C 6 Clusters to Nanowires

P. S. Bechthold

Institut f ¨ur Festk ¨orperforschung Forschungszentrum J ¨ulich GmbH

1

Contents

1 Introduction 2

2 Magnetic Particles in Contact with a Heat Bath 4

3 Stern-Gerlach Experiments 7

4 Clusters on Surfaces 16

5 Co Monatomic Chains on Pt(997) 21

6 Conclusion 23

1slightly corrected version of lecture notes presented in: S. Bl¨ugel, T. Br¨uckel, C.M. Schneider (eds.), Mag- netism goes Nano, Electron Correlations, Spin Transport, Molecular Magnetism, Lecture Manuscripts of the 36^th Spring School of the Institute of Solid State Research, Matter and Materials Vol. 26, Forschungszentrum J¨ulich (2005), ISSN 1433-5506, ISBN 3-89336-381-5; see also: http://hdl.handle.net/2128/560

1 Introduction

Clusters are aggregates of atoms or molecules with a well-defined number of constituents [1].

As such they are the fundamental building blocks of nanostructured matter and bridge the gap between atoms or molecules and the bulk. They provide the unique opportunity to study the development of a physical property from the atom towards the solid, i.e., from a ”zero”- dimensional to a three-dimensional system. This concerns, for example, the formation of an electronic band structure or the development of many-electron phenomena, such as solid state magnetism or superconductivity.

In this contribution we will be concerned with the magnetism of small clusters [2] and nanowires of 3d-, 4d-, and 4f-transition metals. Like the corresponding atoms these small particles have significant spin and orbital contributions to their magnetic moment. This is in contrast to the respective solids, where the orbital moment is almost completely quenched by crystal fields and where in most cases the exchange energy that could be gained by spin alignment is exceeded by the increase in kinetic energy that according to the Pauli principle is required to promote the electrons to empty orbitals. Most solids therefore are nonmagnetic. Only when the bandwidth is small and correspondingly the density of statesD(EF)is large at the Fermi energy like in some d- and f- metals is the gain in exchange energy larger than the expense of increasing kinetic energy. Only then magnetism will survive. This corresponds to the Stoner criterion, which states that for a material to be magnetic the product of the exchange integral I and D(EF) needs to be larger than 1 [3].

For later use, in section 2 we will introduce the concepts of superparamagnetism, ”locked mo- ment”, the magnetic anisotropy energy and the blocking temperature. In this context we will also introduce the Stoner-Wohlfahrt and the N´eel-Brown model. These terms are usually used for particles which are somewhat larger then the ones in the center of interest here [4], i.e. the particles here are so small that molecular beam methods can be applied. In fact, these methods offer the unique opportunity to study a series of clusters atom by atom without the disturbing interactions with any environment. A very sensitive molecular beam method for studying the magnetic moments of metal clusters is based on the classic Stern-Gerlach experiment. This will be the topic of section 3. In this experiment, a gradient-field magnet induces small deflections of a beam of magnetic clusters traveling through a high vacuum machine. These tiny deflec- tions are only a fraction of a millimeter to millimeters in magnitude, but can be measured with high precision by a position-sensitive time-of-flight technique. From the magnitude of these deflections we are able to determine the magnetic moments of the clusters, which is the most important quantity we want to know. In section 4 we will discuss the properties of clusters attached to some surface. We will study how clusters can be grown by aggregation or how preselected clusters can be ”soft landed”. X-ray magnetic circular dichroism (XMCD) [5] then allows us to distinguish between spin and orbital contributions to the magnetic moment. Finally in section 5 an experiment is presented, where an array of Co monoatomic chains is grown on a vicinal Pt(997) surface and also investigated by XMCD.

Before we proceed we need to recapitulate some elementary properties of clusters that we should know. Clusters are not simply pieces of the respective solids, but may have very dif- ferent and often very individual physical and chemical properties, e.g. the chemical reactivity can change by orders of magnitude from one cluster size to the next. Therefore, clusters may form specific very selective catalysts.

Clusters may have decahedral or icosahedral structures which follow their own individual growth patterns. Due to their fivefold symmetry axes they cannot grow into periodic lattice structures.

Fig. 1 compares the smallest icosahedron (13 atoms, Ih-symmetry) with the smallest cube- octahedron (13 atoms,Oh-symmetry). While the cube-octahedron can grow directly into a face centered cubic (fcc) lattice by successively filling additional geometric shells, the icosahedron cannot. At any size an icosahedron has a well-defined center atom. At some size it must per- form a structural transition, before it can grow into a periodic lattice. Metal clusters, e.g. N in

(n= 13,. . .), often follow an icosahedral growth pattern because this optimizes the number of nearest neighbors and thus the number of direct metal-metal bonds, which in turn can optimize the total binding energy. However, because of internal stress²the icosahedra cannot grow arbi- trarily large. The transition to the cubic structure has to occur at some cluster sizen, presumably somewhere betweenn = 1000andn= 10000.

Fig. 1: Comparison of an icosahedron and a cube-octahedron. The cube-octahedron can grow directly into a face centered cubic (fcc) lattice, the icosahedron with its fivefold symmetry axes cannot.

Clusters are characterized by a large surface to volume ratio, that is, a large fraction of the constituents occupies low coordinated surface sites, e.g. for a cluster of 2000 atoms a fraction of≈ 30 % is still at the surface. This is displayed in Fig.2, where we plotted the percentage of surface atoms overn^−1/3. Since the cluster volume scales with the number of atoms it is evident that this quantity is proportional to the inverse of the mean cluster radiusR. The large fraction of surface atoms obviously makes the clusters receptive to chemical reactions. But it also has consequences for magnetism. The surface atoms are lower coordinated than the bulk. They therefore have a more atom like character than interior atoms and - in accordance with findings for thin films and surfaces - contribute more to the total magnetic moment of the cluster. The result is that the magnetic properties of clusters are very sensitive to their chemical environment. Magnetism may be quenched, e.g. by charge transfer into empty d-states. On the other hand, it is also possible to even increase the magnetic moment of a cluster by the correct choice of a reactant. An example will be reported below. It has been recently shown that even gold nanoparticles can be remanently magnetized, when capped with thiol ligands, even at room temperature [6]. Of course this sensitivity also has implications to deposited clusters.

2In an icosahedron the bonds to the center atom are shorter than the bonds in the surface

n

2869

n_s n

n

n

2869

n_s n

n

n

2869

n n

2869

n_s n

n

Fig. 2: Percentage of surface to bulk atoms as a function ofn^−1/3 ∝1/R(bottom scale), where n(top scale) is the total number of atoms in a cluster and R is the cluster radius. The numbers - often referred to as magic numbers - denote closed shell icosahedral and cube-octahedral clusters. Note, that the shell closings occur at the same magic numbers for both structures. The inset shows the three smallest closed shell icosahedra and a scheme of a growing icosahedron.

Even clusters with 2000 atoms still have 30 % of all atoms at the surface.

2 Magnetic Particles in Contact with a Heat Bath

A macroscopic magnetic sample tends to reduce its stray field energy by forming magnetic domains. But, due to the different orientations of the magnetic moments in the domain walls, this costs exchange energy and magnetic anisotropy energy. The energy balance determines the final size and shape of the domains. When the sample becomes smaller and smaller a point is finally reached where the energy gain by the formation of domains equals the energy consumption to build domain walls. Thus, below this critical size the particle becomes single domain. The smallest size for this to happen is at a particle diameter comparable to the thickness of a domain wall. The point of transition depends on the material considered and occurs in the size range of 20 - 2000 nm [7-9]. This transition from a multidomain to a single domain particle is accompanied by an increase of the coercive field.

The direction of the easy axis in a small particle is determined by its shape anisotropy and its magnetostructural anisotropy. The shape anisotropy is due to the external geometry of a par- ticle and favors longer over shorter axes (e.g. for ellipsoidal particles) because the magnetic field energy is smaller for this orientation of the magnetization. The structural anisotropy - for crystallographic symmetries also called magnetocrystalline anisotropy - depends on the struc- tural arrangement of the atoms in the particle and prefers certain structural orientations of the magnetization. For a single domain particle with uniaxial anisotropy the anisotropy energy can be written (to lowest order) as: E_a = KV sin²θ,whereK is the magnetic anisotropy constant (the anisotropy energy per unit volume), V is the particle volume and θ measures the angle between the easy axis and the orientation of the magnetic moment. The two minima atθ = 0^o andθ = 180^o are separated by an energy barrier of height KV. The barrier height is propor-

tional to the particle volume and decreases further with decreasing particle size. Eventually the anisotropy energy may become comparable to or smaller than the thermal energy - even at low temperatures. Then the thermal fluctuations of the magnetization are sufficient to overcome the barrier and the particle becomes superparamagnetic [10,11]. In this state the spins of the parti- cle are still coupled by exchange forces but the total magnetic moment fluctuates freely within the particle geometry. In a magnetic field superparamagnetic particles show magnetization sat- uration but no hysteresis. They behave like paramagnets with large magnetic moments, hence the name superparamagnetism.

The transition from a locked moment state - where the total magnetic moment fluctuates about the easy axis - to superparamagnetism is a thermally activated stochastic process. At thermal contact with a heat bath of temperature T the energy is distributed among an ensemble of particles according to the Boltzmann distribution, i.e. the probability to find a particle with internal energyE isp(E) = _k¹

BTe^−E/kBT, wherekBT is the normalization factor³. kB is the Boltzmann constant andT the temperature of the heat bath. IfEb is the height of an energy barrier then the fraction of particles with an energy in excess ofEb is e^−Eb/kBT,because p = R^∞

Eb p(E)dE = _k¹

BT

R^∞

Ebe^−E/kBTdE =e^−Eb/kBT.

Thus, due to the N´eel-Brown model [12,13] the transition rate can be described by an Arrhenius type equation.

fJ =f_J⁰exp(− Eb

kBT) = f_J⁰exp(−KV

kBT), (1)

where the attempt frequencyf_J⁰ is of the order of the gyromagnetic precession frequency and is in the range of 10⁹ - 10¹² s⁻¹,Eb =KV is the magnetic anisotropy barrier. The indexJ refers to the (magnetically relevant) angular momentum quantum number. Puttingτ = 1/f_J ≈ τ_exp a characteristic time of the experiment and τ⁰ = 1/f_J⁰ the precession time we can define a

”blocking temperature”

Tbl = KV

kB ln(τexp/τ⁰). (2)

ForT > T_bl the magnetization of a cluster may fluctuate on the time scale of the experiment while forT < Tbl it will reside in one energy minimum and therefore is locked to the cluster geometry. In this case the fluctuations are too slow to be observed on the time scale of the experiment. The blocking temperature will depend on the experimental conditions, e.g. for data storage we will requireτexp ≥10awhile for a susceptibility measurement a few seconds might be sufficient. For data storage purposes values ofKV /kBT ≈ 40−60are considered to be acceptable [14].

The appearance of the particle volumeV in the exponential factor of equation (1) results in a fairly sharply defined particle size for the locked moment to superparamagnetic transition. As an example, a spherical iron particle of 23 nm diameter will relax in10⁻¹sat room temperature and hence will reach thermal equilibrium very fast. In contrast, a particle with a diameter of 30 nm will have a relaxation time of10⁹sand hence will be quite stable [11].

At zero magnetic field the two energy minima of the uniaxial system are equivalent. When a field is applied opposite to the magnetization of a uniaxial particle the energy barrier to magnetization reversal is reduced to E_B⁺ = KV(1− B/B0)², while the barrier to thermal switching against the direction of the applied field increases toE_B⁻ =KV(1 +B/B0)²,where B0 = 2K/Ms is the anisotropy field [see appendix]. This results in two different switching

3R^∞

0 e^−E/kBTdE=kBT

rates for forward and backward switching:

fJ,± =f_J⁰exp(−KV(1∓B/B0)²

kBT ), (3)

with which the particles try to establish thermal equilibrium.

In the absence of an external magnetic field the magnetic moments of superparamagnetic par- ticles are randomly oriented⁴. If an external field is applied the magnetic moments tend to align parallel to the field while the thermal fluctuations counteract this alignment. A dynamical equilibrium is established. One then measures an average magnetization that is related to the intrinsic magnetic moment via the Brillouin function.

Mz =N < µz >=N µef f =N µBJ(x)

whereN is the number of particles per unit volume,B_J(x)is the Brillouin function B_J(x) = 2J + 1

2J coth(2J+ 1

2J x)− 1

2J coth( x

2J). (4)

µis the magnetic moment of each particle, N µ = Ms the saturation magnetization, and x = µB/kBT [15]. For superparamagnetic particles J often gets very large. Then, the Brillouin function approaches the Langevin functionL. Thus, using the expansion

coth y ≈1/y+y/3−... (5)

we get

Mz =N µef f =N < µz >=N µL(µB

k_BT) =N µ[coth µB

k_BT − kBT

µB ], (6) If µB ≪ kBT, i.e. for small fields or high temperatures we can once more make use of the expansion (5) and the Langevin function can be replaced by

Mz =N µ²B

3kBT (7)

and the observed effective magnetic moment per particle becomes (Curie-Law):

µef f = µ²B

3k_BT. (8)

When the temperature increases more and more, eventually the individual atomic spins may uncouple. The internal magnetic order breaks down and the particle reaches a paramagnetic state.

4for simplicity we assume that the magnetic moments are all identical andkBT ≫Eb.

3 Stern-Gerlach Experiments

The magnetic moments of mass selected clusters are measured with adequately adapted Stern- Gerlach deflection experiments. Fig. 3 shows a scheme of a modern experimental arrangement.

It consists of three main constituents:

1. a chamber where the clusters are generated in a laser vaporization source,

2. the inhomogeneous Stern-Gerlach deflection field, and a subsequent drift region,

3. a position-sensitive time-of-flight mass-spectrometer (TOF) where the beam deflection is measured for mass-selected clusters.

The mass separation is necessary because the cluster beam always contains a mixture of clus-

Fig. 3: Scheme of a modern Stern-Gerlach deflection experiment. A cluster beam is generated in the variable temperature laser-vaporization source and deflected in the Stern-Gerlach magnet.

The deflections of size selected clusters are measured with a time of flight mass spectrometer [16].

ters of various sizes. The advantage of the time-of-flight mass-spectrometer is that it allows simultaneous determination of the mass and the position of deflected clusters. Moreover, a dis- tribution of clusters can be measured simultaneously because clusters of different masses are clearly separated by their times-of-flight whereas the magnetic deflections induce only small changes of the mass peaks [16]. Cluster deflections can be measured with a resolution of about 10µm.

Essentially the cluster source consists of a metal block with three perpendicular drill-holes. In the first hole a target rod is put forward and backward in a spiral motion⁵. An intense laser pulse is focused through the second hole and vaporizes a small amount of material from the target rod into a helium gas pulse which is synchronously injected into the third hole. This third hole is joined to a drift tube which at a nozzle opens into the source chamber. For reasons that will become clear below, it is necessary to carefully control the temperature of the drift tube and the nozzle.

5This prevents the target from being drilled by the laserablation.

Clusters are grown inside the drift tube by multiple atomic collisions ⁶. The helium supports the condensation and cooling⁷of the clusters and simultaneously serves as a transport medium.

A waiting room may be added to the drift region to provide a more effective thermalization.

At the end of the tube the cluster-helium mixture expands adiabatically into the vacuum vessel and a supersonic molecular beam is formed. During the expansion the clusters are further cooled because part of the random thermal motion is transformed to directed center-of-mass motion of the cluster-helium bunch. However, the clusters do not reach thermal equilibrium in the adiabatic expansion. We cannot, therefore, attribute a unique temperature to the particle bunch. Instead, translational, rotational, and vibrational temperatures (Ttrans, Trot, and Tvib) are defined by the population of the respective degrees of freedom. The expansion cooling is most effective for the translational and second for the rotational degrees of freedom, so that Ttrans < Trot < Tvib. The vibrational temperature is only little effected and is assumed to be close to the temperature of the nozzle. Parallel to the temperature the corresponding speed of sound as = (^γkT_m )^1/2 also decreases during the expansion process and correspondingly the Mach numberM = v/asincreases while the mean velocity of the clusters is only moderately enhanced. Here γ = Cp/Cv is the specific heat ratio and and v is the flow velocity (ca.

1200m/sfor a room temperature source [16]). By increasing the residence time in the source the cluster temperatures are closer to the source temperature when the clusters leave the nozzle.

Mild expansion conditions will then keep the vibrational temperature close to that of the source.

This is an important issue as we shall see below. An alternative cluster source with a low pressure (10 mbar) continuous He-flow provides better control of thermodynamic parameters and leads to milder expansion conditions but requires stronger vacuum pumps [17,18]. After the expansion the particle bunch is skimmed and a molecular beam is formed. The clusters then constitute an ensemble of mutually independent particles.

After passing the skimmer the molecular beam is collimated using 0.3 to 0.8 mm slits and enters the second part of the experiment with the inhomogeneous deflection field. The field gradient should be as homogeneous as possible across the beam cross section. Two type of deflection fields have been used [19-21]: In the ”two wire configuration” the field poles of the iron magnets reshape the magnetic equipotential lines that would be created by two strong currents of equal magnitude flowing in opposite directions through two parallel wires. The maximum separation of the pole pieces is smaller than 3 mm. The other field option uses one hyperbolic and one rectangular pole piece to model one quarter of a quadrupole magnetic field.

Behind the deflection field the clusters traverse the drift region. Finally, in the mass spectrometer they are ionized by an ArF-excimer laser (λ = 193nm, hν = 6.42eV) and mass separated.

Simultaneously the total deflection of a specific cluster is converted to a small difference in the time-of-flight. Alternatively the beam of the ionizing laser can be scanned back and forth, thereby mapping out the profile of the deflected cluster beam.

Now, consider an atomic beam of open shell atoms with magnetic momentµ. Inside the magnet the magnetic moment of an atom interacts with the field and possesses the potential energy (Zeeman energy)Eint = −µ·B. The inhomogeneous Field B = B(z)ez therefore exerts a force F = µzdB

dze_z which - depending on the sign of µz - deflects the beam toward the high or low field direction. For an atomic beam this leads to a symmetric and quantized deflection pattern as is observed in classical Stern-Gerlach experiments.

6Note, that a three body collision is needed to form a stable dimer.

7Heat transport to the tube wall.

Fig. 4: Schemes of the two type of Deflection magnets in use. Left: ”two wire magnet”. The location of the hypothetical two wires is at the intersection of the two circles. Right: quadrupole sector magnet. The numbers indicate the relative strength of the field gradient [21].

The deflection inside the magnet is given by d = 1

2at² = 1 2

F

mt² = µz

2m dB

dz (l

v)² = µz

2mv² dB

dz l² = µz

4Ekin

dB

dzl², (9) whereµ_z is thez-component of the magnetic moment.E_kinis the particle’s kinetic energy, and lis the length of the magnet poles. After passing the deflection field the particles enter a drift region of length D, so that the total deflection will become:

dtot = µ_z mv²

dB dz(1

2l²+l·D) = µ_z 2Ekin

dB dz (1

2l²+l·D) = C µ_z Ekin

dB

dz, (10) whereC is simply an instrumental constant.

In their original experiment Stern and Gerlach [22] measured the deflection of a beam of silver atoms to proof Bohr’s prediction of space quantization of atomic angular moments in a magnetic field. Fig. 5 reproduces a post card sent to Bohr by Gerlach where he congratulates him for the correct prediction. This referred to the old quantum theory. The existence of a spin was not even postulated at the time of the experiment. For an amusing historical account see the article by Friedrich and Hershbach[23]. Today, we interpret the experiment not only as a proof of space quantization but also as a proof of up and down spin, since the5s- electron of a silver atom in its ground state has zero orbital angular momentum. Accordingly, for an atom with total angular momentumJ we expect the splitting into2J+ 1peaks in a Stern-Gerlach experiment, provided the resolution is sufficient to separate the peaks. In this case one could simply determine the magnetic moment through the number of splittings.

Thus, for a cluster with total angular momentumJ one is tempted to also expect a separation into 2J + 1 peaks. However, that is not what is measured. Instead one observes a single sided deflection into one single peak for almost all the clusters studied today. The clusters are exclusively deflected toward the high field region [16,24]. Obviously the cluster moments are able to relax toward the direction of lowest energy which corresponds to alignment of the magnetic moments with the applied field, thereby causing a force in the direction of increasing

Fig. 5: Postcard by Walther Gerlach displaying the results of the historic Stern-Gerlach ex- periment. It was sent to Nils Bohr to congratulate him for the correct prediction of space quantization. Note the length scale 1.0 mm [23].

field strength. The relaxation occurs in a time much shorter than the transit time through the Stern-Gerlach magnet which is in the 10µs regime. This surprising result is due to internal thermal relaxation of the clusters and has been interpreted by the superparamagnetism of the clusters [25-27]. It is assumed that the clusters are single domain and the atomic moments are ferromagnetically aligned. It is further assumed that the thermal energy of the cluster is much larger than the magnetic anisotropy energy. In fact, the superparamagnetic clusters fluctuate on aps time scale which is short compared to the passage time through a Stern-Gerlach magnet.

The reorientation of the magnetic moment is accompanied by a change of angular momentum which has to be compensated by a change of the rotational angular momentum of the cluster. In contrast to single atoms, the clusters are able to transfer angular momentum to cluster rotation.

For the atoms the conservation of the angular momentum prevents any relaxation. But the cluster in addition to its electronic angular momentum has an angular momentum due to its own rotation. These angular momenta are coupled by spin-orbit effects. The coupling is responsible for the relaxation of the cluster-magnetic moment. The total energy including the Zeeman-term and the angular momentum squaredJ² (electronic + rotational) andJ_z have to be conserved in the process. The mechanisms of this spin rotation coupling are fairly complicated and have been addressed by several authors [25-37]. According to the proposed model of superparamagnetic relaxation the effective magnetic moment measured in the deflection experiment is given by (see section 2):

µef f =< µz >=nµnL(nµ_nB

kBT ) =nµn[cothnµ_nB

kBT − k_BT

nµnB] (11) where µn is the magnetic moment per atom andn is the number of atoms of the cluster, i.e.

the total magnetic moment of the cluster is µ = nµn. In the limit nµnB ≪ kBT, i.e. for small fields or high temperatures we can once more make use of the expansion of the Langevin

function (see section 2) and get the Curie law in the form:

µef f/n =< µz > /n ≈n µ²_nB

3kBT. (12)

These two equations are usually used to derive the reported magnetic moments per atom µ_n from the measured Stern-Gerlach deflections.

According to section 2 the Langevin model of paramagnetism requires that the blocking tem- perature is exceeded and that the cluster has contact to a heat bath at temperatureT. However, in the beam the clusters are completely isolated from their environment. Thus, here each cluster is its own heat bath. Therefore, the vibrational temperature Tvib is taken as the temperature of the heat bath. This assumption together with the difficulties to precisely determine T_vib is presumably the biggest source of uncertainty in the whole procedure. Here it becomes clear why the nozzle temperature and the expansion conditions have to be so carefully adjusted in the experiments. The above assumptions might be sufficient for larger clusters which have a large number of vibrational degrees of freedom, are sufficiently hot (heat bath), and have large angular momentaJ (respective large magnetic moments) but might lead to problems when the clusters become small. Knickelbein [41] has recently addressed this point and shown that for a total spin S = 4one might overestimate the magnetic moment by ≈ 10% and forS = 10 by ≈ 4%. On the other hand, in accordance with the model it is found that for small fields µ_{ef f} increases linearly with the applied field and approaches a saturation value for larger fields.

Moreover, for cobalt clusters the measured moment per atom increases linearly with the factor nB/T which is in complete agreement with equation (12)[42].

Thus, using the superparamagnetic model a variety of transition metal clusters have been inves- tigated, namely: F e, Co, N i [38-40,42-47],Rh, Ru, P d [48,49], V, N b [50], Cr [51], M n [52,53] , and mixedBinCom[54] clusters.

Gd[21,42,43,55,56] , T b [57] ,Dy [58], have to be treated differently. For these clusters the superparamagnetic model may break down (see below).

Fe-, Co-, and Ni-clusters were the first transition metal clusters to be studied. Fig. 6(left) shows the magnetic moments as a function of cluster size. With some oscillations the magnetic mo- ments/atom decrease from the atomic toward the bulk value (2.2µB, 1.65µB and 0.6 µB for iron cobalt and nickel, respectively) which is approached at cluster sizes of about 600 atoms.

The values for the smallest clusters are close to the maximum spin values predicted by the first Hund’s rule, i.e. 3µB, 2µB and 1µB with 7, 8, and 9 d-electrons for iron, cobalt, and nickel, respectively. Note, that the observation of this evolution from atom like toward the bulk like values for larger clusters also supports the validity of the superparamagnetic model. A mag- netic shell model has been proposed to explain the contribution of the various geometric shells surrounding the central atom of highly symmetric clusters predicting RKKY-like oscillations toward the interior. Surface atoms usually make a larger contribution due to their lower coordi- nation and larger localization giving the surface atoms a more atomic character than the interior (bulk) atoms [39,59]. More recent measurements of nickel clusters show more fine structure [44,46](see also Fig. 10).

Up to now we have considered all the spins to be exchange coupled and completely aligned.

However spin fluctuations at higher temperatures reduce the spin alignment and with it the magnetic moment. Fig. 6(right) compares for several cluster sizes the decrease of the mag- netic moments with increasing temperature with that of the corresponding bulk material (dashed curves). Bulk materials lose their ferromagnetic order at the Curie temperature where thermal fluctuations destroy the magnetic order. For Ni and Co clusters the magnetization converges

Fig. 6: Magnetic moments of Fe-,Co-, and Ni-clusters as a function of cluster size (left) and as a function of temperature (right). The magnetic moments of the clusters exceed that of the bulk materials. This is due to the lower coordination of the surface atoms which favors magnetic order and incomplete quenching of orbital contributions. The oscillations are presumably due to structural effects. The temperature dependences of Ni- and Co- clusters converge to the bulk magnetization curves with increasing cluster size. The strong deviations for the iron clusters are presumably associated with structural changes.The curves H1 and H2 are simulations in terms of a Heisenberg model [38-40].

toward that of the bulk. However magnetic moments of the clusters also persist above the bulk Curie temperature. This is also confirmed by theory [60]. The Fe clusters show a strange behav- ior which might occur due to structural differences with the bulk material. These temperature dependent measurements also tell us that the temperature must be properly chosen if one wants to measure the maximal spin coupling. On the one hand the temperature must be sufficiently high that the superparamagnetic state is realized with certainty and on the other it must be low enough not to lead to spin uncoupling.

Cluster magnetism is not limited to clusters of magnetic solids. An example is rhodium. The bulk material is known to be a Pauli paramagnet at all temperatures. In contrast, small rhodium clusters in the size rangeRh9 toRh60were found to show magnetic order with magnetic mo-

ments/atom up to0.8µBforRh9 andRh10[48,49].

Superparamagnetism does not require that all local moments are parallel aligned, there may as well be some local antiferromagnetic coupling or spin canting be present in the clusters. It is only required that the effective moment behaves superparamagnetically. Indeed, this seems to be the case in chromium clusters where several magnetic isomers have been observed simulta- neously in the deflection profile [51].

Fig. 7: Magnetic Moments per atom for manganese clusters [52,53].

Fig. 8: Comparison of magnetic moments of hydrogen-saturated iron clusters with those of the unreacted clusters [47].

Fig. 7 shows superparamagnetic moments for manganese clusters. Bulk (α- phase) manganese is an antiferromagnet at temperatures below its N´eel temperature of 95 K and is a simple Pauli

paramagnet above 95 K - in either case resulting in only slight magnetism for the bulk metal. In contrast, the manganese clusters fromM n12 toM n99exhibit large magnetic moments charac- teristic of ferromagnetic spin ordering or ferrimagnetic spin ordering (more spins pointing one way than in the other). It shows distinct minima atn=13, 19,≈35, 57 and maxima atn=12, 15,

≈24, and≈55. The minima at 13 and 19 suggest an icosahedral growth sequence. The 13 atom icosahedron and the 19 atom double icosahedron are geometrically closed shell and therefore highly coordinated species. This higher coordination may result in relatively smaller magnetic moments. Fig. 7 shows magnetic moments per atom for M nn clusters from n = 11−99.

Later the experiments could be extended also to clusters withn= 5−10. All the clusters with n ≥ 7showed the well known single sided deflection when the magnetic field is applied. In contrast,M n5 andM n6only show a symmetric broadening of the peak with no net deflection.

This could be an indication of free-spin behavior where an angular momentumJ is split into 2J + 1unresolved beamlets or of locked moment behavior, provided thatµB ≪ kBTrot [53].

Knickelbein prefers to interpret his data the latter way. Based on density functional calculations Jones et al. [61] have recently proposed a third analysis. They believe that the exchange cou- pling between the different atoms in the cluster is so weak that at the experimental temperatures their spins behave like individual atomic spins when the clusters enter the deflection field. Due to their initial random orientation this results in a pure broadening of the beam without any net deflection. It is interesting to recognize that certain adsorbates can significantly increase the magnetic moments of clusters. Fig. 8 compares the magnetic moments of hydrogen-saturated iron clusters,F enHm, with those of the corresponding bare clusters. In contrast to iron clusters, the magnetic moments of nickel clusters are completely quenched when they are saturated with hydrogen.

So far we have mainly discussed clusters that show a single-sided deflection in a Stern-Gerlach experiment and can be interpreted in terms of superparamagnetism at the experimental tem- peratures. The situation is different for rare earth clusters [21,42,43,55-58]. Rare earth atoms and bulk materials are known to exhibit large spin-orbit interaction which is a prerequisite for a large magnetic anisotropy energy. Therefore, rare earth clusters are promising candidates to observe locked moment behavior. In fact, for gadolinium clusters both superparamagnetic and locked moment behavior has been found. For the locked moment clusters the beam spreads asymmetrically around the zero deflection as is seen forGd21in fig. 9 [55]. The asymmetry is due to spin - rotation coupling and arises when the Zeeman energyµB exceeds the rotational energykBTrot[26,27,53,55]. In fact, the rotational temperature of the clusters can be deduced from the deflection curves. Superparamagnetic gadolinium clusters have internal magnetic mo- ments per atom that are substantially less than the bulk value of 7.63µB per atom. This might be due to antiferromagnetic or canted spin structures of the respective clusters [62]. Some ex- perimental discrepancies between the results of the Virginia and the Lausanne group [56] may be due to different isomers present in the two cluster beam experiments [63]. Like gadolinium terbium clusters show superparamagnetic as well as locked moment behavior. Interestingly, T b22Oshows a locked moment up to temperatures of 250 K [57].

So far, we have focused on experimental aspects of cluster magnetism. It is clear, however, that the experimental efforts always have to be accompanied by calculations of cluster structures and their magnetic properties. In fact, many properties reported in the experiments have earlier been predicted by theory. While ab initio methods can only be used for the smaller clusters, the tight-binding approach is usually utilized for the bigger ones. The theoretical developments have been nicely reviewed by Alonso [64] and I want to refer the reader to his article and the references cited therein. I only want to exemplarily show one recent result for nickel clusters

Fig. 9: Measured deflection profiles ofGd21clusters as a function of the applied magnetic field (dots). The solid lines are calculated profiles for locked moment clusters in terms of a spin- rotation coupling model for a body fixed magnetic moment. The small peak near zero deflection is due to a small superparamagnetic contribution [55] .

where the data of two different experiments are compared with a tight-binding calculation (Fig.

10 [70]). Although there are still differences in the finer details within the experiments as well as with the theory the general trends and also most of the oscillations are nicely reproduced. The authors conclude that the orbital moment is responsible for the enhancement of the magnetic moments with respect to the bulk as well as for the oscillations of the magnetic moments.

Another aspect that has been omitted here is cluster beam spectroscopy, and I simply want to refer the reader to some references where he might find further hints [65-69].

Wan et al.

Apsel et al.

Knickelbein

Fig. 10: Comparison of calculated magnetic moments (black squares) with experimental data for small nickel clusters [70].

4 Clusters on Surfaces

For potential applications in nanoelectronic and data-storage devices one has to accumulate the clusters on a surface [71,72]. This can be done in two ways. One can grow the clusters on a sur- face by statistical agglomeration of atoms. This of course produces a certain size distribution of clusters so that the detailed information on individual particle sizes cannot be gained. The clus- ter size can be controlled by carefully choosing the deposition rate and the temperature of the substrate. The alternative method is to deposit preformed and possibly mass-selected clusters.

Both methods rely on ultrahigh vacuum (UHV) and cryophysical techniques. (UHV) conditions are needed to avoid adsorption of residual gas molecules on the clusters. Low temperatures are required to hinder the clusters from diffusion and further aggregation.

Atoms and clusters on surfaces tend to stick on dislocations and other surface defects. This property has been utilized for more than 40 years by crystallographers to decorate and visualize surface inhomogeneities and dislocations and may be exploited to generate quite regular arrays of clusters on surfaces. Free surfaces may reconstruct to minimize their free energy thereby producing periodic patterns on a nanometer scale. Such reconstructions may also artificially be stimulated by hetero-epitaxial growth of a few monolayers on a free surface. The strain relaxation induced by the lattice mismatch between the substrate and the film may then also lead to periodic patterns [2,71]. Such structures can be decorated with clusters forming peri- odic arrays [74]. Fig 11 shows as an example an array of self assembled cobalt clusters on a herringbone reconstruction of a Au(111) surface [75]. Such arrays of particles are believed to have great potential for applications in spin electronics, for future data storage, and as sensors.

In fact, using the magnetooptic Kerr effect [76] with somewhat larger particles corresponding

Fig. 11: STM image (100×100nm) of self-assembled Co clusters, approximately 200 atoms each, on a zigzag reconstruction of the Au(111) surface[75].

to a coverage of 1.7 monolayers (1 ML =1.5·10¹⁵atoms/cm²) Chado et al. [75] could verify an out of plane magnetization when the particles were grown at room temperature. For smaller particles grown at 30 K they found an in plane magnetization. For potential applications and ex situ measurements the cobalt clusters can be covered with a protective gold film of a few monolayers thickness. When clusters are grown on surfaces by atomic aggregation or deposited from the gas phase some surface specific experimental techniques become available that are not available for gas phase clusters. The magnetooptic Kerr effect was already mentioned. Others are scanning tunneling microscopy (STM) and spectroscopy (STS), X-ray absorption (XAS) and X-ray photoemission (XPS) ⁸ and X-ray magnetic circular dichroism (XMCD) [5]. The latter technique is particularly useful for deposited magnetic clusters because it allows to mea- sure the magnetization and to distinguish between spin and orbital contributions to the magnetic moment. XMCD is the difference in the absorption coefficients for circularly polarized X-rays, when the magnetization of the sample is aligned parallel and antiparallel to the helicity of the incident photons [5]. Because it uses resonant optical transitions XMCD is very selective to the element. Therefore it is ideally suited to study small amounts of deposited magnetic materials down to coverages of3·10¹²atoms cm⁻². By means of dipole sum rules [77-79] one can sep- arately determine spin and orbital contributions to the magnetic moment for a given element.

Moreover, the technique allows one to identify the magnetization direction and strength.

Gambardella et al. [80] have recently studied the evolution of the magnetization, the magnetic anisotropy energy, and the spin and orbital contributions to the magnetic moment for cobalt atoms and small clusters on a clean Pt(111) surface. They used the XMCD technique at the Co L2,3 edges (2p to 3d transitions) using left and right circularly polarized synchrotron light in the total electron yield mode. Particles of various sizes were produced by molecular beam epitaxy, statistical growth on the surface, and for larger particles by diffusion controlled aggregation at different temperatures. Isolated atoms were obtained at coverages of less then 0.03 monolayers.

The particle size distribution was determined by scanning tunneling microscopy (STM). Fields

8the cluster density in the gas phase is to small

Fig. 12: Left: (A) XMCD spectra as a function of the average particle size n. The spectra¯ are normalized to theL2 intensity to show the decrease of theL3 contribution with increasing particle size. (B) magnetization atT = 10K of clusters withn¯=4 atθ0 = 0^o (black) and70^o (gray). Solid lines are fits of a uniaxial anisotropy model. (C) The same forn¯ = 8clusters.

Right: (A) orbital contribution to the magnetic momentsLas a function of average cluster size measured along the easy magnetization direction θ0 = 0^o. Inset: Anisotropy constant as a function ofL. (B) Anisotropy constant as a function ofn. The dashed and dashed-dotted lines¯ show for comparison the values of theL10CoP talloy and hcp-cobalt, respectively [80].

up to 7T were used to magnetize the sample at anglesθ= 0^oand70^owith respect to the surface normal. The easy axis is parallel to the surface normal. Fig. 12 shows the XMCD spectra as a function of the average particle size. The intensity of theL3peak determines the magnetization curves as function of the applied field. The magnetization is fit to a uniaxial anisotropy model, from which the anisotropy constant K ( here referred to the atom, not to the volume) can be deduced. The XMCD-sum rules are used to discriminate the spin and orbital contributions to the magnetic moment. As can be seen in Fig. 12 (right) the anisotropy constant and the orbital contribution to the magnetic moment are very high for the atom and decrease very rapidly with increasing particle size. The large atomic values are explained by low coordination of the isolated atom on top of the flat surface, which favors d-electron localization and therefore the survival of the atomic like character of the d-orbitals. The large anisotropy is explained by the reduced symmetry of the Co adatoms at the surface and a contribution of the platinum 5d- states which exhibit a large spin-orbit coupling. These results confirm theoretical predictions and are of fundamental value to understand how the magnetic anisotropy develops in finite-size magnetic particles.

The particles discussed so far have been grown by surface aggregation and therefore exhibit a certain size distribution. However, for very small clusters and for direct comparison with gas phase data it is desirable to have size-selected clusters on the surface. To achieve this, Lau et al. have performed an experiment where they deposited small mass-separated iron clusters on an ultrathin nickel film on a Cu(100) surface [81-85]. The nickel film had a thickness between

Fig. 13: XMCD in the 2p-3d x-ray absorption of size selected F e7 clusters on a Ni/Cu(001) substrate at a coverage of 0.03 monolayers [81,82].

20 and 40 monolayers. Such films have a perpendicular magnetic anisotropy and can be mag- netized to remanence perpendicular to the film surface. By exchange coupling to the film the magnetic moments of the iron clusters are then also aligned perpendicular to the surface. An additional magnetic field is not needed.

The clusters were produced with a high energy sputter source (28 keV Xe⁺ ions). Positively charged iron clusters were accelerated to 500 eV kinetic energy and mass-selected in a magnetic sector field. Prior to deposition the selected clusters were decelerated to energies of less than 1 - 2eV per atom. To avoid fragmentation of the clusters upon surface impact a ”soft landing”

technique was used in addition [86-89]. For this purpose the nickel film is covered by≈ 20 layers of protective argon-buffer gas at 20K into which the clusters are deposited. The argon layers are afterwards desorbed by heating the sample to 100 K leaving behind the iron clusters on the Ni/Cu(100) surface. To suppress cluster diffusion and coagulation the sample was then cooled down to below 20K again. In addition, the coverage was kept below 0.03 monolayers.

Note, even if the clusters survive this procedure it does not mean that they are identical to those in the gas phase or among each other. They are still subject to the interaction with the surface of the film, may structurally relax and may very well form various isomers at different trapping sites. Thus, the magnetic properties of deposited clusters are governed by the interplay of cluster-specific properties on the one hand and cluster-substrate interactions on the other. Size dependent variations of magnetic moments may be modified upon contact with the substrate.

The so prepared samples were studied by XMCD at the 2p to 3d Fe-core level absorption edges.

Fig. 13 shows the XMCD spectrum forF e7 clusters on Ni/Cu(100). By comparison with the XMCD Signal of the Ni substrate it is found that all clusters are coupled ferromagnetically to the underlayer. Again with the aid of sum rules, orbital and spin magnetic moments have been extracted from the spectra. The spin and orbital magnetic moments per unoccupied 3d state of the deposited iron clusters are shown in figure 14. Both, orbital and spin magnetic moments are significantly enhanced in small clusters as compared to bulk iron. These observations are

Fig. 14: Normalized spin and orbital magnetic moment per unoccupied 3d state as a function of cluster size forF enclusters on Ni/Cu(001)[81,82]. For comparison the bulk values are 0.65 per unoccupied 3d state for the spin moment and 0.04 for the orbital moment which in the bulk is quenched by the crystal field.

in accordance with calculated magnetic moments for small iron clusters on Ag(100) which also predict enhanced magnetic moments [90].

5 Co Monatomic Chains on Pt(997)

For vicinal stepped surfaces, if the separation of steps becomes smaller than the mean free path of atoms on the surface, the atoms will gather on the surface steps before they can coagulate to form bigger clusters. In such cases quite regular arrays of monoatomic chains may be formed on the surface [91]. This occurs due to the increased binding energy at the step site. Fig. 15 shows

Fig. 15: STM image showing monatomic Co wires at the step edges of a vicinal Pt(997) surface (The vertical scale has been streched). The terrace width is 2.02nm which allows an atomic wire density up to5·10⁶ cm⁻¹.

as an example the formation of monatomic Co chains on a carefully prepared Pt(997) vicinal surface. To achieve this, well defined thermal growth conditions have to be adjusted. Fig. 16 il- lustrates the temperature dependence of various growth modes on a stepped surface. Below 250 K the formation of monatomic wires is limited by the too slow edge diffusion and at high tem- peratures by interlayer diffusion and eventually by alloying with the substrate. The temperature window for optimum growth of monatomic chains is between 250 and 300 K. Monatomic wires are obtained at coverages of 0.13 monolayer (ML) for Pt(997) (1ML = 1.5·10¹⁵ atoms/cm⁻²).

Within the wires Co-Co distances are expanded by about 10% with respect to bulk hcp Co. An exchange splitting of≈2.1 eV was observed in angular resolved photoemission spectra of the Co monatomic nanowires clearly exceeding the one of Co thin films (1.4-1.9 eV) and bulk Co≈ 1.4 eV, thus suggesting an enhanced magnetic moment of≈2.1µ_B[92]. The exchange splitting decreases with increasing Co atomic coordination. This is due to the 3d band broadening with larger coordination and is seen in the photoemission spectra at coverages larger than the cover- age of 0.13 ML for the ideal monatomic chains. In contrast to Co, Cu chains on Pt(997) only show a single 3d feature at 2.3 eV binding energy in their photoemission spectra and therefore lack any evidence of magnetic moments.

As for the deposited small clusters the magnetic properties of the nanowires can be measured by soft-x-ray magnetic circular dichroism (XMCD) in the energy range of the2p→3dtransitions (photon energy ca. 800 eV). The amplitude of the dichroic signal measures the strength of the magnetization. The background due to the platinum substrate does not contribute to the dichroic

Fig. 16: (a) Different growth modes on a vicinal stepped surface. (b) temperature dependence of the various growth modes for Co on Pt(997). The formation of monatomic chains is limited by slow edge diffusion at low temperatures and by interlayer diffusion at high temperatures [91]. Similar schemes applay to the growth of Ag and Cu wires.

signal. The easy axis of the magnetization is found to be in the plane perpendicular to the axes of the nanowires tilted by 43^owith respect to the Pt(111) surface normal. The inset of Fig. 17 shows a sketch of this geometry. The [111] surface normal is indicated by the small dotted line.

Using the orbital sum rule, the orbital magnetic moment can be derived from the integrated

XMCD signal: Z

L3+L2

(µ++µ⁻)dε = c 2µB

µL,

whereεis the photon energy and c is an experimental constant that is derived from the known bulk valueµ_L = 0.15µ_B/atomand the bulk integrated XMCD signal. For the nanowires one gets µL = 0.68±0.05µB/atom a value that exceeds that of the bulk by more than a factor of 4. Larger values are only found for adsorbed isolated atoms and small clusters [see above;

93,94]. Fig. 17 displays the magnetization of the nanowire array at T = 45 K as a function of the applied field B with the magnetization oriented parallel to the easy axis (43^o) and with the magnetization tilted by an angle of 80^o to this (-57^o). A strong magnetic anisotropy is obvious.

The missing remanence indicates one- dimensional superparamagnetism. Fits of the magneti- zation to a classical model [95] indicate that the spins ofN = 15±1Co atoms are exchange coupled and that an anisotropy energy Ea = 2.0± 0.2meV /atom corresponds to each spin block. Lowering the sample temperature below the blocking temperature to 10 K induces a

Fig. 17: Magnetization curves of an array of monatomic Co wires on Pt(997) as a function of the applied field B. The data points represent the strength of the CoL3XMCD signal at 779 eV atT = 45K (above)(a) and at T = 10K (below the blocking temperature) (b), respectively, at angles43^o (filled squares; easy axis) and −57^o (open circles) with respect to the Pt[111]

direction (see inset)[91].

ferromagnetically ordered state with a finite remanence. The change in free energy at a temper- ature T due to excitations of an Ising chain was estimated to be∆G = 2J −kBT ln(N −1).

ForN → ∞∆Ggets negativ at any finite temperature. Then the ferromagnetic state becomes unstable against thermal fluctuations, however, for(N −1) < e^2J/kBT the ferromagnetic state gets stabilized.

6 Conclusion

Small transition metal clusters usually exhibit magnetic moments/atom which are intermediate between the atom and the bulk. While in the bulk the orbital contribution to the magnetic moment is almost completely quenched, there might be a significant contribution in the small clusters. Due to their size the particles are single domain and mostly superparamagnetic, at least at room temperature. Due to their large surface to volume ratio and their specific and selective reactivity they may play a significant role in catalysis [96].

To be useful for data storage one has to overcome superparamagnetism. Since the anisotropy energy is increasing with the particle volume one can try to increase the volume in the vertical direction without loosing horizontal resolution. Thus, one can generate vertical nanowire arrays [97,98] or pile up vertical pillars [99]. Another option is to take advantage of exchange bias [100].

Appendix:

Stoner-Wohlfarth Model of a Single Domain Particle [101]

Consider a monodomain particle with uniaxial anisotropy atT = 0.Assume the magnetic mo- mentµ=MsV is aligned parallel to the easy axis atθ= 0. Msis the saturation magnetization andV the volume of the particle. Apply a magnetic field at an angleφwith respect to the easy axis. The field will pull the magnetic moment toward the field direction by an angleθ(see figure 18). Then the total energy of the particle is the sum of the anisotropy energyEa =KV sin²θ and the Zeeman energy

−µ·B=−MsV Bcos(φ−θ). (13) E(φ, θ) =KV sin²θ−M_sV B cos(φ−θ) (14) Like in the N´eel-Brown model (section 2) it is assumed that all magnetic moments in the particle are aligned at all times and will rotate coherently. Now, put the field to φ = 180^o trying to reverse the particle’s magnetic moment. We then get:

E(180^o, θ) = KV sin²θ−MsV B(−cos θ).

At zero magnetic field we have two equivalent minima E_min = 0 at θ = 0 and θ = 180^o separated by the anisotropy barrierKV atθ= 90^o.With the field applied the situation changes.

To find the new extrema we setdE/dθ = 0and get

dE/dθ = (2Kcos θ−MsB)V sin θ = 0.

Again we find two minima at θ = 0 and θ = 180^o (sin θ = 0), but the potential depth has changed toE(180^o,0) = MsV B andE(180^o,180^o) = −MsV B,respectively. The maximum has shifted to a smaller angle defined bycos θmax =MsB/2Kand its energy isE(180^o, θmax) = KV (1−cos²θmax)−MsV B(−cos θmax) = KV(1 + (MSB/2K)²).The barrier height that hinders the magnetic moment from switching is given by:E_B⁺ =E(180^o, θmax)−E(180^o,0) = KV(1 + (MSB/2K)²)−MsV B = KV(1−MsB/2K)² = KV(1−B/B0)², whereB0 = 2K/Msis called the ”anisotropy field”⁹. At this field the barrier disappears and the angleθmax

gets zero, that is the first minimum and the maximum coincide. The second minimum gets

−2KV. At any higher field the magnetization switches spontaneously, even at T = 0. The barrier height for the reverse direction is: E_B⁻ = E(180^o, θmax)−E(180^o,180^o) = KV(1 + (MSB/2K)²) +MsV B = KV(1 +MsB/2K)² = KV(1 +B/B0)². Thus the effect of an applied weak field is essentially a reduction of the height of the switching barrier for the an- tiparallel orientation and an increase of the barrier height for the parallel orientation.

Thermal agitation as described by the the N´eel-Brown model allows jumps across the barrier from both sides. So we have to consider two switching rates τ±⁻¹ = τ0⁻¹exp(−^E

± B

kBT). The switching probability is then given by p^± = exp(−t/τ^±). Thermal equilibrium is reached, when, on a statistical basis, the net forward and reverse ”flow” across the barrier is zero. The time evolution into that state is governed by an effective rateτ_{ef f}⁻¹ =τ+⁻¹ −τ−⁻¹ which goes to zero when thermal equilibrium is reached.

9The exponent 2 is only valid for direct reversalφ= 180^o. In general it will depend on the angleφand is often approximated byg(φ) = 0.86 + 1.14F(φ), whereF(φ) = (cos^2/3φ+sin^2/3φ)^−3/2[14,102-106].

Fig. 18: Magnetic potential energy of an isolated Stoner-Wohlfarth particle as described in the text [14].

Data storage of course requires operation far from equilibrium . In that case the reverse reaction rate (transition from the global minimum) is ignored when storage times are calculated.[14].

Wernsdorfer et al. [107-109] have recently confirmed the magnetization reversal by uniform rotation on small nanoparticles and a nickel nanowire using a micro SQUID (superconducting quantum interference device) technique at temperatures between 0.2 and 6 K [see also the re- cent review: 110]. Indications of quantum tunneling which plays an important role in cluster compounds - so called single molecule magnets - were also reported for very low temperatures [108].

References

[1] Clusters of Atoms and Molecules I, Ed.: H. Haberland, Series in Chemical Physics, Vol.

52, Springer, 1994.

[2] For a recent review see: J.P. Bucher, In: S.N.Khanna, A.W. Castleman(Eds.), Quantum Phenomena in Clusters and Nanostructures, Springer Verlag, Berlin, Heidelberg (2003) [3] see the contribution by R.Zeller

[4] see the contribution by M. Farle [5] see the contribution by U.Hillebrecht

[6] P. Crespo, R. Litr´an, T. C. Rojas, M. Multigner, J. M. de la Fuente, J. C. S´anchez-L´opez, M. A. Garc´ıa, A. Hernando, S. Penad´es, and A. Fern´andez, Phys. Rev. Lett. 93, 087204 (2004)

[7] For a recent review, see: J.L. Dormann, D.Fiorani, E. Tronc, Adv. in Chem. Phys. 98, I.

Prigogine, S. A. Rice (eds.), J. Wiley & Sons, Inc. (1997), p.283 ff.

[8] R.H. Kodama, J. Magn. Magn. Mater. 200, 359 (1999)

[9] S. Majetich, in: Advances in Metal and Semiconductor Clusters, Vol. 4, 35 JAI Press Inc.

(1998) ISBN: 0-7623-0058-2

[10] C.P. Bean, J.D. Livingston, J. Appl. Phys. 30, 120S (1959)

[11] I.S. Jacobs, C. P. Bean, in:G.T. Rado, H. Suhl, Magnetism, Vol. III, Academic Press, New York (1963), p. 271

[12] L. N´eel, Ann. Geophys. 5, 99 (1949)

[13] W.F. Brown, J. Appl. Phys. 34, 1319 (1963)

[14] D. Weller, A. Moser, IEEE Trans. Magn., 35, 4423 (1999)

[15] For a derivation see: Frederick Reif, Statistische Physik und Theorie der W¨arme, Walter de Gruyter, Berlin, New York (1987), p. 302 ff; English version: Fundamentals of statistical and thermal physics, McGraw Hill, Singapore (1985)

[16] W.A. de Heer, P. Milani, Rev. Sci. Instr. 62, 670 (1991)

[17] P.S. Bechthold, E.K. Parks, B.H. Weiller, L.G. Pobo, and S.J. Riley, Z. Phys. Chemie N.F., 169, 101 (1990)

[18] M. B. Knickelbein, Phys.Rev. Lett. 86, 5255 (2001).

[19] D. McColm, Rev. Sci. Instrum. 37, 1115 (1966);

[20] F. Ramsey, Molecular Beams (Oxford University Press, Oxford (1956)

[21] L. A. Bloomfield, J. P. Bucher, and D.C. Douglas , in: On Clusters and Clustering, P.J.Reynolds (ed.) Elsevier Science Publishers, 193 (1993).

[22] W. Gerlach and O. Stern, Z. Phys. 8, 110 (1992); ibid. 9, 349 (1922).

[23] B. Friedrich and D. Herschbach, ”Stern and Gerlach: How a bad cigar helped reorient atomic physics,” Physics Today 56,(December) 53 (2003).

[24] W. A. de Heer, P. Milani, and A. Chtelain, Phys. Rev. Lett. 65, 488 (1990).

[25] S.N. Khanna and S. Linderoth, Phys. Rev. Lett. 67, 742 (1991)

[26] P.J. Jensen, K.H. Bennemann, Ber. Bunsenges. phys. Chem. 96, 1233 (1992) [27] A. Maiti, L.M. Falicov, Phys. Rev. B 48, 13596 (1993)

[28] P. Jensen and K.H. Benneman, Z. Phys. D 26, 246 (1993); ibid. D 29, 67 (1994) [29] P. Milani, Z. Phys. D 28, 163 (1993)

[30] J.A. Becker, Z. Phys. D 29, 299 (1994)

[31] G. F. Bertsch and K. Yabana, Phys. Rev. A 49, 1930 (1994).

[32] G. Bertsch, N. Onishi, and K. Yabana, Z. Phys. D 34, 213 (1995).

[33] V. Visuthikraisee and G. Bertsch, Phys. Rev. A 54, 5104 (1996).

[34] G. Bertsch, N. Onishi, and K. Yabana, Surf. Rev. Lett. 3, 435 (1996).

[35] N. Hamamoto, N. Onishi, and G. Bertsch, Phys. Rev. B 61, 1336 (2000).

[36] S. Dattagupta, and S.D. Mahanti, Phys. Rev. B 57, 10244 (1998).

[37] Z.Y. Liu, P.A. Dowben, A.P.Popov, D.P. Pappas,Phys. Rev. A 67, 033202 (2003).

[38] I.M.L. Billas, J.A. Becker, A. Chˆatelain, W.A. de Heer, Phys. Rev. Lett. 71, 4067 (1993).

[39] I. M. L. Billas, A. Chˆatelain, and W. A. de Heer, Science 265, 1682 (1994).

[40] I. M. L. Billas, A. Chˆatelain, and W. A. de Heer, J. Magn. Magn. Mater. 2168, 64 (1997) [41] M.B. Knickelbein, J. Chem. Phys. 121, 5281 (2004)

[42] J.P. Bucher and L.A. Bloomfield, Int. J. Mod. Phys. B 7, 1079 (1993)

[43] D. C. Douglass, A. J. Cox, J. P. Bucher, and L. A. Bloomfield, Phys. Rev. B 47, 12 874 (1993)

[44] S. E. Apsel, J. W. Emmert, J. Deng, and L. A. Bloomfield, Phys. Rev. Lett. 76, 1441 (1996).

[45] M.B. Knickelbein, J. Chem. Phys. 115, 1983 (2001) [46] M.B. Knickelbein, J. Chem. Phys. 116, 9703 (2002) [47] M.B. Knickelbein, Chem. Phys. Lett. 353, 221 (2002)

[48] A. J. Cox, J. G. Louderback, and L. A. Bloomfield, Phys. Rev. Lett. 71, 923 (1993).

[49] A. J. Cox, J. G. Louderback, S. E. Apsel, and L. A. Bloomfield, Phys. Rev. B 49, 12 295 (1994)

[50] R. Moro, S. Yin, X. Xu, W.A. de Heer, Phys. Rev. Lett. 93, 86803 (2004)

[51] L.A. Bloomfield, J. Deng, H. Zhang, J. W. Emmert, p.131 in: P.Jena, S.N. Khanna, B.K.Rao, Cluster and Nanostructure Interfaces, World Scientific, Singapore (2000) [52] M.B. Knickelbein,Phys. Rev. Lett. 86, 5255 (2001)

[53] M.B. Knickelbein, Phys. Rev. B 70, 014424 (2004)

[54] T.Hihara, S. Pokrant, J.A. Becker Chem. Phys. Lett. 294, 357 (1998)

[55] D.C. Douglass, J.P. Bucher and L.A. Bloomfield, Phys. Rev. Lett. 68, 1774 (1992).

[56] D. Gerion, A, Hirt, A. Chˆatelain, Phys. Rev. Lett. 83, 532 (1999)

[57] D.C. Douglass, J.P. Bucher, D.B. Haynes and L.A. Bloomfield, in ”From Clusters to Crys- tals”, Eds. P. Jena, S.N. Khanna and B. Rao (Kluwer, Boston, 1992)

[58] S. Pokrant, Phys. Rev. B 49, 12 295 (1994)

[59] P.J. Jensen, H.K. Bennemann, Z. Phys. D 35, 273 (1995)

[60] G.M. Pastor, J. Dorantes-D´avila, and K.H. Bennemann, Phys. Rev. B 70, 064420 (2004) [61] N.O.Jones, S.N. Khanna, T. Baruah, and M.R. Pederson,Phys. Rev. B 70, 045416 (2004) [62] V. Z. Cerovski, S.D. Mahanti, and S.N. Khanna, Eur. Phys. J. D 10, 119(2000)

[63] F. L´opez-Ur´ıas, A. D´ıaz-Ortiz, and J.L. Mor´an-L´opez, Phys. Rev. B 66, 144406 (2002) [64] J.A. Alonso, Chem. Rev. 100, 637 (2000)

[65] J. Ho, M. L. Polak, K. M. Ervin, and W. C. Lineberger, J. Chem. Phys. 99, 8542 (1993).

[66] H. Yoshida, A. Terasaki, K. Kobayashi, M. Tsukada, and T. Kondow, J. Chem. Phys. 102, 5960 (1995).

[67] G. Gantef¨or and W. Eberhardt, Phys. Rev. Lett. 76, 4975 (1996).

[68] J. Morenzin, H. Kietzmann, P.S. Bechthold, G. Gantef¨or, and W. Eberhardt,Pure and Ap- plied Chemistry 72, 2149 (2000)

[69] S.-R. Liu, H.-J. Zhai, and L.-S. Wang, Phys. Rev. B 65, 113401 (2002)

[70] X. Wan, L. Zhou, J. Dong, T. K. Lee, and D.-S. Wang, Phys. Rev. B 69, 174414 (2004) [71] K.H. Meiwes Broer (Ed.), Metal Clusters at Surfaces; Springer Verlag, Berlin Heidelberg

(2000)

[72] C. Binns, Surf. Science Reports 44, 1 (2001)