1. Magnetic Nanoclusters
By: Adam Krause
2/27/07
Physics 672

2. Nanocluster Quick Introduction
From a few atoms to several thousand atoms
High fraction of atoms on the surface
Different elements form different bonds and different nanocluster structures
1.) Nanoclusters are particle that range in size from a few atoms to several thousand atoms.
2.) Their high fraction of surface atoms give them properties different from bulk material properties.
3.) Different elements form different bonds and different nanocluster structures.
4.) These bonds and structures contribute to their unique properties.
5.) I will give a few examples of nanocluster types

3. A Few Types of Nanoclusters
Van der Waals Nanoclusters
Binding energy: < 0.3 eV / atom
Balance between induced dipole force and quantum closed shell interaction
Noble gases form icosahedral Van der Waals clusters
1.) Long Range Attraction: Induced dipole Force
2.) Short Range Repulsion: Quantum Closed Shell electronic interactions
3.) Van der Waals Binding energy: < 0.3 eV per atom
4.) It has been shown that noble gases form icosahedral shapes
Figure above from: Alonso, J. A., Structure and Properties of Atomic Nanoclusters, 2005

4. A Few Types of Nanoclusters
Van der Waals Nanoclusters
The drops at 148 and 309 atoms correspond to completed icosahedra.
1.) The mass spectrum of Argon nanoclusters shows the completing of icosahedra structures.
2.) This suggests a higher stability for closed icosahedral clusters.
Figure above from: Echt, O., et al., J. Chem. Soc. Faraday Trans., 86 (1990) 2411

5. A Few Types of Nanoclusters
Ionic Nanoclusters
Bond Strength: 2-4 eV / atom
Tend to form boxes
1.) Formed by electrostatic force
2.) NaCl is a typical example of ionic cluster
3.) The bonds are around 2-4 eV per atom
4.) Ionic clusters tend to form cubic shapes
5.) This is shown by the mass spectrum for n = 62, 171, 364, 665, 1098
6.) This shows that completed boxes are more stable than partially completed boxes
NaCl Cluster
Graph above from: Martin, T. P., Physics Reports, 273 (1996) 199

6. A Few Types of Nanoclusters
Metal Nanoclusters
Metal clusters have complicated bonding that varies from metal to metal
Due to this variation the bond strength varies from around 0.5 eV to 3 eV per atom
1.) Some metals bond primarily with their outer valence sp electrons
2.) Others; like Fe, Co, and Ni addressed in this presentation; bond with their d orbitals
3.) Due to this variation the bond strength varies from around 0.5 eV to 3 eV per atom

7. Metal Nanoclusters Produced By Laser Vaporization
1.) The metal nanoclusters considered here were made using the laser vaporization technique.
2.) This technique involves focusing a laser beam onto a metal sample.
3.) Metal atoms evaporate and are cooled with a flow of inert gas.
4.) As they cool the atoms combine into nanoclusters of varying sizes.
5.) They are then expanded through a nozzle into a vacuum to further cool them.
Figure above from: Billas et al., J. Magn. Magn. Mater. 168 (1997) 64

8. Stern-Gerlach Apparatus
1.) The Stern-Gerlach apparatus was used to measure the magnetic moment of the nanoclusters created by laser vaporization.
2.) The beam of clusters is deflected by the non-uniform magnetic field as a function of the cluster magnetic moment.
3.) The mass of the clusters are then measured by the Time-of-Flight mass spectrometer.
Figure above from: Billas et al., J. Magn. Magn. Mater. 168 (1997) 64

9. Description of magnetic particles
1.) Magnetic moments arise in atoms from the net electron spin z component of the electron angular momentum.
2.) Hund’s rule states that electrons tend to fill their orbitals in such a way as to maximize their net spin.
3.) The total magnetic moment of the atom comes from the coupling of the electronic spin with the z-angular momentum.
4.) When these atoms combine to form nanoclusters, the atomic magnetic moments can align to form a net magnetic moment for the cluster.

10. Band Structure Evolution
1.) As the dimensionality increases from the atom to surfaces to bulk matter, the atomic magnetic spins do not simply add.
2.) Due to an increases coordination number or number of nearest neighbors, the 3d orbitals which are responsible for the magnetic moment overlap which causes an energy band structure.
3.) This energy banding serves to decrease the total magnetic moment per atom of the material
4.) This overlapping of orbitals delocalizes the electrons and is responsible for the mutual alignment of magnetic moments and the ferromagnetic behavior seen in many metals.
Increasing Coordination Number
Figure above from: Billas et al., J. Magn. Magn. Mater. 168 (1997) 64

11. Magnetic Moment vs. Cluster Size
1.) a = Ni, b = Co, c = Fe. These graphs show how smaller nanoclusters of Ni, Co, and Fe have higher magnetic moments per atom.
2.) As the nanoclusters increase in size, the magnetic moments approach the bulk limit.
3.) As stated before this phenomenon is attributed to the increased coordination number of the atoms and the resulting delocalization of the electrons.
4.) Note the oscillatory behavior in the graphs.
5.) It has been proposed that this is due to the closing of magnetic shells.
Figure above from: Billas et al., J. Magn. Magn. Mater. 168 (1997) 64

12. Closed Shell Cluster Size vs. Magnetic Moment Minima.
1.) Jensen and Bennemann attempted to model the magnetic shell structure by assuming the shapes: cube, octahedron, or a cubo-octahedron. These are regular shapes that minimize the surface energy.
2.) They calculated the number of atoms in the closed shells of the f.c.c.-cube, f.c.c.-octahedron, f.c.c.-cubo-octahedron, b.c.c.-cube, and b.c.c.-octahedron.
3.) Experimental data were then compared with these numbers and are shown in the table.
4.) The table shows that the Iron cluster data fit the b.c.c.-cube calculations, the Ni cluster data fit the f.c.c.-cube calculation, and the Co cluster data fits closest to the f.c.c.-octahedron calculation.
Table above from: Jensen, P. J., and K. H. Bennemann, Z. Phys. D. 35 (1995) 273

13. Magnetic Shell Model (1) (2)
1.) By assuming the cluster shapes for Fe, Ni, and Co indicated in the previous table a shell model was formed.
2.) The equation 1 is the magnetic moment per atom as a function of cluster size, N.
3.) N0 and N1 are the number of atomic sites in the two outer most cluster shells. These numbers depend on the cluster shape which is why it is important to know that from the table.
4.) The mu0, mu1, and mu-bulk are the magnetic moments per atom in the first shell, second shell and bulk respectively.
5.) xo is the statistical concentration of occupied sites in the outermost shell.
6.) This equation 1 gives the oscillatory behavior seen above.
7.) The solid line shows the model given by Equation 1.
8.) An even better approximation comes from adding a further constraint on the values that mu-I can take. This is shown in Equation 2.
9.) This constraint says that the magnetic moment at any given lattice site is a function of the number of nearest neighbors for that site.
Graphs from: Jensen, P. J., and K. H. Bennemann, Z. Phys. D. 35 (1995) 273

14. Magnetic Moment vs. TemperatureCo Ni
1.) Another interesting phenomenon is how the magnetic moment in a magnetic field changes with temperature for various cluster sizes.
2.) The trend drops much faster as the temperature increases for the bulk matter than for the clusters.
3.) This smooth transition can be modeled with the Heisenberg Model shown in the Ni graph. Fe
Graphs from: Billas, M. L., A. Chatelain, and W. A. de Heer, Science 265 (1994) 1682

15. Monte Carlo Simulation of Magnetization vs. 1/Temp
1.) Using Monte Carlo simulations of the Heisenberg Model Binder’s group calculated the behavior shown in the previous slide.
2.) Their calculations are shown in the graph assuming spherical clusters.
3.) In the simulation, as the size of the nanocluster increases the transition sharpens at the curie temperature.
4.) The smaller particles maintain a higher magnetic order at higher temperatures than the bulk material.
Graph from: Binder, K., et al., J. Phys. Chem. Solids, 31 (1970) 391

16. Superparamagnetism Magnetization Loops of Fe Nanoclusters
1.) Most bulk magnetic materials exhibit ferromagnetic behavior below the Curie temp and paramagnetic behavior above the Curie temp.
2.) However, when the particle being considered is sufficiently small, some materials that are normally ferromagnetic behave paramagnetic even below the Curie temp.
3.) This is called superparamagnetism.
4.) It is the tendency of the particle to lose its net magnetic moment due to ambient thermal conditions randomly reorienting spins after it is removed from an external magnetic field.
5.) Note: This is not contrary to the above slides where the magnetic moment persists past the bulk Curie Temperature. It has been proposed by Billas et al. that since their clusters were cooled and expanded supersonically, the superparamagnetic model is limited due rotational effects.
6.) Jackson’s iron nanocluster are stationary and embedded in a silver matrix.
7.) The graph shows magnetization loops of Fe nanoclusters for various temperatures between 5 K and 50 K.
8.) The 5 K magnetization loop shows a hysteresis broadening as the magnetic field is changed. At this temperature the clusters are ferromagnetic.
9.) At the measured temperatures above 5 K, superparamagnetism keeps the loops on top of each other as the magnetic field is reversed.
10.) The solid line is a 10 K Langevin function prediction.
Graph from: Jackson, T. J., et al., J. Phys.: Condens. Matter, 12 (2000) 1399

17. Summary
Metal nanoclusters of an element behave differently than bulk matter of the same element.
d-orbital overlap reduces magnetic moment per atom.
Metal nanoclusters exhibit magnetic shell phenomenon
Metal nanoclusters do not lose their magnetization as quickly above the Curie temp.
Metal nanoclusters exhibit superparamagnetic behavior.
Superparamagnetism provides a theoretical minimum size per bit in magnetic moment based memory systems.
1.) It has been shown that magnetic nanoclusters behave differently than bulk matter made of the same element.
2.) As the electronic orbitals overlap, they delocalize causing a reduction of the net magnetic moment per atom.
3.) Nanoclusters exhibit magnetic shell phenomenon
4.) As a function of temperature, the nanoclusters behave differently as well. When expanded supersonically they do not lose their magnetization as readily above the Curie temperature.
5.) They also exhibit superparamagnetism, a phenomenon unique to nanoclusters.
6.) Superparamagnetism poses a problem in magnetic moment based memory systems.
7.) It provides a theoretical minimum size per bit below which thermal effects would randomize the magnetic moment.

18. References
Alonso, J. A., Structure and Properties of Atomic Nanoclusters (Imperial College Press, London, 2005).
Echt, O., et al., J. Chem. Soc. Faraday Trans., 86 (1990) 2411
Martin, T. P., Physics Reports, 273 (1996) 199
Dietz, T. G., et al., J. Chem. Phys., 74 (1981) 6511
Bondybey, V. E., and J. H. English, J. Chem. Phys., 76 (1982) 2165
Billas, M. L., A. Chatelain, and W. A. de Heer, J. Magn. Magn. Mater. 168 (1997) 64
Cox, D. M., et al, Phys. Rev. B., 32 (1985) 7291
Billas, M. L., A. Chatelain, and W. A. de Heer, Science 265 (1994) 1682
Jensen, P. J., and K. H. Bennemann, Z. Phys. D. 35 (1995) 273
Billas, M. L., et al., Phys. Rev. Lett., 71 (1993) 4067
Binder, K., et al., J. Phys. Chem. Solids, 31 (1970) 391
Jackson, T. J., et al., J. Phys.: Condens. Matter, 12 (2000) 1399