever, which is of the order of 5 kHz. The applied rf field was 100% modulated with a depth of 0.2 mT and varied between 700 and 1000 MHz. In Charbois et al. [269], the microwaves were supplied by a microwave synthesizer, at a frequency of around 10.5 GHz, coupled to a stripline resonator [269]. Signal to noise, S/N, was found to dramatically improve when measurements were performed in vacuum since the reduction in pressure increases the factor Q of the cantilever, where the authors indicate a pffiffiffiffi signal-to-noise ratio S=N ∝ Q [265]. In the work of both Zhang et al. and Charbois et al. [265, 269], a sample of single-crystal Y3Fe5O12 (yttrium iron garnet or YIG) was studied. Since the samples in both these studies were of tens of microns in size, the results are fairly similar, showing a series of resonances corresponding to the magnetostatic modes as first evaluated by Damon and Eshbach [270]. The form of the spectra depends on the tip and sample interaction, where it should be noted that the tip can perturb the internal field of the sample. Klein and coworkers [266, 268] have studied micron and submicron disks of permalloy which have been compared with cavity and stripline FMR techniques.

ARTICLE IN PRESS 402

A

David S. Schmool and Hamid Kachkachi

B

Figure 49 MRFM spectra for (a) a 1-μm disk and (b) a 500-nm disk, for excitation frequencies at 4.2, 5.6, 7.0, and 8.2 GHz. The lines indicate the analytical prediction of the locus of the (l, m) modes (see text). Source: Reprinted (figure) with permission from Ref. [268]. Copyright (2008) by the American Physical Society.

Figure 49 shows some representative spectra for 1 μm and 500 nm diameter disks under different applied frequencies. The interpretation of the MRFM spectra can be performed using the basic FMR theory, which is based on the Landau–Lifshitz equation of motion (62) for the magnetization vector M with the effective field expressed as the sum of contributions due to the external field, the exchange field, and the dipolar field: Heff ¼ Hext + Hexch + Hdip. The exchange field depends on the exchange length, Λexch, which can be written as Hexch ¼ 4πΛ2exch r2 M:

(94)

The sample shape is taken into account using the dipolar field which can be expressed in the form of a Green’s tensor, which for a disk of radius R takes the form: R Hdip ¼ 4π r 0
(96)

Taking into account the shape of the disks along with the demagnetization factors leads to a resonance condition which can be expressed as [268, 271, 272] ω2s ¼ ω1 ω2 ,

(97)

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

403

where ω1 ¼ Hext cos ðθ  θH Þ  2πMs ½Nzz + Nxx + ðNzz  Nxx Þcos ð2θÞ + 4πMs Λ2 k2l, m , γ   ω2 ω1 ¼ + 2πMs 1  2Gl?, m cos ð2θÞ: γ γ

Here θ and θH refer to the directions of the magnetization, and the applied magnetic field with respect to the disk normal, Nςς, are the demagnetizing factors for ς ¼ x, y, z. kl, m are the wave vectors for the l, m mode with the corresponding perpendicular Green’s tensor Gl?, m , which depend on the relevant Bessel function, Jl(x). The lines in Fig. 49 are obtained from Eq. (97). Magnetocrystalline anisotropies, which are in general important, can be included as an additional term in the effective field. This is neglected here since permalloy has a negligible anisotropy for the current study. In fact, the shape anisotropy is much stronger than the magnetocrystalline anisotropy, as was remarked in Ref. [273]. In this work, the FMR characteristics were obtained for a pair of (vertical or horizontal) nanodisks coupled by dipolar interaction (beyond the dipole–dipole approximation), taking account of their finite size [274]. The MRFM technique lends itself to the study of many nanometric magnetic systems, such as coupled nanodisks, exchange-coupled magnetic layers, nanopillars, as well as spin torque oscillators in which a spin current is passed through a multilayered nanopillar [275–277]. The sensitivity of the MRFM technique was shown in the extreme in the magnetic resonance study of a single-electron spin [267], as illustrated in Fig. 50 for two different applied magnetic fields. Another method related to MRFM is scanning thermal microscopy ferromagnetic resonance (SThM-FMR). The technique employs a microwave cavity which is used to excite the magnetic response in the sample, and a thermal sensor is then used to detect small changes in local temperature which occur when the sample absorbs microwave power as the sample is driven through resonance. The spatial resolution is quoted as 30 nm 100 nm with a thermal sensitivity of around 1 mK and an overall sensitivity of about 106 spins [278, 279]. A review of this and related techniques has been recently published by Meckenstock [280]. By way of illustration, we consider the study of locally resolved FMR in 1.5-mm Co stripes using this technique [279]. A schematic illustration of the experimental setup

ARTICLE IN PRESS 404

David S. Schmool and Hamid Kachkachi

Figure 50 Spin signal as the sample was scanned laterally in the x direction for two values of external field: (a) Bext ¼ 34 mT and (b) Bext ¼ 30 mT. The smooth curves are Gaussian fits that serve as guides to the eye. The 19-nm shift in peak position reflects the movement of the resonant slice induced by the 4-mT change in external field. Source: Reprinted by permission from Macmillan Publishers Ltd: Nature [267]. Copyright (2004).

with sample orientation is shown in Fig. 51a. The sample demagnetization tensor will produce a uniaxial anisotropy in the plane of the sample with nonuniform modes of excitation expected due to nonellipsoidal stripe shape. These may take the form of rim (edge) and backward volume spin-wave modes. Conventional FMR and SThM-FMR measurements exhibit at least three SW modes (I, II, and III), as shown in Fig. 51b. These have been identified as (I) rim mode, (II) backward volume modes, and (III) uniform modes, with the aid of OOMMF9 micromagnetic calculations. This is also shown by the thermal images in Fig. 51c, where for applied fields of 85 and 120 mT, different areas of the Co stripe are heated. This means that the heat generated due to the absorption of microwave power can be spatially resolved and can be seen to locally drive the spin dynamics of the system. This is particularly evident for the rim mode, which is restricted to a region of about 150 nm at the edge of the Co stripe.

9

http://www.ctcms.nist.gov/rdm/mumag.org.html.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

405

Figure 51 (a) Schematic diagram of SThM-FMR setup and stripe orientation. (b) FMR spectrum of the Co stripe taken in the SThM-FMR setup with microwave amplitude modulation. Inset: In-plane angle-dependent conventional FMR showing the amplitude of the FMR field derivative as grayscale versus applied field angle (x-axis) and external field (y-axis). I–III mark the resonances further discussed by OOMMF and locally resolved SThM-FMR. (c) Topography and thermal amplitude image of SThM-FMR with external field perpendicular to the Co stripes at 85 mT (upper images corresponding to mode III in (b)) and topography thermal amplitude and phase images for rim resonance with external field perpendicular at 120 mT (lower images corresponding to mode I in (b)). Source: Reprinted with permission from Ref. [279]. Copyright (2007), AIP Publishing LLC.

In a related work, Rod et al. [281] have used a bolometer detection method for FMR in nanostructures. A bolometer is a detector for electromagnetic radiation caused by the dissipative heating of the detector material [282]. The spatial resolution of the Au–Pd thermocouple device is dependent on the size of the (hot) junction, which is fabricated using electron beam lithography (EBL), with the smallest device being 230 85 nm2. The sample, Py stripe with the thermocouple fabricated in close proximity, is placed in the center of a microwave cavity at a position of maximum rf field amplitude. The Seebeck voltage acts as the measurement of the

ARTICLE IN PRESS 406

David S. Schmool and Hamid Kachkachi

resonance absorption. The device has a sensitivity of a few mK, though improvements are expected to increase this to the μK scale. As in the typical FMR experiment, modulation and lock-in techniques are used to improve the signal-to-noise ratio.

4. SUMMARY In this contribution, we have tried to provide the reader with an overview of the main phenomena, issues, and techniques, in both theory and experiments, that are relevant to the field of nanomagnetism. The latter refers to the study of nanoscaled magnetic systems, especially magnetic nanoparticles which, when compared to the bulk materials, exhibit amazing novel properties such as enhanced remanence and giant coercivity, as well as an exponentially slow dynamics at very low temperature due to anisotropy barriers. On the other hand, these nanomagnets become superparamagnetic at finite temperature for very small sizes. Superparamagnetism, which is the key feature of these systems, consists in a fast thermally activated shuttling of the magnetization between the various effective-energy minima. It has opened a rich and broad area to the application of out-of-equilibrium statistical mechanics, and is the main reason for the extensive work done in experiments and theory to adapt to these nanoscaled systems the various tools that were initially developed for bulk materials. Today, there are mainly two prototypes of nanoparticle samples: (i) There are assemblies of metallic and oxide particles which are in general polydispersed and present the additional distribution of their anisotropy axes. In this case, measurements are available for broad ranges of temperature and applied magnetic field, but the magnetization is obtained as an average over the whole assembly. (ii) Isolated single particles, or what can be considered as such, provide in principle the possibility to avoid the volume and anisotropy distributions, as well as the interparticle interactions. However, the experimental techniques developed so far do not, in general, have a sufficient space-time resolution to probe the magnetic response of a single particle. From the point of view of both experiments and theory, this poses the problem of bridging the physics of isolated particles with that of particle assemblies. On the theory side, the calculations have been done while clearly separating the particle and the assembly scales. On the experimental side, however, such a separation is not so clear and the single-particle physics has been mainly approached using assemblies of very low densities. It is only over recent decades that experiments have shown a new impetus for developing single-particle-focused techniques.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

407

With this in mind, we have tried in this first contribution to review the progress made in single-particle magnetism and nanomagnetism that has been made over the last two decades. In discussing the theoretical considerations of nanomagnetism, we have presented the principal approaches used for dealing with the properties of isolated nanoparticles, both at equilibrium and out of equilibrium. We have discussed the macroscopic approach that represents the magnetic state of a nanoparticle using its net magnetic moment, according to the models of Stoner–Wohlfarth at zero temperature and the Ne´el–Brown model for finite temperature. The validity of these models has been confirmed by μ-SQUID measurements of cobalt nanoparticles of a diameter that is of a few tens of nanometers. As the size of the nanoparticles becomes smaller, the surface contribution induces new effects that cannot be dealt with by these macrospin models. We have shown that in this case, one has to resort to atomic approaches that are capable of accounting for the atomic environment and the associated effective fields. We have then presented and reviewed the main physical phenomena that are described by this approach. We have also given a survey of the various computing methods and precisely explained how they have to be adapted to the nanoscale before they can be applied to the calculation of the physical observables, such as the magnetization and susceptibility as functions of temperature and applied magnetic field, the spin-wave spectrum and FMR characteristics, and finally the relaxation rate of the thermally activated barrier crossing. With regard to experiments, there has also been much progress in the various aspects of single-particle magnetism. In this work, we have tried to outline the main tools that have been used to study single element or single magnetic nanoparticles. In general, these techniques have been developed from preestablished methods. For example, the micro-SQUID and micro-Hall methods have used a miniaturization of the active device to increase their sensitivity to be able to measure the stray fields emanating from nanometric magnetic objects. The microscopy techniques that have been employed to study single-particle magnetism are already adapted to measuring objects at the nanoscale. The Lorentz microscopy (LTEM) technique is a direct application of the scanning electron microscope and relies on the deflection of the electron beam by the magnetic forces that come into play when the electrons pass through a thin magnetic sample and are subject to its magnetic induction. Magnetic force microscopy also has a relatively high resolution, though is more generally used for studying the magnetic domain structure in thin films and nanostructures typically in the 100 nm range. This

ARTICLE IN PRESS 408

David S. Schmool and Hamid Kachkachi

can also be said of LTEM, though both methods have been, as we have discussed, shown to be sensitive to single magnetic nanoparticles. Photoemission electron microscopy (PEEM) is a specialized method that employs soft X-rays to excite electrons from core states to free band states in the solid, where the band-edge absorptions are exploited to gain element specificity. Furthermore, in ferromagnetic materials, the spin-split bands have different absorption cross sections for left and circular polarized light, due to the difference in their spin populations. This allows a magnetic signature to be obtained. In terms of single-particle measurements, X-PEEM shows enormous promise and has shown its sensitivity for such applications. Ferromagnetic resonance is a sensitive tool for magnetic thin film-based samples. Dynamic measurements of magnetization have seen something of a revolution in recent decades and have become increasingly popular. This is partly because of the wealth of information that they can provide, such as the sample magnetization and g-factor. However, FMR and related techniques have really come into their own when characterizing magnetic anisotropies, whether they are magnetocrystalline, surface, or shape in origin. In addition to this, FMR is also extremely well adapted to study relaxation processes in magnetic systems. This provides information on intrinsic as well as extrinsic damping mechanisms. In recent years, it was also found to be sensitive to spin currents and in particular in relation to interface effects, such as spin pumping and spin accumulation. With regard to single-particle magnetic measurements, the development of coplanar waveguide and microresonators in conjunction with vector network analyzers and microwave synthesizers has shown significantly greater sensitivity than conventional microwave spectroscopy. This is mainly due to the enhanced filling factors that can be obtained in the former. While no true single-particle measurements have been made as yet, the technique has been shown to have the necessary sensitivity for such a purpose. One of the difficulties with the measurement is the ability to place a single magnetic nanoparticle inside the microresonator itself. Magnetic resonance force microscopy (MRFM) combines the sensitivity of the scanning probe methods with that of ferromagnetic resonance. Since the resonance condition is extremely specific and depends on just the right conditions being satisfied, this makes it exceptionally sensitive. In fact, so sensitive is the MRFM technique, it has been shown to be able to detect a single-electron spin. Given that MRFM provides a local measurement of FMR, it can also provide much of the same information, though with spatial resolution. It may, however, be argued that linewidth information it provides will be in some modified form due to the experimental setup.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

409

Magnetic nanoparticles have found a number of practical applications, most notably in nanomedicine, such as hyperthermia, magnetic filtering, and magnetic labeling. However, there seems to be good potential for drug delivery applications and cancer treatments using magnetic nanoparticles. Magnetic recording is a multibillion dollar industry and nanomagnetism has found several applications in this area. One of the principal goals would be to reduce the bit size to a single magnetic element. However, stability becomes a key issue and superparamagnetic effects impair the thermal stability as we have seen. Increasing the magnetic anisotropy would therefore be of crucial importance to offset the size reduction inherent in this problem; shape and surface anisotropies seem unlikely at present to resolve this issue. However, increasing the anisotropy implies an increase of the writing magnetic fields and this leads to a new technological challenge. In other developments, recent research in the optical properties of nanosystems has shown how size and shape are also fundamental parameters for defining the energy (wavelength) of plasmonic excitations [283–285]. Proximity effects between neighboring particles have demonstrated that interactions between particles can shift plasmon resonances, and enhanced electric fields (or “hot spots”) are also generated where the magnitude of the fields can be increased by several orders of magnitude [286–288]. This effect has been exploited, for example, in surface-enhanced Raman spectroscopy (SERS) [289]. The use of metallic nanoparticles in photovoltaic cells has also attracted much attention since it may provide a physical mechanism for increasing the energy transfer from solar radiation to the cell, significantly improving its quantum efficiency [290]. The coupling of magnetic properties with those studied in plasmonics may provide yet another way for manipulating the physical properties of nanosized objects [291–293]. One goal of magnetoplasmonics would be to exploit any coupling of these properties to tune plasmon frequencies and magnetic excitations.

REFERENCES [1] R. Skomski, Simple Models of Magnetism, Oxford University Press, Oxford, 2008. [2] D.S. Schmool, Introductory Solid State Physics: From the Material Properties of Solids to Nanotechnologies, The Pantaneto Press, UK, 2014. [3] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett. 76 (1996) 4250. [4] C. Boeglin, E. Beaurepaire, V. Halte´, V. Lo´pez-Flores, C. Stamm, N. Pontius, H.A. Du¨rr, J.-Y. Bigot, Distinguishing the ultrafast dynamics of spin and orbital moments in solids, Nature 465 (2010) 458.

ARTICLE IN PRESS 410

David S. Schmool and Hamid Kachkachi

[5] C. Stamm, N. Pontius, T. Kachel, M. Wietstruk, H.A. Du¨rr, Femtosecond x-ray absorption spectroscopy of spin and orbital angular momentum in photoexcited Ni films during ultrafast demagnetization, Phys. Rev. B 81 (2010) 104425. [6] J.-Y. Bigot, M. Vomir, E. Beaurepaire, Coherent ultrafast magnetism induced by femtosecond laser pulses, Nat. Phys. 5 (2009) 515. [7] A.R. Khorsand, M. Savoini, A. Kirilyuk, A.V. Kimel, A. Tsukamoto, A. Itoh, T. Rasing, Element-specific probing of ultrafast spin dynamics in multisublattice magnets with visible light, Phys. Rev. Lett. 110 (2013) 107205. [8] A. Kirilyuk, A.V. Kimel, T. Rasing, Ultrafast optical manipulation of magnetic order, Rev. Mod. Phys. 82 (2010) 2731. [9] S.H. Baker, S.C. Thornton, K.W. Edmonds, M.J. Maher, C. Norris, C. Binns, The construction of a gas aggregation source for the preparation of size-selected nanoscale transition metal clusters, Rev. Sci. Instrum. 71 (2000) 3178. [10] G.N. Iles, S.H. Baker, S.C. Thornton, C. Binns, Enhanced capability in a gas aggregation source for magnetic nanoparticles, J. Appl. Phys. 105 (2009) 024306. [11] J. Bansmanna, et al., Magnetic and structural properties of isolated and assembled clusters, Surf. Sci. Rep. 56 (2005) 189. [12] M. Brust, M. Walker, D. Bethell, D.J. Schiffrin, R. Whyman, Synthesis of thiolderivatised gold nanoparticles in a two-phase liquid-liquid system, J. Chem. Soc. Chem. Commun. 7 (1994) 801. [13] S.R.K. Perala, S. Kumar, On the mechanism of metal nanoparticle synthesis in the Brust-Schiffrin method, Langmuir 29 (2013) 9863. [14] J. Pe´rez-Juste, I. Pastoriza-Santos, L.M. Liz-Marza´n, P. Mulvaney, Gold nanorods: synthesis, characterization and applications, Coord. Chem. Rev. 249 (2005) 1870. [15] N. Fontaı´n˜a-Troitin˜o, S. Lie´bana-Vin˜as, B. Rodrı´guez-Gonza´lez, Z.-A. Li, M. Spasova, M. Farle, V. Salgueirin˜o, Room-temperature ferromagnetism in antiferromagnetic cobalt oxide nanooctahedra, Nano Lett. 14 (2014) 640. [16] X.C. Jiang, M.P. Pileni, Gold nanorods: influence of various parameters as seeds, solvent, surfactant on shape control, Coll. Surf. A 295 (2007) 228. [17] Matthias, et al., Monolayer and multilayer assemblies of spherically and cubic-shaped iron oxide nanoparticles, J. Mater. Chem. 21 (2011) 16018. [18] V. Sharma, K. Park, M. Srinivasarao, Colloidal dispersion of gold nanorods: historical background, optical properties, seed-mediated synthesis, shape separation and selfassembly, Eng. Mat. Sci. R 65 (2009) 1. [19] A. Lo´pez-Ortega, et al., Strongly exchange coupled inverse ferrimagnetic soft/hard, MnxFe3-xO4/FexMn3-xO4, core/shell heterostructured nanoparticles, Nanoscale 4 (2012) 5138. [20] F. Wetz, K. Soulantica, A. Falqui, M. Respaud, E. Snoeck, B. Chaudret, Hybrid Co/Au nanorods: controlling Au nucleation and location, Angew. Chem. Int. Ed. 46 (2007) 7079. [21] M.P. Ferna´ndez-Garcı´a, et al., Microstructure and magnetism of nanoparticles with γ-Fe core surrounded by α-Fe and iron oxide shells, Phys. Rev. B 81 (2010) 094418. [22] D.S. Schmool, et al., The role of dipolar interactions in magnetic nanoparticles: ferromagnetic resonance in discontinuous magnetic multilayers, J. Appl. Phys. 101 (2007) 103907. [23] A. Garcı´a-Garcı´a, et al., Magnetic properties of Fe/MgO granular multilayers prepared by pulsed laser deposition, J. Appl. Phys. 105 (2009) 063909. [24] Z. Cui, Nanofabrication: Principles, Capabilities and Limits, Springer, New York, 2008. [25] H.J. Levinson, Principles of Lithography, The International Society for Optical Engineering, Bellingham, Washington (USA), 2005.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

411

[26] A.A. Tseng, Nanofabrication: Fundamentals and Applications, World Scientific, Singapore, 2008. [27] A. Priimagi, A. Shevchenko, Azopolymer-based micro- and nanopatterning for photonic applications, J. Polym. Sci. B 52 (2013) 163. [28] J. Berendt, J.M. Teixeira, A. Garcı´a-Garcı´a, M. Raposo, P.A. Ribeiro, J. Dubowik, G.N. Kakazei, D.S. Schmool, Tunable magnetic anisotropy in permalloy thin films grown on holographic relief gratings, Appl. Phys. Lett. 104 (2014) 082408. [29] J. Orloff, M. Utlaut, L. Swanson, High Resolution Focused Ion Beams: FIB and Its Applications, Kluwer Academic/Plenum Publishers, New York, 2003. [30] N. Yao, A.K. Epstein, Surface nanofabrication using focused ion beam, in: A. Me´ndezVilas, J. Dı´az (Eds.), Microscopy: Science, Technology, Applications and Education, Formatex Research Center, Badajoz (Spain), 2010, p. 2190. [31] C.T. Sousa, D.C. Leitao, M.P. Proenca, J. Ventura, A.M. Pereira, J.P. Arau´jo, Nanoporous alumina as templates for multifunctional applications, Appl. Phys. Rev. 1 (2014) 031102. [32] P.R. Evans, G. Yi, W. Schwarzacker, Current perpendicular to plane giant magnetoresistance of multilayered nanowires electrodeposited in anodic aluminum oxide membranes, Appl. Phys. Lett. 76 (2000) 481. [33] D.S. Schmool, H. Kachkachi, Intrinsic and collective properties of magneticnanosystems. Part II: nanoparticle assemblies and interactions, in: R.L. Stamps, R.E. Camley (Eds.), Solid State Phys, Elsevier, 2016. [34] A.H. Morrish, K. Haneda, X.Z. Zhou, in: G.C. Hadjipanayis, R.W. Siegel (Eds.), Nanophase Materials, NATO ASI Series, vol. 260, Springer, Netherlands, 1994, pp. 515–535. [35] O. Eriksson, A.M. Boring, R.C. Albers, G.W. Fernando, B.R. Cooper, Spin and orbital contributions to surface magnetism in 3 d elements, Phys. Rev. B 45 (1992) 2868. [36] P. Fulde, Electron Correlations in Molecules and Solids, Springer, Berlin, 1995. [37] E.C. Stoner, E.P. Wohlfarth, A Mechanism of Magnetic Hysteresis in Heterogeneous Alloys, Philos. Trans. R. Soc. Lond. A 240 (1948) 599. [38] E.C. Stoner, E.P. Wohlfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, IEEE Trans. Magn. 27 (1991) 3475. [39] L. Ne´el, Influence des fluctuations thermiques sur l’aimantation de grains ferromagne´tiques tre´s fins, C. R. Acad. Sci. 228 (1949) 664. [40] L. Ne´el, Anisotropie magne´tique superficielle et surstructures d’orientation, J. Phys. Radium 15 (1954) 225. [41] W.F. Brown, Thermal fluctuations of a single-domain particle, Phys. Rev. 135 (1963) 1677. [42] W.F. Brown, Thermal fluctuations in fine ferromagnetic particles, IEEE Trans. Magn. 15 (1979) 1196. [43] A. Aharoni, Effect of surface anisotropy on the exchange resonance modes, J. Appl. Phys. 81 (1997) 830. [44] W. Wernsdorfer, Classical and quantum magnetization reversal studied in nanometersized particles and clusters, Adv. Chem. Phys. 118 (2001) 99. [45] E. Bonet, W. Wernsdorfer, B. Barbara, A. Benoıˆt, D. Mailly, A. Thiaville, Threedimensional magnetization reversal measurements in nanoparticles, Phys. Rev. B 83 (1999) 4188. [46] H. Kachkachi, D.A. Garanin, in: D. Fiorani (Ed.), Surface Effects in Magnetic Nanoparticles, Springer, Berlin, 2005, p. 75. [47] W.F. Brown Jr., Micromagnetics, Interscience, New York, 1963. [48] A. Aharoni, Introduction to the Theory of Ferromagnetism, Oxford Science Pubs, Oxford, 1996.

ARTICLE IN PRESS 412

David S. Schmool and Hamid Kachkachi

[49] L. Ne´el, The´orie du trainage magne´tique des ferromagne´tiques au grains fin avec applications aux terres cuites, Ann. Geophys. 5 (1949) 99. [50] M. Dimian, H. Kachkachi, Effect of surface anisotropy on the hysteretic properties of a magnetic particle, J. Appl. Phys. 91 (2002) 7625. [51] D.A. Dimitrov, Wysin, Effects of surface anisotropy on hysteresis in fine magnetic particles, Phys. Rev. B 50 (1994) 3077. [52] H. Kachkachi, M. Dimian, Hysteretic properties of a magnetic particle with strong surface anisotropy, Phys. Rev. B 66 (2002) 174419. [53] H. Kachkachi, D.A. Garanin, Boundary and finite-size effects in small magnetic systems, Phys. A 300 (2001) 487. [54] H. Kachkachi, D.A. Garanin, Spin-wave theory for finite classical magnets and superparamagnetic relation, Eur. Phys. J. B 22 (2001) 291. [55] M.E. Fisher, V. Privman, First-order transitions breaking O(n) symmetry: finite-size scaling, Phys. Rev. B 32 (1985) 447. [56] M.E. Fisher, V. Privman, First-order transitions in spherical models: finite-size scaling, Commun. Math. Phys. 103 (1986) 527. [57] L. Ne´el, L’anisotropie superficielle des substances ferromagne´tiques, C. R. Acad. Sci. 237 (1953) 1468. [58] D.A. Garanin, H. Kachkachi, Surface contribution to the anisotropy of magnetic nanoparticles, Phys. Rev. Lett. 90 (2003) 65504. [59] H. Kachkachi, H. Mahboub, Surface anisotropy in nanomagnets: Ne´el or transverse? J. Magn. Magn. Mater. 278 (2004) 334. [60] J.P. Chen, C.M. Sorensen, K.J. Klabunde, G.C. Hadjipanayis, Enhanced magnetization of nanoscale colloidal cobalt particles, Phys. Rev. B 51 (1995) 11527. [61] A. Ezzir, Proprie´te´s magne´tiques d’une assemble´es de nanoparticules: mode´lisation de l’aimantation et de la susceptibilite´ superparamagne´tique et applications, Ph.D. thesis, Universite´ Paris-Sud, Orsay, 1998. [62] M. Respaud, et al., Surface effects on the magnetic properties of ultrafine cobalt particles, Phys. Rev. B 57 (1998) 2925. [63] A.Z. Patashinskii, V.L. Pokrovskii, Fluctuation Theory of Phase Transitions, Pergamon Press Ltd., Oxford, 1979. [64] W.T. Coffey, Y.P. Kalmykov, J. Waldron, The Langevin Equation, World Scientific, Singapore, 2005. [65] H. Kachkachi, E. Bonet, Surface-induced cubic anisotropy in nanomagnets, Phys. Rev. B 73 (2006) 224402. [66] H. Kachkachi, Effects of spin non-collinearities in magnetic nanoparticles, J. Magn. Magn. Mater. 316 (2007) 248. [67] H. Kachkachi, D.A. Garanin, Unpublished . [68] R. Yanes, O. Fesenko-Chubykalo, H. Kachkachi, D.A. Garanin, R. Evans, R.W. Chantrell, Effective anisotropies and energy barriers of magnetic nanoparticles within the Ne´el’s surface anisotropy, Phys. Rev. B 76 (2007) 064416. [69] F. Gazeau, E. Dubois, M. Hennion, R. Perzynski, Y. Raikher, Quasi-elastic neutron scattering on γ-Fe2O3 nanoparticles, Europhys. Lett. 40 (1997) 575. [70] M.F. Hansen, S. Morup, Models for dynamics of interacting magnetic nanoparticles, J. Magn. Magn. Mater. 184 (1998) 262. [71] E. Tronc, A. Ezzir, R. Cherkaoui, C. Chane´ac, M. Nogue`s, H. Kachkachi, D. Fiorani, A.M. Testa, J.M. Grene`che, J.P. Jolivet, Surface-related properties of γ-Fe2O3 nanoparticles, J. Magn. Magn. Mater. 221 (2000) 63. [72] S. Rohart, V. Repain, A. Thiaville, S. Rousset, Limits of the macrospin model in cobalt nanodots with enhanced edge magnetic anisotropy, Phys. Rev. B 76 (2007) 104401. [73] T. Holstein, H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev. 58 (1940) 1098.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

413

[74] D.N. Zubarev, Double-time temperature-dependent Green functions, Sov. Phys. Usp. 3 (1960) 320. [75] R. Bastardis, U. Atxitia, O. Chubykalo-Fesenko, H. Kachkachi, Unified decoupling scheme for exchange and anisotropy contributions and temperature-dependent spectral properties of anisotropic spin systems, Phys. Rev. B 86 (2012) 094415. [76] J.L. Garcia-Palacios, F.J. Lazaro, Langevin-dynamics study of the dynamical properties of small magnetic particles, Phys. Rev. B 58 (1998) 14937. [77] J.L. Garcia-Palacios, On the statics and dynamics of magnetoanisotropic nanoparticles, Adv. Chem. Phys. 112 (2000) 1. [78] A. Lyberatos, R.W. Chantrell, Thermal fluctuations in a pair of magnetostatically coupled particles, J. Appl. Phys. 73 (1993) 6501. [79] U. Nowak, R.W. Chantrell, E.C. Kennedy, Monte Carlo simulation with time step quantification in terms of Langevin dynamics, Phys. Rev. Lett. 83 (2000) 163. [80] D.V. Berkov, N.L. Gorn, Susceptibility of the disordered system of fine magnetic particles: a Langevin-dynamics study, J. Phys. Condens. Matter 13 (2001) 9369. [81] N. Kazantseva, D. Hinzke, U. Nowak, R.W. Chantrell, U. Atxitia, O. ChubykaloFesenko, Towards multiscale modeling of magnetic materials: simulations of FePt, Phys. Rev. B 77 (2008) 184428. [82] D.A. Garanin, Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Phys. Rev. B 55 (1997) 3050. [83] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer-Verlag, Berlin, 1996. [84] J.L. Garcia-Palacios, D. Zueco, Solving spin quantum master equations with matrix continued-fraction methods: application to superparamagnets, J. Phys. A Math. Gen. 42 (2006) 13243. [85] J.L. Garcia-Palacios, Perturbative continued-fraction method for weakly interacting Brownian spins and dipoles, arXiv:cond-mat/0602072 . [86] S. Titov, H. Kachkachi, Y. Kalmykov, W.T. Coffey, Magnetization dynamics of two interacting spins in an external magnetic field, Phys. Rev. B 72 (2005) 134425. [87] K. Binder, D. Heermann, Monte Carlo Simulation in Statistical Physics, SpringerVerlag, Berlin, 1992. [88] H. Kachkachi, D.A. Garanin, Magnetic free energy at elevated temperatures and hysteresis of magnetic particles, Phys. A 291 (2001) 485. [89] J.H. Van’t Hoff, in: F. Muller, Co. (Eds.), Etudes de Dynamiques Chimiques, North Holland, Amsterdam, 1884. [90] S. Arrhenius, Ueber die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Sa¨uren, Z. Phys. Chem. (Leipzig) 4 (1889) 226. [91] H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica (Amsterdam) 7 (1940) 284. [92] M. Bu¨tticker, E.P. Harris, R. Landauer, Thermal activation in extremely underdamped Josephson-junction circuits, Phys. Rev. B 28 (1983) 1268. [93] V.I. Mel’nikov, S.V. Meshkov, Theory of activated rate processes: exact solution of the Kramers problem, J. Chem. Phys. 85 (1986) 1018. [94] P. Ha¨nggi, P. Talkner, M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys. 62 (1990) 251. [95] A. Aharoni, Effect of a magnetic field on the superparamagnetic relaxation time, Phys. Rev. 177 (1969) 793. [96] W.T. Coffey, Finite integral representation of characteristic times of orientational relaxation processes: application to the uniform bias force effect in relaxation in bistable potentials, Adv. Chem. Phys. 103 (1998) 259. [97] L.J. Geoghegan, W.T. Coffey, B. Mulligan, Differential recurrence relations for nonaxially symmetric rotational Fokker-Planck equations, Adv. Chem. Phys. 100 (1997) 475.

ARTICLE IN PRESS 414

David S. Schmool and Hamid Kachkachi

[98] E.C. Kennedy, Relaxation times for single-domain ferromagnetic particles, Ph.D. thesis, The Queen’s University of Belfast, Belfast, 1997. [99] H. Kachkachi, W.T. Coffey, D.S.F. Crothers, A. Ezzir, E.C. Kennedy, M. Nogue`s, E. Tronc, Field dependence of the temperature at the peak of the zero-field-cooled magnetization, J. Phys. Condens. Matter 48 (2000) 3077. [100] D.A. Garanin, E.C. Kennedy, D.S.F. Crothers, W.T. Coffey, Thermally activated escape rates of uniaxial spin systems with transverse field: uniaxial crossovers, Phys. Rev. E 60 (1999) 6499. [101] W.T. Coffey, et al., Interpolation formulae between axially symmetric and non-axially symmetric Kramers escape rates for single-domain ferromagnetic particles in the intermediate to high-damping limit, J. Magn. Magn. Mater. 221 (2000) 110. [102] E. Fermi, Notes on Quantum Mechanics, University of Chicago Press, Chicago, IL, 1965. [103] P.M. De´jardin, D.S.F. Crothers, W.T. Coffey, D.J. McCarthy, Interpolation formula between very low and intermediate-to-high damping Kramers escape rates for singledomain ferromagnetic particles, Phys. Rev. E 63 (2001) 021102. [104] J.S. Langer, Theory of Nucleation Rates, Phys. Rev. Lett. 21 (1968) 973. [105] J.S. Langer, Statistical theory of the decay of metastable states, Ann. Phys. (N.Y.) 54 (1969) 258. [106] H. Kachkachi, Effect of exchange interaction on superparamagnetic relaxation, Europhys. Lett. 62 (2003) 650. [107] H. Kachkachi, Dynamics of a nanoparticle as a one-spin system and beyond, J. Mol. Liq. 114 (2004) 113. [108] R.H. Kodama, A.E. Berkovitz, Atomic-scale magnetic modeling of oxide nanoparticles, Phys. Rev. B 59 (1999) 6321. [109] J.T. Richardson, D.I. Yiagas, B. Turk, J. Forster, Origin of superparamagnetism in nickel oxide, J. Appl. Phys. 70 (1991) 6977. [110] P.-M. De´jardin, H. Kachkachi, Y. Kalmykov, Thermal and surface anisotropy effects on the magnetization reversal of a nanocluster, J. Phys. D 41 (2008) 134004. [111] D. Ledue, R. Patte, H. Kachkachi, Dynamical susceptibility of weakly interacting ferromagnetic nanoclusters, J. Nanosci. Nanotechnol. 12 (2012) 4953. [112] G. Margaris, K.N. Trohidou, H. Kachkachi, Surface effects on the magnetic behavior of nanoparticle assemblies, Phys. Rev. B 85 (2012) 024419. [113] Z. Sabsabi, F. Vernay, O. Iglesias, H. Kachkachi, Interplay between surface anisotropy and dipolar interactions in an assembly of nanomagnets, Phys. Rev. B 88 (2013) 104424. [114] F. Vernay, Z. Sabsabi, H. Kachkachi, ac susceptibility of an assembly of nanomagnets: combined effects of surface anisotropy and dipolar interactions, Phys. Rev. B 90 (2014) 094416. [115] J.M.D. Coey, Noncollinear spin arrangement in ultrafine ferrimagnetic crystallites, Phys. Rev. Lett. 27 (1979) 1140. [116] K. Haneda, Recent advances in the magnetism of fine particles, Can. J. Phys. 65 (1987) 1233. [117] S. Morup, E. Tronc, Superparamagnetic relaxation of weakly interacting particles, Phys. Rev. Lett. 72 (1994) 3278. [118] J.-L. Dormann, D. Fiorani, E. Tronc, Magnetic relaxation in fine particle systems, Adv. Chem. Phys. 98 (1997) 283. [119] R. Skomski, J.M.D. Coey, in: Permanent Magnetism Studies in Condensed Matter Physics>, vol. 1, IOP Publishing, London, 1999. [120] K.B. Urquhart, B. Heinrich, J.F. Cochran, A.S. Arrott, K. Myrtle, Ferromagnetic resonance in ultrahigh vacuum of bcc Fe(001) films grown on Ag(001), J. Appl. Phys. 64 (1988) 5334.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

415

[121] R. Perzynski, Y.L. Raikher, in: D. Fiorani (Ed.), Surface Effects in Magnetic Nanoparticles, Springer, Berlin, 2005, p. 141. [122] O. Iglesias, A. Labarta, Finite-size and surface effects in maghemite nanoparticles: Monte Carlo simulations, Phys. Rev. B 63 (2001) 184416. [123] Y. Labaye, O. Crisan, L. Berger, J.M. Greneche, J.M.D. Coey, Surface anisotropy in ferromagnetic nanoparticles, J. Appl. Phys. 91 (2002) 8715. [124] O. Iglesias, A. Labarta, Influence of surface anisotropy on the hysteresis of magnetic nanoparticles, J. Magn. Magn. Mater. 290-291 (2005) 738. [125] C. Va´zquez-Va´zquez, M.A. Lo´pez-Quintela, M.C. Buja´n-Nu´n˜ez, J. Rivas, Finite size and surface effects on the magnetic properties of cobalt ferrite nanoparticles, J. Nanopart. Res. 13 (2011) 1663. [126] N. Friedenberger, Single Nanoparticle Magnetism: Hysteresis of Monomers, Dimers and Many-Particle Ensembles, Univ. of Duisburg-Essen, Duisburg-Essen, 2011. [127] A. Cehovin, C.M. Canali, A.H. MacDonald, Orbital and spin contributions to the g tensors in metal nanoparticles, Phys. Rev. B 69 (2004) 045411. [128] P. Jeppson, et al., Cobalt ferrite nanoparticles: achieving the superparamagnetic limit by chemical reduction, J. Appl. Phys. 100 (2006) 114324. [129] Y.A. Koksharov, in: S.P. Gubin (Ed.), Magnetic Nanoparticles, Wiley-VCH, Weinheim, 2009, p. 197. [130] H. Kachkachi, A. Ezzir, M. Nogue`s, E. Tronc, Surface effects in nanoparticles: Monte Carlo simulations, Eur. Phys. J. B 14 (2000) 681. [131] B. Koopmans, M. van Kampen, J.T. Kohlhepp, W.J.M. de Jonge, Ultrafast magnetooptics in nickel: magnetism or optics? Phys. Rev. Lett. 85 (2000) 844. [132] M. van Kampen, C. Jozsa, J.T. Kohlhepp, P. LeClair, W.J.M. de Jonge, B. Koopmans, All-optical probe of coherent spin waves, Phys. Rev. Lett. 88 (2002) 227201. [133] B. Koopmans, J.J.M. Ruigrok, F. Dalla Longa, W.J.M. de Jonge, Unifying ultrafast magnetization dynamics, Phys. Rev. Lett. 95 (2005) 267207. [134] C.D. Stanciu, F. Hansteen, A.V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, T. Rasing, All-optical magnetic recording with circularly polarized light, Phys. Rev. Lett. 99 (2007) 047601. [135] C. Thirion, W. Wernsdorfer, D. Mailly, Switching of magnetization by nonlinear resonance studied in single nanoparticles, Nature 2 (2003) 524. [136] Raufast, C.,Etude par magne´tome´trie à micro-SQUID d’agre´gats à base de cobalt, Ph.D. thesis. Universite´ Claude Bernard Lyon I, 2007. [137] A. Tamion, C. Raufast, E. Bonet, V. Dupuis, T. Fournier, C. Crozes, E. Bernstein, W. Wernsdorfer, Magnetization reversal of a single cobalt cluster using a RF field pulse, J. Magn. Magn. Mater. 322 (2010) 1315. [138] S.I. Denisov, T.V. Lyutyy, P. Ha¨nggi, Magnetization of nanoparticle systems in a rotating magnetic field, Phys. Rev. Lett. 97 (2006) 227202. [139] G. Bertotti, I. Mayergoyz, C. Serpico, Analysis of instabilities in nonlinear LandauLifshitz-Gilbert dynamics under circularly polarized fields, J. Appl. Phys. 91 (2002) 7556. [140] Z.Z. Sun, X.R. Wang, Magnetization reversal through synchronization with a microwave, Phys. Rev. B 74 (2006) 132401. [141] I. Mayergoyz, M. Dimian, G. Bertotti, C. Serpico, Inverse problem approach to the design of magnetic field pulses for precessional switching, J. Appl. Phys. 95 (2004) 7004. [142] N. Barros, M. Rassam, H. Jirari, H. Kachkachi, Optimal switching of a nanomagnet assisted by microwaves, Phys. Rev. B 83 (2011) 144418. [143] N. Barros, M. Rassam, H. Kachkachi, Microwave-assisted switching of a nanomagnet: analytical determination of the optimal microwave field, Phys. Rev. B 88 (2013) 014421.

ARTICLE IN PRESS 416

David S. Schmool and Hamid Kachkachi

[144] W.-K. Liu, B. Wu, J.-M. Yuan, Nonlinear dynamics of chirped pulse excitation and dissociation of diatomic molecules, Phys. Rev. Lett. 75 (1995) 1292. [145] J.-M. Yuan, W.-K. Liu, Classical and quantum dynamics of chirped pulse dissociation of diatomic molecules, Phys. Rev. A 57 (1998) 1992. [146] G. Marcus, L. Friedland, A. Zigler, From quantum ladder climbing to classical autoresonance, Phys. Rev. A 69 (2004) 013407. [147] G. Marcus, A. Zigler, L. Friedland, Molecular vibrational ladder climbing using a subnanosecond chirped laser pulse, Europhys. Lett. 74 (2006) 43. [148] B. Meerson, L. Friedland, Strong autoresonance excitation of Rydberg atoms: the Rydberg accelerator, Phys. Rev. A 41 (1990) 5233. [149] S. Chelkowski, A. Bandrauk, P.B. Corkum, Efficient molecular dissociation by a chirped ultrashort infrared laser pulse, Phys. Rev. Lett. 65 (1990) 2355. [150] S.G. Schirmer, H. Fu, A.I. Solomon, Complete controllability of quantum systems, Phys. Rev. A 63 (2001) 063410. [151] D.A. Garanin, H. Kachkachi, L. Reynaud, Magnetization reversal and nonexponential relaxation via instabilities of internal spin waves in nanomagnets, Europhys. Lett. 82 (2008) 17007. [152] D.A. Garanin, H. Kachkachi, Magnetization reversal via internal spin waves in magnetic nanoparticles, Phys. Rev. B 80 (2009) 014420. [153] A. Kashuba, Domain instability during magnetization precession, Phys. Rev. Lett. 96 (2006) 047601. [154] W. Wernsdorfer, K. Hasselbach, D. Mailly, B. Barbara, A. Benoit, L. Thomas, G. Suran, DC-SQUID magnetization measurements of single magnetic particles, J. Magn. Magn. Mater. 145 (1995) 33. [155] B. Barbara, Single-particle nanomagnetism, Solid State Sci. 7 (2005) 668. [156] J.-P. Cleziou, W. Wernsdorfer, V. Bouchiat, T. Ondarcuhu, M. Monthioux, Carbon nanotube superconducting quantum interference device, Nat. Nanotechnol. 1 (2006) 53. [157] W. Wernsdorfer, From micro-to nano-SQUIDs: applications to nanomagnetism, Supercond. Sci. Technol. 22 (2009) 064013. [158] W. Wernsdorfer, D. Mailly, A. Benoit, Single nanoparticle measurement techniques, J. Appl. Phys. 87 (2000) 5094. [159] W. Wernsdorfer, E. Boner Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. Demoncy, A. Loiseau, H. Pascard, D. Mailly, Experimental evidence of the Ne´elBrown model of magnetization reversal, Phys. Rev. Lett. 78 (1997) 1791. [160] M. Martin, L. Roschier, P. Hakonen, U. Parts, M. Paalanen, B. Schleicher, E.I. Kauppinen, Manipulation of Ag nanoparticles utilizing noncontact atomic force microscopy, Appl. Phys. Lett. 73 (1998) 1505. [161] B. Schleicher, U. Tapper, E.I. Kauppinen, M. Martin, L. Roschier, M. Paalanen, W. Wernsdorfer, A. Benoit, Magnetization reversal measurements of size-selected iron oxide particles produced via an aerosol route, Appl. Organomet. Chem. 12 (1998) 315. [162] M. Jamet, V. Dupuis, P. Me´linon, G. Guiraud, A. Pe´rez, W. Wernsdorfer, A. Traverse, B. Baguenard, Structure and magnetism of well defined cobalt nanoparticles embedded in a niobium matrix, Phys. Rev. B 62 (2000) 493. [163] M. Jamet, W. Wernsdorfer, C. Thirion, V. Dupuis, P. Me´linon, A. Pe´rez, D. Mailly, Magnetic anisotropy of a single cobalt nanocluster, Phys. Rev. B 69 (2004) 024401. [164] L. Bogani, C. Danieli, E. Biavardi, N. Bendiab, A.L. Barra, E. Dalcanale, W. Wernsdorfer, A. Cornia, Single-molecule-magnet carbon-nanotube hybrids, Angew. Chem. Int. Ed. Engl. 48 (2009) 746. [165] C. Chapelier, M. El Khatib, P. Perrier, A. Benoit, D. Mailly, in: H. Koch, H. Lebbig (Eds.), Superconducting Devices and Their Applications, Springer, Berlin, 1991, p. 286.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

417

[166] W. Wernsdorfer, Magne´tome´trie à micro-SQUID pour l’e´tude de particules ferromagne´tiques isole´es aux e´chelles, Ph.D. thesis, Joseph Fourier University, 2009. [167] W. Wernsdorfer, B. Doudin, D. Mailly, K. Hasselbach, A. Benoit, J. Meier, J.-P. Ansermet, B. Barbara, Nucleation of magnetization reversal in individual nanosized nickel wires, Phys. Rev. Lett. 77 (1996) 1873. [168] W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. Demoncy, A. Loiseau, D. Boivin, H. Pascard, D. Mailly, Experimental evidence of the Ne´el-Brown model of magnetization reversal, Phys. Rev. Lett. 78 (1997) 1791. [169] W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, D. Mailly, O. Kubo, H. Nakano, B. Barbara, Macroscopic quantum tunneling of magnetization of single ferrimagnetic nanoparticles of barium ferrite, Phys. Rev. Lett. 79 (1997) 4014. [170] M. Jamet, W. Wernsdorfer, C. Thirion, D. Mailly, V. Dupuis, P. Me´linon, A. Pe´rez, Magnetic Anisotropy of a Single Cobalt Nanocluster, Phys. Rev. Lett. 86 (2001) 4676. [171] E. Bonet, W. Wernsdorfer, B. Barbara, A. Benoit, D. Mailly, A. Thiaville, Threedimensional magnetization reversal measurements in nanoparticles, Phys. Rev. Lett. 83 (1999) 4188. [172] A. Thiaville, Coherent rotation of magnetization in three dimensions: a geometrical approach, Phys. Rev. B 61 (2000) 12221. [173] A.D. Kent, S. von Molna´r, S. Gider, D.D. Awschalom, Properties and measurement of scanning tunneling microscope fabricated ferromagnetic particle arrays, J. Appl. Phys. 76 (1994) 6656. [174] M. Rahm, J. Biberger, D. Weiss, Micro-Hall magnetometry, in: U. Sivan, T. Chakroborty, F. Peeters (Eds.), Nano-Physics and Bio-Electronics: A New Odyssey, Elsevier, Amsterdam, 2002, p. 257. [175] F.M. Peeters, X.Q. Li, Hall magnetometer in the ballistic regime, Appl. Phys. Lett. 72 (1998) 572. [176] M. Rahm, J. Raabe, R. Pulwey, J. Biberger, W. Wegscheider, D. Weiss, C. Meier, Planar Hall sensors for micro-Hall magnetometry, J. Appl. Phys. 91 (2002) 7980. [177] F.J. Castan˜o, C.A. Ross, C. Frandsen, A. Eilez, D. Gil, H.I. Smith, M. Redjdal, F.B. Humphrey, Metastable states in magnetic nanorings, Phys. Rev. B 67 (2003) 184425. [178] F.Q. Zhu, D. Fan, X. Zhu, J.-G. Zhu, R.C. Cammarata, C.-L. Chien, Ultrahighdensity arrays of ferromagnetic nanorings on macroscopic areas, Adv. Mater. 16 (2004) 2155. [179] C.C. Chen, J.Y. Lin, L. Horng, J.S. Yang, S. Isogami, M. Tsunoda, M. Takahashi, J.C. Wu, Investigation on the magnetization reversal of nanostructured magnetic tunnel junction rings, IEEE Trans. Magn. 45 (2009) 3546. [180] G. Li, V. Joshi, R.L. White, S.X. Wang, J.T. Kemp, C. Webb, R.W. David, S. Sun, Detection of single micron-sized magnetic bead and magnetic nanoparticles using spin valve sensors for biological applications, J. Appl. Phys. 93 (2003) 7557. [181] A.K. Petford-Long, J.N. Chapman, Lorentz microscopy, in: H. Hopster, H.P. Oepen (Eds.), Magnetic Microscopy of Nanostructures, Springer, Berlin, Heidelberg, 2005, p. 67. [182] W. Kuch, Magnetic imaging, Lect. Notes Phys. 697 (2006) 275. [183] D. Petit, et al., Magnetic imaging of the pinning mechanism of asymmetric transverse domain walls in ferromagnetic nanowires, Appl. Phys. Lett. 97 (2010) 233102. [184] M.A. Basith, S. McVitie, D. McGrouther, J.N. Chapman, J.M.R. Weaver, Direct comparison of domain wall behavior in permalloy nanowires patterned by electron beam lithography and focused ion beam milling, J. Appl. Phys. 110 (2011) 083904. [185] H.W. Fuller, M.E. Hale, Determination of magnetization distribution in thin films using electron microscopy, J. Microsc. 31 (1960) 238.

ARTICLE IN PRESS 418

David S. Schmool and Hamid Kachkachi

[186] A. Hu¨tten, J. Bernardi, C. Nelson, G. Thomas, Lorentz microscopy of giant magnetoresistive Au-Co alloys, Phys. Status Solidi A 150 (1995) 171. [187] C. Salling, S. Schultz, I. McFayden, M. Ozaki, Measuring the coercivity of individual sub-micron ferromagnetic particles by Lorentz microscopy, IEEE Trans. Magn. 27 (1991) 5184. [188] S.A. Majetich, Y. Jin, Magnetization directions of individual nanoparticles, Science 284 (1999) 470. [189] K. Yamamoto, C.R. Hogg, S. Yamamuro, T. Hirayama, S.A. Majetich, Dipolar ferromagnetic phase transition in Fe3O4 nanoparticle arrays observed by Lorentz microscopy and electron holography, Appl. Phys. Lett. 98 (2011) 072509. [190] G. Binnig, H. Rohrer, C. Gerber, E. Weibel, Tunneling through a controllable vacuum gap, Appl. Phys. Lett. 40 (1982) 178. [191] G. Binnig, H. Rohrer, C. Gerber, E. Weibel, Surface studies by scanning tunneling microscopy, Phys. Rev. Lett. 49 (1982) 57. [192] G. Binnig, H. Rohrer, Scanning tunneling microscopy: from birth to adolescence, Rev. Mod. Phys. 59 (1987) 615. [193] G. Binnig, H. Rohrer, In touch with atoms, Rev. Mod. Phys. 71 (1999) S324. [194] P. Jaschinsky, P. Coenen, G. Pirug, B. Voigtla¨nder, Design and performance of a beetle-type double-tip scanning tunneling microscope, Rev. Sci. Instrum. 77 (2006) 093701. [195] W.J. Kaiser, R.C. Jaklevic, Scanning tunneling microscopy study of metals: spectroscopy and topography, Surf. Sci. 181 (1987) 55. [196] H.J.W. Zandvliet, A. van Houselt, Scanning tunneling spectroscopy, Ann. Rev. Anal. Chem. 2 (2009) 37. [197] G. Binnig, C.F. Quate, C. Gerber, Atomic force microscope, Phys. Rev. Lett. 56 (1986) 930. [198] Y. Martin, H.K. Wickramasinghe, Magnetic imaging by force microscopy with 1000 A˚ resolution, Appl. Phys. Lett. 50 (1987) 1455. [199] T. Kaneko, H. Imamura, Effect of the tip-contact interaction on the MFM image of magnetic nanocontact, J. Appl. Phys. 109 (2011) 07D356. [200] A. Winkler, T. Mu¨hl, S. Menzel, R. Kozhuharova-Koseva, S. Hampel, A. Loenhardt, B. Bu¨chner, Magnetic force microscopy sensors using iron-filled carbon nanotubes, J. Appl. Phys. 99 (2006) 104905. [201] A. Hendrych, R. Kubı´nek, A.V. Zhukov, The magnetic force microscopy and its capability for nanomagnetic studies—the short compendium, in: A. Me´ndez-Vilas, J. Dı´az (Eds.), Modern Research in Educational Topics in Microscopy, Formatex Research Center, Badajoz (Spain), 2007, p. 805. [202] A. Fernandez, P.J. Bedrossian, S.L. Baker, S.P. Vernon, D.R. Kania, Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography, IEEE Trans. Magn. 32 (1996) 4472. [203] M. Ras¸a, B.W.M. Kuipers, A.P. Philipse, Atomic force microscopy and magnetic force microscopy study of model colloids, J. Colloid Interface Sci. 250 (2002) 303. [204] B. Torre, G. Bertoni, D. Fragouli, A. Falqui, M. Salerno, A. Diaspro, R. Cingolani, A. Athanassiou, Magnetic force microscopy and energy loss imaging of superparamagnetic iron oxide nanoparticles, Sci. Rep. 202 (2011), http://dx.doi.org/ 10.1038/srep00202. [205] T. Okuno, K. Shigeto, T. Ono, K. Mibu, T. Shinjo, MFM study of magnetic vortex cores in circular permalloy dots: behavior in external field, J. Magn. Magn. Mater. 240 (2002) 1. [206] J. Chang, H. Yi, H.C. Koo, V.L. Mironov, B.A. Gribkov, A.A. Fraerman, S.A. Gusev, S.N. Vdovichev, Magnetization reversal of ferromagnetic nanoparticles under inhomogeneous magnetic field, J. Magn. Magn. Mater. 309 (2007) 272.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

419

[207] V.L. Mironov, B.A. Gribkov, A.A. Fraerman, S.A. Gusev, S.N. Vdovichev, I.R. Karetnikova, I.M. Nefedov, I.A. Shereshevsky, MFM probe control of magnetic vortex chirality in elliptical Co nanoparticles, J. Magn. Magn. Mater. 312 (2007) 153. [208] C. Dietz, E.T. Herruzo, J.R. Lozano, R. Garcia, Nanomechanical coupling enables detection and imaging of 5 nm superparamagnetic particles in liquid, Nanotechnology 22 (2011) 125708. [209] X.M. Cheng, D.J. Keavney, Studies of nanomagnetism using synchrotron-based x-ray photoemission electron microscopy (X-PEEM), Rep. Prog. Phys. 75 (2012) 026501. [210] J. St€ ohr, Y. Wu, M.G. Samant, B.B. Hermsmeier, G. Harp, S. Koranda, D. Dunham, B.P. Tonner, Element specific magnetic microscopy using circularly polarized x-rays, Science 259 (1993) 658. [211] J. St€ ohr, H.A. Padmore, S. Anders, T. Stammler, M.R. Sheinfein, Principles of X-ray magnetic dichroism spectromicroscopy, Surf. Rev. Lett. 5 (1998) 1297. [212] G. Sch€ onhense, H.J. Elmers, PEEM with high time resolution imaging of transient processes and novel concepts of chromatic and spherical aberration correction, Surf. Interface Anal. 38 (2006) 1578. [213] S.-B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J. St€ ohr, H.A. Padmore, Vortex core-driven magnetization dynamics, Science 304 (2004) 420. [214] J. Miguel, et al., Time-resolved magnetization dynamics of cross-tie domain walls in permalloy microstructures, J. Phys. Condens. Matter 21 (2009) 496001. [215] D.P. Woodruff, T.A. Delchar, Modern Techniques of Surface Science, Cambridge University Press, Cambridge, 1994. [216] G. Schu¨tz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, G. Materlik, Absorption of circularly polarized x rays in iron, Phys. Rev. Lett. 58 (1987) 737. [217] C.T. Chen, Y.U. Idzerda, H.-J. Lin, N.V. Smith, G. Meigs, E. Chaban, G.H. Ho, E. Pellegrin, F. Sette, Experimental confirmation of the X-ray magnetic circular dichroism sum rules for iron and cobalt, Phys. Rev. Lett. 75 (1995) 152. [218] B.T. Thole, P. Carra, F. Sette, G. van der Laan, X-ray circular dichroism as a probe of orbital magnetization, Phys. Rev. Lett. 68 (1992) 1943. [219] P. Carra, B.T. Thole, M. Altarelli, X. Wang, X-ray circular dichroism and local magnetic fields, Phys. Rev. Lett. 70 (1993) 694. [220] R. Wu, D. Wang, A.J. Freeman, First principles investigation of the validity and range of applicability of the x-ray magnetic circular dichroism sum rule, Phys. Rev. Lett. 71 (1993) 3581. [221] R. Wu, A.J. Freeman, Limitation of the magnetic-circular-dichroism spin sum rule for transition metals and importance of the magnetic dipole term, Phys. Rev. Lett. 73 (1994) 1994. [222] J.-S. Kim, et al., Current-induced vortex dynamics and pinning potentials probed by homodyne detection, Phys. Rev. B 82 (2010) 104427. [223] X.M. Cheng, K.S. Buchanan, R. Divan, K.Y. Guslienko, D.J. Keavney, Nonlinear vortex dynamics and transient domains in ferromagnetic disks, Phys. Rev. B 79 (2009) 172411. [224] D.J. Keavney, X.M. Cheng, K.S. Buchanan, Polarity reversal of a magnetic vortex core by a unipolar, nonresonant in-plane pulsed magnetic field, Appl. Phys. Lett. 94 (2009) 172506. [225] Y. Acremann, Magnetization dynamics: ultra-fast and ultra-small, C. R. Phys. 9 (2008) 585. [226] F. Wegelin, A. Krasyuk, H.-J. Elmers, S.A. Nepijko, C.M. Schneider, G. Sch€ onhense, Stroboscopic XMCD-PEEM imaging of standing and propagating spinwave modes in permalloy thin-film structures, Surf. Sci. 601 (2007) 4694.

ARTICLE IN PRESS 420

David S. Schmool and Hamid Kachkachi

[227] A. Fraile Rodrı´guez, A. Kleibert, J. Bansmann, A. Voitkans, L.J. Heyderman, F. Nolting, Size-dependent spin structures in iron nanoparticles, Phys. Rev. Lett. 104 (2010) 127201. [228] N de Sousa, et al., Spin configurations in hard/soft coupled bilayer systems: transitions from rigid magnet to exchange-spring, Phys. Rev. B 82 (2010) 104433. [229] E.E. Fullerton, J.S. Jiang, M. Grimsditch, C.H. Sowers, S.D. Bader, Exchange-spring behavior in epitaxial hard/soft magnetic bilayers, Phys. Rev. B 58 (1998) 12193. [230] P. Bruno, Geometrically constrained magnetic wall, Phys. Rev. Lett. 83 (1999) 2425. [231] A. Balan, P.M. Derlet, A. Fraile Rodrı´guez, J. Bansmann, R. Yanes, U. Nowak, A. Kleibert, F. Nolting, Direct observation of magnetic metastability in individual iron nanoparticles, Phys. Rev. Lett. 112 (2014) 107201. [232] C.P. Poole, Electron Spin Resonance, A Comprehensive Treatise on Experimental Techniques, Wiley, New York, 1983. [233] D.J.E. Ingram, Spectroscopy at Radio and Microwave Frequencies, Butterworths, London, 1967. [234] D.S. Schmool, Spin dynamics in nanometric magnetic systems, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials>, vol. 18, Elsevier Science, 2009, p. 111. [235] D.S. Schmool, Spin dynamic studies in ferromagnetic nanoparticles, in: S. Thomas, N. Kalarikkal, A. Manuel Stephan, B. Raneesh, A.K. Haghi (Eds.), Advanced Nanomaterials: Synthesis, Properties and Applications, Apple Academic Press, New York, 2015. [236] G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto, Y. Otani, Spin wave contributions to the high-frequency magnetic response of thin films obtained with inductive methods, J. Appl. Phys. 95 (2004) 5646. [237] Y. Ding, T.J. Klemmer, T.M. Crawford, A coplanar waveguide permeameter for studying high-frequency properties of soft magnetic materials, J. Appl. Phys. 96 (2004) 2969. [238] S.S. Kalarichal, P. Krivosik, M. Wu, C.E. Patton, M.L. Schneider, P. Kabos, T.J. Silva, J.P. Nibarger, Ferromagnetic resonance linewidth in metallic thin films: comparison of measurement methods, J. Appl. Phys. 99 (2006) 093909. [239] O. Mosendz, B. Kardasz, D.S. Schmool, B. Heinrich, Spin dynamics at low microwave frequencies in crystalline Fe ultrathin film double layers using co-planar transmission lines, J. Magn. Magn. Mater. 300 (2006) 174. [240] C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, P.P. Freitas, Vector network analyzer ferromagnetic resonance of thin films on coplanar waveguides: comparison of different evaluation methods, J. Appl. Phys. 101 (2007) 074505. [241] R. Narkowicz, D. Suter, I. Niemeyer, Scaling of sensitivity and efficiency in planar microresonators for electron spin resonance, Rev. Sci. Instrum. 79 (2008) 084702. [242] W. Barry, A broad-band, automated, stripline technique for the simultaneous measurement of complex permittivity and permeability, IEEE Trans. Microwave Theory Tech. 34 (1986) 80. [243] D.K. Ghodgaonkar, V.V. Varadan, V.K. Varadan, Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies, IEEE Trans. Instrum. Meas. 39 (1990) 387. [244] H. Kachkachi, D.S. Schmool, Ferromagnetic resonance in systems with competing uniaxial and cubic anisotropies, Eur. Phys. J. B 56 (2007) 27. [245] E. Montoya, T. McKinnon, A. Zamani, E. Girt, B. Heinrich, Broadband ferromagnetic resonance system and methods for ultrathin magnetic films, J. Magn. Magn. Mater. 356 (2014) 12. [246] M. Helsen, A. Gangwar, A. Vansteenkiste, B. Van Waeyenberge, Magneto-optical spectrum analyzer, Rev. Sci. Instrum. 85 (2014) 083902.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

421

[247] R. Narkowicz, D. Suter, R. Stonies, Planar microresonators for EPR experiments, J. Magn. Res. 175 (2005) 275. [248] A. Banholzer, R. Narkowicz, C. Hassel, R. Meckenstock, S. Stienen, O. Posth, D. Suter, M. Farle, J. Lindner, Visualization of spin dynamics in single nanosized magnetic elements, Nanotechnology 22 (2011) 295713. [249] C. Schoeppner, K. Wagner, S. Stienen, R. Meckenstock, M. Farle, R. Narkowicz, D. Suter, J. Lindner, Angular dependent ferromagnetic resonance analysis in a single micron sized cobalt stripe, J. Appl. Phys. 116 (2014) 033913. [250] M. Farle, Private Communication, 2014. [251] D.S. Schmool, M. Schmalzl, Ferromagnetic resonance in magnetic nanoparticle assemblies, J. Non Cryst. Solids 353 (2007) 738. [252] G de Loubens, G.D. Chaves-O’Flynn, A.D. Kent, C. Ramsey, E. del Barco, C. Beedle, D.N. Hendrickson, Magnetization and EPR studies of the single molecule magnet Ni4 with integrated sensors, J. Appl. Phys. 101 (2007) 09E104. [253] G de Loubens, D.A. Garanin, C.C. Beedle, D.N. Hendrickson, A.D. Kent, Magnetization relaxation in the single-molecule magnet Ni4 under continuous microwave irradiation, Eur. Phys. Lett. 83 (2008) 37006. [254] H.M. Quddusi, et al., On-chip integration of high-frequency electron paramagnetic resonance spectroscopy and Hall-effect magnetometry, Rev. Sci. Instrum. 79 (2008) 074703. [255] E. del Barco, A.D. Kent, E.C. Yang, D.N. Hendrickson, Quantum superposition of high spin states in the single molecule magnet Ni 4, Phys. Rev. Lett. 93 (2004) 157202. [256] Y. Twig, E. Dikarov, W.D. Hitchinson, A. Blank, High sensitivity pulsed electron spin resonance spectroscopy with induction detection, Rev. Sci. Instrum. 82 (2011) 076105. [257] A. Blank, E. Suhovoy, R. Halevy, L. Shtirberg, W. Harneit, ESR imaging in solid phase down to sub-micron resolution: methodology and applications, Phys. Chem. Chem. Phys. 11 (2009) 6689. [258] U. Wiedwald, et al., Ratio of orbital-to-spin magnetic moment in Co core-shell nanoparticles, Phys. Rev. B 68 (2003) 064424. [259] G. Ghiringhelli, L.H. Tjeng, A. Tanaka, O. Tjernberg, T. Mizokawa, J.L. de Boer, N.B. Brookes, 3 d spin-orbit photoemission spectrum of nonferromagnetic materials: the test cases of CoO and Cu, Phys. Rev. B 66 (2002) 075101. [260] D. Rugar, C.S. Yannoni, J.A. Sidles, Mechanical detection of magnetic resonance, Nature 360 (1992) 563. [261] J.A. Sidles, Noninductive detection of single-proton magnetic resonance, Appl. Phys. Lett. 58 (1991) 2854. [262] J.A. Sidles, Folded Stern-Gerlach experiment as a means for detecting nuclear magnetic resonance in individual nuclei, Phys. Rev. Lett. 68 (1992) 1124. [263] J.A. Sidles, J.L. Garbini, K.L. Bruland, D. Rugar, O. Zu¨ger, S. Hoen, C.S. Yannoni, Magnetic resonance force microscopy, Rev. Mod. Phys. 67 (1995) 249. [264] J.A. Sidles, J.L. Garbini, G.P. Drobny, The theory of oscillator-coupled magnetic resonance with potential applications to molecular imaging, Rev. Sci. Instrum. 63 (1992) 3881. [265] Z. Zhang, P.C. Hammel, P.E. Wigen, Observation of ferromagnetic resonance in a microscopic sample using magnetic resonance force microscopy, Appl. Phys. Lett. 68 (1996) 2005. [266] G de Loubens, V.V. Naletov, O. Klein, J. Ben Youssef, F. Boust, N. Vukadinovic, Magnetic resonance studies of the fundamental spin-wave modes in individual submicron Cu/NiFe/Cu perpendicularly magnetized disks, Phys. Rev. Lett. 98 (2007) 127601.

ARTICLE IN PRESS 422

David S. Schmool and Hamid Kachkachi

[267] D. Rugar, R. Budaklan, H.J. Mamin, B.W. Chui, Single spin detection by magnetic resonance force microscopy, Nature 430 (2004) 329. [268] O. Klein, et al., Ferromagnetic resonance force spectroscopy of individual submicronsize samples, Phys. Rev. B 78 (2008) 144410. [269] V. Charbois, V.V. Naletov, J. Ben Youssef, O. Klein, Influence of the magnetic tip in ferromagnetic resonance force microscopy, Appl. Phys. Lett. 80 (2002) 4795. [270] R.W. Damon, J.R. Eshbach, Magnetostatic modes of a ferromagnet slab, J. Phys. Chem. Solids 19 (1961) 308. [271] B.A. Kalinikos, A.N. Slavin, Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions, J. Phys. C Solid State Phys. 19 (1986) 7013. [272] B.A. Kalinikos, M.P. Kostylev, N.V. Kozhus’, A.N. Slavin, The dipole-exchange spin wave spectrum for anisotropic ferromagnetic films with mixed exchange boundary conditions, J. Phys. Condens. Matter 2 (1990) 9861. [273] A.F. Franco, J.L. De´jardin, H. Kachkachi, Ferromagnetic resonance of a magnetic dimer with dipolar coupling, J. Appl. Phys. 116 (2014) 243905. [274] M. Beleggia, S. Tandon, Y. Zhu, M. De Graef, On the computation of the demagnetization tensor for particles of arbitrary shape, J. Magn. Magn. Mater. 272 (2004) e1197. [275] V.V. Naletov, et al., Identification and selection rules of the spin-wave eigenmodes in a normally magnetized nanopillar, Phys. Rev. B 84 (2011) 224423. [276] A. Hamadeh, G de Loubens, V.V. Naletov, J. Grollier, C. Ulysse, V. Cros, O. Klein, Autonomous and forced dynamics in a spin-transfer nano-oscillator: quantitative magnetic-resonance force microscopy, Phys. Rev. B 85 (2012) 140408(R). [277] B. Pigeau, et al., Measurement of the dynamical dipolar coupling in a pair of magnetic nanodisks using a ferromagnetic resonance force microscope, Phys. Rev. Lett. 109 (2012) 247602. [278] R. Meckenstock, I. Barsukov, C. Bircan, A. Remhoff, D. Dietzel, D. Spoddig, Imaging of ferromagnetic-resonance excitations in Permalloy nanostructures on Si using scanning near-field thermal microscopy, J. Appl. Phys. 99 (2006) 08C706. [279] R. Meckenstock, I. Barsukov, O. Posth, J. Lindner, A. Butko, D. Spoddig, Locally resolved ferromagnetic resonance in Co stripes, Appl. Phys. Lett. 91 (2007) 142507. [280] R. Meckenstock, Microwave spectroscopy based on scanning thermal microscopy: resolution in the nanometer range, Rev. Sci. Instrum. 79 (2008) 041101. [281] I. Rod, R. Meckenstock, H. Za¨hres, C. Derricks, F. Mushenok, N. Reckers, P. Kijamnajsuk, U. Wiedwald, M. Farle, Bolometer detection of magnetic resonances in nanoscaled objects, Nanotechnology 25 (2014) 425302. [282] J. Schmidt, I. Solomon, High-sensitivity magnetic resonance by bolometer detection, J. Appl. Phys. 37 (1966) 3719. [283] W.A. Murray, W.L. Barnes, Plasmonic materials, Adv. Mater. 19 (2007) 3771. [284] M.I. Stockman, The physics behind the applications, Phys. Today 64 (2011) 39. [285] M.I. Stockman, Nanoplasmonics: past, present, and glimpse into future, Opt. Express 19 (2011) 22029. [286] H. Duan, A.I. Ferna´ndez-Domı´nguez, M. Bosman, S.A. Maier, J.K.W. Yang, Nanoplasmonics: classical down to the nanometer scale, Nano Lett. 12 (2012) 1683. [287] D.J. Bergman, M.I. Stockman, Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems, Phys. Rev. Lett. 90 (2003) 027402. [288] J.N. Li, T.Z. Liu, H.R. Zheng, F. Gao, J. Dong, Z.L. Zhang, Z.Y. Zhang, Plasmon resonances and strong electric field enhancements in side-by-side tangent nanospheroid homodimers, Phys. Rev. Lett. 21 (2013) 17176.

ARTICLE IN PRESS Single-Particle Phenomena in Magnetic Nanostructures

423

[289] V. Amendola, S. Scaramuzza, S. Agnoli, S. Polizzi, M. Meneghetti, Strong dependence of surface plasmon resonance and surface enhanced Raman scattering on the composition of Au-Fe nanoalloys, Nanoscale 6 (2014) 1423. [290] H.A. Atwater, A. Polman, Plasmonics for improved photovoltaic devices, Nat. Mater. 9 (2010) 205. [291] G. Armelles, A. Cebollada, A. Garcı´a-Martı´n, M.U. Gonza´lez, Magnetoplasmonics: combining magnetic and plasmonic functionalities, Adv. Opt. Mater. 1 (2013) 10. [292] V. Bonanni, S. Bonetti, T. Pakizeh, Z. Pirzadeh, J. Chen, J. Nogues, P. Vavassori, R. Hillenbrand, J. Akerman, A. Dmitriev, Designer magnetoplasmonics with nickel nanoferromagnets, Nano Lett. 11 (2011) 5333. [293] S. Peng, C. Lei, Y. Ren, R.E. Cook, Y. Sun, Plasmonic/magnetic bifunctional nanoparticles, Angew. Chem. Int. Ed. Engl. 50 (2011) 3158.