MagnetismMagnetism inin NanomaterialsNanomaterials

Advanced Reading Principles of Nanomagnetism A.P. Guimarães Springer-Verlag, Berlin, 2009. Introduction to Magnetism

Magnetism is virtually universal.  Coherent magnetic fields have been found at the scale of the galaxies and cluster of galaxies.

 Earth's magnetic field has a strength of about 1G and reverses itself with an average period of about 2105 years.

 Magnetic nanoparticles are found in some rocks and can be used to determine the earth's magnetic field (strength and direction) in the past.

 Magnetotactic bacteria have nanometer-sized magnets, which they use for alignment with the earth's magnetic field.

 Many birds (e.g. the homing pigeon) and other living creatures have clusters of nanoparticles (~2-4 nm in the pigeon) in their beak area, which helps them with

their homing ability.

Origin of Magnetism

Macroscopic Microscopic (Charge currents) (Atomic scale)

 If a loop of area A is carrying a current I, the intrinsic intensity of the magnetic field is given by the magnetic moment vector (m or ) directed from the north pole to the south

pole; such that the magnitude of m is given by: m = IA (units: Am2).

 The magnetic moment is the measure of the strength of the magnet and is the ability to

produce (and be affected by) a magnetic field.

Important quantities in magnetism

 Magnetic Moment Vector (m or ). |m| = IA, Units: [Am2] or equivalently [Joule/Tesla]. Measure of the strength of the magnet.

 Magnetic field strength/Magnetizing force (H). Units: [A/m] Measure of the strength of the externally applied field.  Magnetization (M) = magnetic moment (m) per unit volume (V). Units: [A/m] m M  V M measures the materials response to the applied field H (of course we know from our experience with permanent magnets that M can exist even if H is removed). M is the magnetization induced by the applied external field H.   = magnetic moment per unit mass = m/mass. Units: [Am2/kg]

 Magnetic induction/Magnetic flux density (B) = Magnetic flux per unit area. Units: [Tesla = Weber/m2 = Vs/m2 = Kg/s2/A] B is the magnetic flux density inside the material.

 B = 0 (H + M) (0 is the magnetic permeability of vacuum = H/m = Wb/A/m = mKg/s2A2)  Permeability ().Units: [dimensionless] B   H

 Magnetic susceptibility (). (volume susceptibility) Units: [dimensionless] M   (the symbol v is also used to emphasize that the quantity is per unit H volume).

m (mass susceptibility) and M (molar susceptibility) are also used. Susceptibility is a better quantity compared to permeability to get a 'feel' and 'physical picture' of the type of magnetism involved (as we will see later). Susceptibility is actually a second

order tensor and should be written as ij.

 Energy of a magnetic moment (E) E = m·B (scalar dot product)

 Magnetic anisotropy Anisotropy means that various directions in the crystals are non-equivalent with respect magnetization (M) and this implies that M may not be in the same direction of the applied field. There are many contributions to this anisotropy as we shall see later, crystalline (magneto-crystalline) anisotropy being the prominent one. Atomic origin of magnetic moments

Origin of Magnetism This is classical way of looking Due to Electrons at a quantum effect ! Spin of the nucleus Orbital motion of electrons Spin of electrons Small effect

i) Nuclear spin (which is slow and has a small contribution to the overall magnetic effect) Note: at very low temperatures magnetism due to nuclear spin may become important

ii) Spin of electrons

iii) Orbital motion of electrons around the nucleus

The magnetic moment due to spin is equal to the magnetic moment due to orbital motion (in the first Bohr orbit) and is approximately expressed in terms of the Bohr magneton

(B):

eh mAm 9.27  1024 2 BB4 m Understanding magnetism (and formulating theories) to understand the effects observed:

 Direct coupling → Moments (spin, orbital motion, nuclear) localized to an atom and their direct interaction with moments in neighbouring atoms

 Mediated interaction →Moments (magnetism) arising from itinerant electrons in the bands of metals (with the possibility of mediation of interaction via free electrons).

 Superexchange → Local magnetic moments interacting with other local moments via the mediation of non-magnetic elements (super-exchange)  e.g. antiferromagnetism in MnO.

From magnetism of the fundamental components to magnetism of devices

Spin Weak Orbital Motion Magnetism of Atoms Fundamental components Magnetism of Weak Strong of Magnetism Solids Lattice Electron,, Electron Nucleus Orbit Spin Spin Magnetism of Molecules Magnetism of Hybrids e.g. Giant Magnetoresistance Other parameters to comprehend magnetism in solids:

 Effect of external magnetic fields (Diamagnetism and Pauli paramagnetism are effects of external magnetic fields and do not arise independently from fundamental atomic entities)

 Effect of temperature (Ferromagnets can become paramagnets. Alignment of magnetic moments in a paramagnet due a field is thermally assisted) MAGNETISM

All matter Arising out of band structure of metals Diamagnetism Arising out of atomic magnetic moments (permanent) (Spin + Oribital) Pauli spin paramagnetism

Non-interacting atomic moments Band ferromagnetism Interacting atomic moments Curie paramagnetism Band antiferromagnetism

Ferromagnetism Ferrimagnetism Antiferromagnetism Diamagnetism

 This is a property of all materials in response to an applied magnetic field and hence there is no requirement for the atoms to have net magnetic moments.

 This is a weak negative magnetic effect ( ~105) and hence may be masked by the presence of stronger effects like ferromagnetism (even though it is still present).

 A simplified understanding of the diamagnetic effect (in a more classical way!) is based

on Lenz's law applied at the atomic scale. Lenz's law states that change in magnetic field will induce a current in a loop of electrical conductor, which will tend to oppose the applied magnetic field. As the electron velocity is a function of the energy of the electronic states, the diamagnetic susceptibility is essentially independent of temperature. A diamagnet tends to exclude lines of force from the material.

 A superconductor (under some conditions) is a perfect diamagnet and it excludes all magnetic lines of force.

 Closed shell electronic configuration leads to a net zero magnetic moment (spin and orbital moments are oriented to cancel out each other). Monoatomic noble gases (e.g. He,

Ne, Ar, Kr etc.) are diamagnetic. In polyatomic gases (e.g. H2, N2 etc.), the formation of the molecule leads to a closed electronic shell configuration, thus making these gases diamagnetic. Many ionically bonded (e.g. NaCl, MgO, etc.) and covalently bonded (C- diamond, Ge, Si) materials also lead to a closed shell configuration, thus making diamagnetism as the predominant magnetic effect. Most organic compounds (involving other types of bonds as well) are diamagnetic.

A simplified understanding of diamagnetism based on Lenz's law: (a) electrons paired in the same orbital moving with a velocity 'v' canceling each others magnetic moments (m), (b) effect of an increasing magnetic field (B) on the magnetic moments. m1 increases and m2 decreases, so that the net magnetic moment opposes the field B.

The M-H plot for a diamagnetic substance

Paramagnetism

 There are two distinct types of paramagnetism: (i) that arising when the atom/molecule has a net a magnetic moment,

(ii) that come from band structure (Pauli spin or weak spin paramagnetism)

 If the net magnetic moments do not cancel out then the material is paramagnetic. Oxygen for example has a next magnetic moment  = 2.85  per molecule. A point to be noted B here is that even if there are many electrons in the atom; most of the moments cancel out,

leaving a resultant of a few Bohr magnetons. In the absence of an external field these magnetic moments point in random directions and the magnetization of the specimen is zero. When a field is applied two factors come into picture: (i) the aligning force of the magnetic field (we have already seen what this alignment

means!) (ii) the disordering tendency of temperature

 The combined effect of these two factors is that only partial alignment of the magnetic moments is possible and the susceptibility of paramagnetic materials has a small value.

6 3 For example Oxygen has a m (20C) = 1.36  10 m /Kg.

 Two types of paramagnets can be differentiated: (i) those which are always paramagnetic with no other details to be considered and (ii) those which are ferromagnetic, ferrimagnetic or anti-ferromagnetic (and become

paramagnetic on heating)  these will have non-zero value for '' in the Curie-Weiss law

(as considered below).  Effect of Temperature  Any magnetic alignment (which is an ordering phenomenon) is always fighting against

the disordering effect of temperature. While mass susceptibility (m) is independent of temperature for a diamagnet for a general paramagnet it follows the Curie-Weiss law

( ~T ): C C   m T  3 Where m is the mass susceptibility [m /Kg], C is the Curie constant and  is in units of temperature and is a measure of the interaction of the atomic magnetic moments (usually

thought of as an internal 'molecular/atomic field'- the concept of exchange integral, which

we will deal with in the context of ferromagnetism, is the quantum mechanical equivalent

of this). Actually, 'molecular field' is a 'force/torque' tending to align adjacent atomic moments. It's typical value is ~ 109 A/m and is much stronger than any continuous filed produced in a

lab.  If there is no interaction between the atomic magnetic moments; then  = 0 and the Curie- Weiss law reduces to the Curie law (e.g. for O ). The variation of the susceptibility for 2 these kinds of behaviour is shown in . '' can be positive (usually with small value) or

negative. Negative values of '' imply that the molecular/atomic filed is opposing the externally imposed field and thus decreasing the susceptibility of the material. In reality the Curie temperature many not be sharp and further aspects come into the picture which we

shall not consider here.

Variation of mass susceptibility with temperature (in Kelvin): the Curie law and the Curie-Weiss law (with a positive value of ). The behaviour of a diamagnetic material is shown for comparison. Diamagnets have small negative susceptibility which essentially does not change with temperature. Ferromagnetism (FM)

 Ferromagnetism, Antiferromagnetism and Ferrimagnetism involve no new types of magnetic moments; but involve the way the magnetic moments are coupled (arranged).

(a) Ferromagnetic (b) Antiferromagnetic (c) Ferrimagnetic

 Two important ways of understanding ferromagnetism in metals is: (as listed in the introduction to the magnetic properties):

(i) assuming that moments are localized to atoms, (ii) using the band structure of metals (giving rise to itinerant electrons).

The former is conceptually easier and has been assumed in the 'molecular field theory' and

the Heisenberg's approach. It should be noted right at the outset that even in metals (e.g. Fe) most of the electrons behave as if they are 'localized' and the number of itinerant electrons is could be a small number.  In Fe there are 8 valence electrons which occupy the (3d + 4s) bands. Out of these 8 electrons only ~0.95 in the 4s band are 'truly' free/itinerant and remaining ~7.05 are

occupy the 'localized' 3d band.

 In Ni the corresponding quantities are: (3d + 4s) = 10, free  4s0.6, localized  3d9.4.

Band Theory to Understand Ferromagnetism

. As mentioned before a correct theory of magnetism in metals has to involve bands as the electrons are not localized to atoms. However, as noted before, most of the electrons

(especially in 3d metals which are elemental magnets) are rather localized and the 'free' electrons (4s) do not contribute to the ferromagnetic behaviour. Truly speaking the 3d

electrons in transition metals are neither fully localized nor fully free.

Band theory is able to explain the non-integral values of magnetic moment per atom; though, the values may often not match exactly. . The density of states varies in a complicated manner.

. In Fe the 3d electrons are all not fully localized and about 5-8% have some itinerant character and these electrons mediate the exchange coupling between the localized moments.

Using the observed magnetic moment per atom (H) of Fe to be 2.2B the up-spin and down- spin occupancy can be calculated as: , 

   NNdd7.05 NNdd2.2 NNdd4.62, 2.42 (a) (b)

Simplified use of band theory to understand ferromagnetism: (a) Fe (inset shows the alignment of up and down spin bands in the absence of exchange coupling), (b) Ni. Two important points to be noted are: (i) the N(E) is actually more complicated than the simplified curve shown, (ii) N(E) is different for Fe and Ni, but has been shown/assumed to be same. 3d band has a high density of states close to Fermi level (EF). . The above discussions can be summarized as a few thumb-rules for existence of ferromagnetism in metals:

(i) the bands giving rise to magnetism must have vacant levels (e.g. 3d bands in Fe, Co, Ni) for unpaired electrons to be promoted to;

(ii) close to the Fermi level the density of states should be high– this ensures that when electrons are promoted to the unfilled higher energy levels the energy cost is small (high

density of states implies a smaller spacing in energy); (iii) assuming direct exchange, the interatomic distance should be correct for exchange forces to be operative (leading to parallel alignment).

Effect of External Magnetic Field

. Important parameters marked on the curve are Saturation Induction (B ), Retentivity (B ) and s r Coercivity (Hc). The coercivity in an M-H plot is

called 'Intrinsic Coercivity' (Mic or Mci). Saturation magnetization is a structure insensitive property

while coercivity is a structure sensitive property (coercivity of nanoparticles is different from that of bulk materials). . In 'permanent magnet' applications a high coercivity value is usually desired. Another

quantity marked in the figure is the permeability (maximum and initial). Permeability (measured as the slope of the line from the origin to a point) is also a structure sensitive property. The field

required to bring a ferromagnet to saturation (Ms) at room temperature is small (~80 kA/m); but, further increase in magnetization would require much stronger fields and this effect is called 'forced B = 0 (H + M) magnetization'.

Alignment of domains leading to magnetization of the sample

. Preferential alignment of domains can be brought about by an external magnetic field.

 During magnetization the domains oriented favourably (along the field direction), grow at the

expense of the unfavourably oriented domains.  This can occur by: (i) domain wall motion (smooth or jerky) and

(ii) by rotation of the magnetization of the domains. The external magnetic field tends to align the misoriented spins in the domain wall- leading to its displacement. These processes can occur

simultaneously as the field increases.  Rotation of spin is opposed by the increase in Domain related mechanisms operative anisotropy energy (magnetocrystalline, shape, during the magnetization process

stress). During rotation all spins need not be parallel to one another and the actual picture may be a little complicated. Effect of Temperature

. Spatially correlated collective quantized modes lead to demagnetization (called spin waves (or magnons)).

. Ferromagnet becomes paramagnet above Curie temperature (Tc). At Tc susceptibility becomes infinite. Even beyond T there are local clusters ('spin clusters') of aligned magnetic c moments.

. Maximum magnetization is obtained when all the moments have parallel orientation– let this state correspond to a magnetization M (or  ). 0 0 . It is expected that a plot of s/0 versus T/Tc will approximately lie on one another for different materials.

Demagnetization curve for a ferromagnet. Domain structure and the Magnetization Process

. The magnetic structure of a ferromagnetic material consists of domains → to reduce magnetostatic energy.

. Domains are separated by domain walls. Broadly two types of domain walls can be differentiated: Bloch walls and Néel walls. Other types of domain walls like cross-tie walls

and more complicated configurations are also possible.

 As shown in in Bloch walls the spin vectors rotate out of plane in the domain wall (while in Néel walls they rotate in plane).

 Néel walls are seen in thin films (they are usually observed in thin films ~40 nm thick).

 Usually the domain wall thickness is few hundred atomic diameters (i.e. it is rather

diffuse). Hence, the domain wall by itself is a nanostructure.

Actual domain structure more complicated than this A Bloch wall (This is a crude schematic as the number of spins involved in the wall is much larger and hence rotations between adjacent spins are usually much smaller). . The domain wall represents a region of high energy as the spin vectors are not in the directions of easy magnetization. Hence thicker walls represent higher energy and in

materials with high magnetocrystalline anisotropy energy (EA; e.g rare-earth metals), the domain walls are thin (~10 atomic diameters).

. Other sources of anisotropy are those due to shape of the particle and due to residual (or

applied) stresses. A competition between the magnetostatic energy and the

magnetocrystalline anisotropy energy, essentially decides the domain size/shape. . The word 'essentially' has been used as other factors like magnetoelastic energy (E =E ) due to magnetostriction (change in dimension due to a magnetic field) Magnetoelastic ME also contribute to the overall energy.

. The total energy (ETotal) can be written as a sum of four terms: EE E  E  E Total Exchange Anisotropy Magnetoelastic External

Wherein, EExternal corresponds to the energy of total magnetic moment in the external magnetic field.

Magnetoresistance  The resistance of a conductor changes when placed in an external magnetic field. This effect is called magnetoresistance.

 The resistance is higher if the field is parallel to the current and lower if the field is perpendicular to the current. In general the resistance depends on the angle between the current and magnetic field and this effect is called Anisotropic Magnetoresistance (AMR).

 The change is usually small (~ 5%; can be as large as 50% as in the case of some ferromagnetic uranium compounds). Magnetoresistance arises from a larger probability of s-d scattering of electrons in a direction parallel to the magnetic field. AMR effect is used in magnetic field sensing devices. Magnetism in Nanomaterials

Magnetic nanostructures in bulk materials

Even in bulk magnetic materials some structures can be in the nanoscale:  Domain walls in a ferromagnet (~60nm for Fe).  Some domains (especially those in the vicinity of the surface or grain boundaries), could

themselves be nanosized.

 Spin clusters above paramagnetic Curie temperature (p) could be nano-sized.

When we go from bulk to ‘nano’ only the structure sensitive magnetic properties (like coercivity) is expected to change significantly.

Some of the possibilities when we go from bulk to nano are:  Ferromagnetic particles becoming single domain

 Superparamagnetism in small ferromagnetic particles (i.e. particles which are

ferromagnetic in bulk)

 Giant Magnetoresistance effect in hybrids (layered structures)  Antiferromagnetic particles (in bulk) behaving like ferromagnets etc.

Dependence of magnetic moment on the dimensionality of the system

 There is a increase in magnetic moment/atom as we decrease the dimensionality of the system.

 This is indicative of fundamental differences in magnetic behaviour between nano- structures and bulk materials.

 This effect is all the more noteworthy as surface spins are usually not ordered along the same directions as the spins in the interior of the material (thus we expect nanocrystals

with more surface to have less B/atom than bulk materials- purely based on surface effect).

Magnetic Moment (B/atom)

0D 1D 2D Bulk

Ni 2.0 1.1 0.68 0.56 Fe 4.0 3.3 2.96 2.27

Increasing magnetic moment/atom

Fe can have a maximum possible moment of 6B/atom (3B orbital + 3B spin)  this implies that in 0D nanocrystals very little of the orbital magnetic moment is quenched

Superparamagnetism

 As the size of a particle is reduced the whole particle becomes a single domain below a critical size.

 This aspect can be understood in two distinct ways: i) a particle smaller than the domain wall thickness cannot sustain a domain wall (noting

that domain wall thickness may not be constant with size), ii) the magnetostatic energy increases as r3 ('r' being the radius of the particle) and the domain wall energy is a function of r2  there must be a critical radius (r ) below which c domain walls are not stable.

(in reality the calculation is complicated by other factors).

 The general trend is: 2    = magnetic moment per unit mass = m/mass. Units: [Am/kg] rfc ~ 2 M s  Ms is saturation magnetization 2-3 orders of magnitude

 Multidomain  increasing coercivity with decreasing size

 Single domain  peak coercivity

 Single domain  decreasing coercivity with decreasing size

 Single domain  zero coercivity Fe Co Ni Fe3O4 D (nm) P 16 8 35 4 (Calc.)

M vs H/T curve for a superparamagnetic material

Comparison between paramagnetism and superparamagnetism

23 2 . Magnetization of oxygen (() = 2.85 B per molecule (= 2.64  10 Am /molecule); Number of oxygen molecules = (6.023  1023)/0.032 per kg,

Magnetic field applied = 20106 A/m;  (20 C) = 1.36  106 m3/Kg). m . What is the magnetizing effect of the strong field?

. If all the magnetic moments of all the molecules are aligned the magnetic moment obtained = ((6.023  1023)/0.032)(2.64  1023) = 497 Am2/kg.    H The actual magnetization in the presence of the field () m 6 6 2 = m H = (1.3610 )(2010 ) = 27.2 Am /kg. Percentage of possible magnetization = (27.2/497)100 ~ 5.5%

Thus, even strong fields are very poor in aligning the magnetic moments of paramagnetic

materials.

. What is the magnetization of Fe nanoparticle (d = 15nm) when saturated (Given: eff(Fe) = 2.2  ; a(Fe) = 2.87 Å). B Volume of the particle = 4(15/2)3/3 = 1767 Å3

Volume per atom in BCC Fe = (2.87)3/2 = 11.82 Å3 (the factor 2 in the denominator is due to 2 atoms/cell in BCC).

Number of atoms of Fe in the particle = 149 atoms

Magnetic moment of the particle under saturation = 329 B (Bohr magnetons)

Change Leads Change in Leads Change in Leads Change in Leads Change in A in size to structure to mechanism to property* to performance A’

Change Leads Change in Leads Change in Leads Change in B in size to mechanism to property* to performance

Change Leads Change in Leads Change in C in size to property* to performance

Change Leads Change in D in size to performance

Reduction in size

 Change in domain structure

 Change in mechanism of magnetization (Superparamagnetism)

 Change in property (Coercivity, Retentivity = 0)

 Performance Magnetism of Clusters

 Like other properties of clusters, magnetic properties of clusters can change with the addition (or removal) of an atom. Clusters considered here have few to a thousand atoms

typically (extending upto about 5 nm).

 Important factors which determine the magnetic behaviour of clusters are:

(i) atomic structure,

(ii) nearest neighbours distance,

(iii) purity and defect structure of the cluster.

Ferromagnetic clusters  In small clusters (with less than 20 atoms) there are large oscillations in the magnetic moment of the cluster (calculated as magnetic moment per atom).

 For more than 600 atoms in the cluster 'bulk-like' behaviour emerges (i.e. with increasing number of atoms the oscillations die down and bulk behaviour emerges).

 Fe can have a maximum possible moment of 6B/atom (3B orbital + 3B spin).  Fe cluster has a moment of 5.4 /atom  practically very little of the orbital moment 12 B is quenched in the cluster.

 Fe13 however has a moment of only 2.44B.  Ni cluster has an abnormally low moment as well and this is attributed to the 13 icosahedral structure of the cluster (which is densely packed).

With larger and larger cluster size the orbital contribution seems to be low; but, there is

still an enhancement of the magnetic moment over the bulk value. Thus structure and packing seem to play an important role in the net magnetic moment obtained.

Variation of Magnetic moment per atom in Fe clusters with cluster size. Enhancement over bulk value is to be noted. Antiferromagnetic clusters  In antiferromagnetic materials we do not expect any net magnetic moment in the bulk. However, there is a possibility that in small clusters 'up' spins do not cancel out the 'down'

spins (leading to a net magnetic moment)  these are anti-ferromagnets behaving as ferromagnets!  Magnetic 'frustration' is also a possibility. (frustration  the spin on a given atom does not 'know' which way to point).

 Small clusters of Cr (one of the few metals which are antiferromagnetic- spin density wave AFM) have an interesting rich set of possibilities (along with allied complications!).

A plot of magnetic moment per atom oscillates with size (as in the case of ferromagnetic

clusters). A given cluster size (e.g. Cr9) is expected to exist in multiple magnetization

states (in the case of Cr9 magnetization can be small (~0.65 B/atom) or as high as ~1.8

B/atom [1]). In addition to the 'multiple magnetization states' there is a possibility of co- existence structural isomers.

 Mn clusters show some similarities with ferromagnetic Fe clusters with regard to cluster size dependence (with more than 10 atoms) [2]. Compact Mn (icosahedral) and Mn 13 19 (double-icosahedral) clusters have very low magnetic moment as compared to

neighbouring clusters. Mn15 has the highest moment of 1.5 B/atom [2].

[1] L. A. Bloomfield, J. Deng, H. Zhang, and J. W. Emmert, in “Clusters and Nanostructure Interfaces” (P. Jena, S. N. Khanna, and B. K. Rao, Eds.), p. 213. World Scientific, Singapore, 2000. [2] M. B. Knickelbein, Phys. Rev. Lett. 86, 5255 (2001).

Mn

Variation of magnetic moment per atom in Mn (which is antiferromagnetic in bulk)

M. B. Knickelbein, Phys. Rev. Lett. 86, 5255 (2001). Next slide inserted on ref’s comments Experimental production of clusters

 A gas phase supersaturated metal vapour is ejected into flowing inert gas (which is cooled).  The metal vapour is produced by: (i) thermal evaporation, (ii) laser ablation, (iii) sputtering, etc.  Most mass separators require the clusters to be charged (the clusters need to be ionized if they are not charged). Examples of mass filters include: Radio Frequency Quadrupole filter, Wien filter, Time-of-flight mass spectrometer, Pulsed field mass selector, etc.  At the end of separation we can get a narrow distribution of masses of particles (in small

clusters we can even get a precise number of atoms in a cluster).

Example of a metal vapour production method Measurement of magnetic moment of clusters

 The experimental results presented for free clusters [Fe (ferromagnetic clusters) and Cr and Mn (antiferromagnetic clusters)] are typically measured using a setup, which is based on the Stern-Gerlach experiment (that detected electron spin) which is typically coupled

with pulsed laser vaporization technique (details in next slide).  A collimated cluster beam is guided into a magnetic field gradient (dB/dz). The field gradient will deflect a cluster with magnetic moment  by a distance ‘d’ given by the equation as below (L  length of the magnet, D  distance from the end of the magnet to

the detector, M  cluster mass, vx  entrance velocity).  For clusters deposited on surfaces other techniques of measurement exist such as: X-Ray Magnetic Circular Dichroism, Dichroism in Photoelectron Spectroscopy, Surface Magneto-Optical Kerr Effect, UHV Vibrating Sample Magnetometry, etc.  For embedded clusters techniques like: Micro-SQUID Measurements, Micro-Hall Probes, etc. can be used to measure the magnetic moments.

dB2 (1 2 D / L ) dL  2 dz2 Mvx Experimental setup for the measurement of magnetic moments

Metal clusters are Cluster+gas Mass dependent produced by pulsed mixture undergoes Magnetic deflection deflection measured laser vaporization of a supersonic Beam is collimated Ionization by Laser of collimated beam perpendicular to the target material into a expansion on beam in a TOFMS jet of helium gas entering vacuum Issues regarding the measurement of magnetic properties of nanomaterials

 The measurement of magnetic properties in clusters and nanostructures is needless to say challenging, as compared to their bulk counterparts.  In clusters as the magnetic moment is a sensitive function of the number of atoms in the cluster- the number of atoms have to be known precisely.  Coagulation or contamination of clusters/nanocrystals- either during production or during measurements has to be avoided. Surface oxidation can also severely alter the magnetic properties (e.g. Co-CoO system to be considered).  Temperature plays a key role in the magnetic behaviour of nanoscale systems and hence temperature has to be precisely controlled.  The spin alignment in nanoscale systems (to be considered in coming slides) could be very different from their bulk counterparts and hence models with which experimental results are compared have to take into account the precise geometry of the system and surface effects.  In the case of particles in a substrate or embedded magnetic nanoparticles, the role of the interface and the substrate could be pronounced (i.e. deducing the properties of the free- standing nanoparticles from those measured could be difficult).

Magnetism in thin films, hybrids Illustrative examples Ni  2D versus 3D behaviour Cu (100) In the case of Ni films on Cu(100) substrates, when the thickness of the Ni film is greater

than 7 monolayers (ML) the systems behaves as a 3D Heisenberg ferromagnet and below 7ML it behaves like a 2D system [1]. In the 2D system all the spins are in the plane, while

in the 3D system out of plane spin orientation is also observed.  Curie Temperature of thin films In the case of Fe(110) films (1-3 monolayers) grown epitaxially on Ag(111) substrates the

Curie temperature reduces from bulk values to ~100K (~10% of the bulk value) for ~1.5

monolayer films [2]. (Thermal disordering effects are becoming prominent).

Fe(110) Ag (111)

[1] F. Huang, M. T. Kief, G. J.Mankey, and R. F.Willis, Phys. Rev. B 49 (1994) 3962 [2] Z.Q. Qin, J. Pearson, and S. D. Bader, Phys. Rev. Lett. 67,1646 (1991). Magnetism of Hybrids: Giant Magnetoresistance (GMR)

 As compared to normal (conventional) magnetoresistance, where the change in resistance due to a magnetic field is ~5%; in GMR the change could be of the order of about 80% (or

more).

weak RKKY type coupling

 Carrying forward the concept of GMR sandwich structures, a spin valve (GMR) has been devised. In a 'spin valve', the two ferromagnetic layers have different coercivities and can

be switched on at difference field strengths.

 An extension of spin vales is obtained by replacing the non-ferromagnetic layer with a thin insulating (tunnel) barrier. This can give rise to an effect known as the Tunnel

Magneto-resistance (TMR); wherein, larger impedance, which can be matched to the circuit impedance, is obtained. In TMR, spins traveling perpendicular to the layers, tunnels through the insulating layer and hence the name of the effect.  Applications of TMR effect include: hard drives (with high areal densities), Magnetoresistive Random Access Memory (MRAM), etc. It also forms a fundamental

unit in spin electronics with applications such as reprogrammable magnetic logic devices.

Magnetism of Hybrids: Exchange anisotropy

 Due to exchange coupling of spins across an interface between a ferromagnetic phase and an antiferromagnetic phase, there is a preferred direction (anisotropy) for the field, which

leads to a shift in the hysteresis (M-H) loop. [E.g. Co particles (ferromagnetic) covered with CoO (antiferromagnetic with large crystal anisotropy)].

 Steps involved in creating exchange anisotropy: • Have a single domain FM particle (say Co) in contact with a AFM layer (CoO)

• Apply a field above the Néel temperature of the AFM phase to saturate the FM phase

• Cool the system below the Néel temperature of the AFM phase to introduce a

preferential alignment of spins across the interface (in the AFM phase). The spins in the

FM phase will maintain their orientation even after the field is removed.

• Construct the usual M-H loop

 If the field is removed the spins in the FM phase will flip to the field-cooled orientation, due to the influence of the AFM phase. As the field direction is reversed, the spins across

the interface in the AFM (with large crystal anisotropy) oppose the reversal of spins in the

FM phase. Hence, the exchange coupling leads to large coercivity value.

(a) Preferential ordering of spins in the antiferromagnetic phase across the interface, (b) application of a field in opposition to the magnetization of the ferromagnetic phase leading to a disturbance of spins across the interface in the AF phase. Nanodiscs Special spin arrangements with no bulk counterparts

 Nanodiscs can exist in vortex spin state. . 15 nm thick permalloy discs show the vortex state when the diameter of the disc is

above 100 nm.

. The spin arrangement consists of concentric arrangement of spins on the outside (in

plane of the disc) and with out of plane component towards the centre of the disc.

. The core radius (wherein the spins are out of plane) is of the order of the exchange length (l , which is about 5nmfor permalloy). Exchange . Other non-equilibrium configurations of spin may also be observed in nanodiscs (e.g. antivortex, double votex states).

Exchange length (lex) is the characteristic length scale of a magnetic material, below which exchange is dominant over magnetostatic effects.

Vortex spin structure of nanodiscs. In the core regions the spins have an out of plane component (the magnitude of which has been shown with an out of plane displacement of vectors). Nanorings

 In nanorings there is no 'core based on spin structure'. . As shown in part from the votex state the nanorings may have onion and twisted states

of magnetization (which are realized in different parts of the hysteresis loop).

Spin structures (states) in nanorings: (a) votex, (b) onion, (c) twisted