Magnetic Materials

1. Magnetic Materials • Pre-requisites • Magnetization and Susceptibility • Type of Magnetism • Ferromagnetic Domains • Soft and Hard Materials

2. Lorentz force When a charge particle moves with velocity v in the presence of a magnetic field, the Lorentz force acting on the charge is given by F=qv×B Example: if electrons move in “–x” direction with drift velocity v in the presence of a magnetic field B in the “+z” direction, the Lorentz force acting on the electron is a force in “–y” direction, given by the right hand rule. Pre-requisites -x z -y • Current carrying conductor • In 1819, Hans Christian Oersted discovered that a magnetic field is present in the vicinity of a current carrying conductor. The direction of the magnetic field is determined by the right hand corkscrew rule.

3. Magnetic Materials • Pre-requisites • Magnetization and Susceptibility • Magnetic Dipole Moment • Atomic Magnetic Moment • Magnetization Vector M • Magnetizing Field or Magnetic Field Intensity H • Magnetic Permeability and Susceptibility • Type of Magnetism • Magnetic Domains • Soft and Hard Materials

4. Definition: Magnetic dipole: A current loop where the circulating current is I, with area A enclosed by the current. Magnetic dipole moment: Note: current I circulates clockwise when looking along un(same as RH rule). Magnetic Dipole Moment un: is a unit vector coming out from area A A: area of the circle mm: dipole moment I: current circulating in the circle Definition of magnetic dipole moment

5. Properties of magnetic dipole: Under external field Bx, magnetic moment experiences a torque which tries to rotate it to align its axis with the field In addition, since a magnetic moment is a current loop, it gives rise to a magnetic field B around it, B ∝ μm/r3. The field lines around the loop is similar to the field around a bar magnet (note that there are no magnetic monopoles). Magnetic dipole in external field Magnetic dipole as a bar magnet So what is a common current loop we encounter?

6. Magnetism is inseparable from quantum mechanics. The magnetic moment of a free atom has 3 principal sources: (1) The orbital angular moment of electrons about the nucleus (the effect gives paramagnetic contribution to the magnetization) (2) the spin with which electrons are endowed (The effect also gives paramagnetic contribution to the magnetization) (3) The change in the orbital moment induced by an applied magnetic field (The effect gives diamagnetic contribution) Atomic Magnetic Moment mspin morb

7. Definition: Orbiting electron in an atom behaves like a current loop and has a orbital magnetic moment associated with it Orbital magnetic moment (μorb): where L is the orbital angular momentum, r the radius of orbit, ω the angular frequency, e the electronic charge and methe rest mass of electron. (Note that electron orbits in an opposite dir compared to I) Magnetic moment proportional to orbital angular momentum by a constant factor which depends on the ratio of the charge to the mass of the electron -- called the gyromagnetic ratio. The‘-‘ indicates μorbis in opposite direction to L due to –ve charge of electron. Orbital Magnetic Moment

8. Electron Spin Magnetic Moment • Spin magnetic moment: electron spinning on its own axis has magnetic momentum mspin (also called b, or μB), given by mspin morb where S is spin angular moment, meis the rest mass of electron, e the electronic charge, h the Planck’s constant. β is called the Bohr magnetonand in SI units has value 9.27 x 10-24 Am2. • Thus the spin of a single electron has a magnetic moment of 1 Bohr magneton along the field.

9. Only unfilled subshells contribute to the overall magnetic moment of an atom. In atoms where electrons are paired, +ve and –ve magnetic moments cancel with each other. Example: unpaired electrons in inner electron shells can have small +ve moments, like the case of 3d electrons in transition metals. Some discussions on atomic magnetic moment Magnetic moments of neutral 3d transition elements

10. Magnetic field inside solenoid B0 (field in free space) depends on current I and # of turns per unit length n as B0=m0nI=m0I’, where I’ is current per unit length and m0 is the absolute permeability of free space. Magnetization Vector M In magnetic media: • Each atom responds to the applied field B0 and acquires a magnetic moment mm along the field, • Medium develops net magnetic moment along field and becomes magnetized. • Magnetic vector M, defined as magnetic dipole moment per unit volume, describes extent of magnetization of medium. • For N atoms in a small volume ΔV and each atom i with magnetic moment mmi, M is defined by • Wherenatis # of atoms per unit volume andmavthe average magnetic moment per atom. In vacuum media: A tightly wound solenoid with (i) vacuum media; and (ii) magnetic media.

11. Some discussions on Magnetization Vector M Side cross section end of solenoid Im • Each magnetic moment can be viewed as an elementary current loop at the atomic scale • Bulk region: All neighboring loops in the bulk have adjacent currents in opposite directions that cancel each other  no net bulk or internal current. • Surface region: current in the surface loops cannot be cancelled and leads to net surface currents in anticlockwise direction when looking from RHS of solenoid. • Total magnetic moment: • From the definition of M, Total magnetic moment= M (Volume) = M A l • From surface current concept, Total magnetic moment = (Total current) (cross-sectional area)=Iml A • Thus, we have M=Im Im: the magnetization current on the surface per unit length

12. B-m0M refers to the contribution of the external currents alone to the magnetic field B0 in the media. Magnetizing field or magnetic field intensity H in Am-1 can be defined as H is the “cause” and B is the “effect”. H depends only on external conduction currents whereas effect B depends on magnetization of matter. For ferromagnetic materials, m0M is often much greater than applied field m0H, so B~m0M Magnetizing Field or Magnetic Field Intensity (H) • Magnetization currents on surface behaves like a solenoid. • Magnetic field within medium arises from (1) the conduction current I’ in the wire and from (2) the magnetization current Imon surface. • Magnetic field B inside solenoid is given by • B in a magnetized media is the sum of the applied field B0, and a contribution from the M of the material. • M is caused by application of B0 due to conduction current in solenoid wires which can be adjusted.

13. The increase in magnetization when a magnetic material is placed in an applied magnetic field is measured by the magnetic permeability It also represents the magnetic field per unit magnetizing field – relates the effect B to the cause H at the same point P. If only vacuum in the applied magnetic field, then m0 the permeability of free space is Also relative permeability for a medium is defined as Analogous to dielectric constant of dielectric materials Magnetic Permeability and Susceptibility • Assuming medium is isotropic, M can also be related to H by • And also the magnetic susceptibility relates to the relative permeability as B-H initial magnetization curve for a ferromagnet. m of a ferromagnetic material is not constant but changes as the material is magnetized = gradient of B-H loops

14. Units for Magnetic Quantities

15. Magnetic Materials • Pre-requisites • Magnetization and Susceptibility • Type of Magnetism • Magnetic Domains • Soft and Hard Materials

16. Diamagnetism Paramagnetism Ferromagnetism Antiferromagnetism Ferrimagnetism Saturation Magnetization and Curie Temperature Type of Magnetism

17. Small –ve χm ~ -10-5 Occurs in all materials, but in many it is shadowed by its +ve magnetic effects. External magnetic field acting on atoms of materials slightly unbalances their orbiting electrons and creates small magnetic dipoles within atoms opposethe applied field  produces “–ve” magnetic effect known as diamagnetism. Magnetization vector M is in opposite direction as m0H and resultant B is smaller than m0H. Substance exhibits diamagnetism whenever constituent atoms in materials have closed subshells and shells. This means each constituent atom has no permanent magnetic moment in absence of applied field. Covalent and many ionic crystals are diamagnetic materials because constituent atoms have no unfilled subshells. Superconductors have χm=-1. Diamagnetic levitation demo Diamagnetism

18. Small +ve χm ~ 10-6 to 10-2 Materials that exhibits small +ve χm in presence of magnetic field are called paramagnetic and the effect is termed paramagnetism. Produced by alignment of individual magnetic dipole moments of atoms/molecules with an applied magnetic field. Since thermal energy randomizes direction of magnetic dipoles, increase in temperature decreases paramagnetic effect. Many metals are also paramagnetic, such as magnesium, with the origin due to the alignment of the majority of spins of conduction electrons with the field (called Pauli spin paramagnetism). Two types of materials: materials with individual magnetic moment and many metals. Paramagnetism

19. Dia- and para-magnetisms are induced by applied field and the magnetization is present only when field is maintained. Ferromagnetic materials can possess large permanent magnetizations even in absence of applied field Large +ve χm, depends on applied field intensity, relation between M and m0H highly non-linear. At sufficiently high fields, M saturates. Ferromagnetism occurs below a critical temperature called Curie temperature, Tc. At T>Tc, ferromagnetism is lost and material becomes paramagnetic. Origin: the quantum mechanical exchange interaction between the constituent atoms that results in regions of the material possessing permanent magnetization; All atomic magnetic moments have been aligned parallel to each other. Example: In neutral atoms of Fe, Co and Ni which are the 3d transition elements, spins of 3d electrons of adjacent atoms align parallel by spontaneous magnetization. Cf: ferroelectrics Ferromagnetism

20. Microscopic region of a ferromagnetic crystal called magnetic domain For all magnetic domains, the magnetic moments have been aligned in the same direction, parallel to each other, and thus it has a net magnetization M. The parallel alignment of magnetic dipoles of atoms is due to +ve exchange energy. Temperature dependence of ferromagnetism

21. Antiferromagnetic materials such as chromium have magnetic ordering in which the magnetic moments of alternating atoms in crystals align in opposite direction Small but +ve χm, and have no magnetization in absence of field. Occurs below a critical temp called Neel temperature TN. Above TN, antiferromagnetic material becomes paramagnetic. Antiferromagnetism

22. In some ceramic materials (ferrites eg Fe3O4), different ions have different magnitudes for their magnetic moments. When these magnetic moments align in an antiparallel manner, there is a net moment in one direction (ferrimagnetism) All A atoms have spins aligned in one direction while all B atoms have spins aligned in opposite direction. As magnetic moment of A atom is greater than B atom, net M exists in crystal. Unlike antiferromagnetic case, oppositely directed moments have different magnitudes and do not cancel. Crystal possess magnetization even in absence of field. Ferimagnetism occurs below a critical temp called Curie temp Tc. At T>Tc, ferrimagnetism is lost and material becomes paramagnetic. Ferrimagnetism

23. Types of Magnetic Materials

24. Maximum magnetization in a ferromagnet when all magnetic moments have been aligned is called saturation magnetization Msat or Ms. As temp T is increased, lattice vibrations become more energetic, leading to frequent disruption of alignments of spins. Spins cannot align perfectly with each other as T increases due to lattice vibrations randomly agitating individual spins. Ferromagnetic behavior disappears at critical T called Curie temperature TC when thermal energy of lattice vibrations in crystal is sufficient to destroy spin alignments. Above TC, crystals behaves as if it were paramagnetic. Msat decreases from its maximum value Msat(0) at absolute temperature to zero at TC, as shown in Fig for iron. When a ferromagnetic is cooled from T>TC, ferromagnetic domains reform and material become ferromagnetic again. Saturation Magnetization and Curie Temperature Saturation magnetization versus reduced temperature

25. Magnetic Materials • Pre-requisites • Magnetization and Susceptibility • Type of Magnetism • Ferromagnetic Domains • Soft and Hard Materials

26. Ferromagnetic Domain Types of Energy Domain Walls Magnetostrictive Energy Origin of Domains Motion of Domain Walls Coercivity and Hysteresis Ferromagnetic Domain

27. Ferromagnetic domains: A typical magnetic material consists of many small regions called magnetic domains. A magnetic domain is a region of crystal in which all the spin magnetic moments are aligned to produce a magnetic moment in one direction only. The domain represents a saturated region of magnetization. The moment in one domain will not be parallel to the moment in a neighbouring domain. Net magnetization of a sample represents the integrated effect of these elemental domains. Domain structure/configuration is a natural consequence of various major contributions to total energy of a ferromagnetic body. Ferromagnetic Domains Ferromagnetic material Co Thin Film

28. Origin of Domains • A single crystal of iron does not necessarily possess a net permanent magnetization in the absence of an applied field, if a magnetized piece of iron is heated to a temperature above its Curie temperature and then allowed to cool in the absence of a magnetic field, it will possess no net magnetization. • The reason for the absence of net magnetization is due to the formation of magnetic domains that effectively cancel each other. Which domain configuration will this ferromagnet assume?

29. In (b): Magnetostatic energy is reduced by about one-half by dividing the crystal into two domains magnetized in opposite directions. One domain wall (Bloch wall) is created, and potential energy increases due to the domain wall energy The creation of domain is spontaneous if the overall energy is reduced. In (c) and (d) Magnetostatic energy can be further reduced by eliminating external field lines by closing the ends with sideway domains (closure domains) with magnetizations at 90º Potential energy in walls increased due to additional walls. The creation of magnetic domains continues spontaneously until the overall potential energy reduction in creating an additional domain is the same as the increase in creating an additional wall. Size, shape and distribution of domain depend on factors such as size, thickness and shape of whole specimen, which are related to different energy terms.

30. Landau and Lifshitz pointed that domain structure of ferromagnetic material is a natural consequence of the various contributions to the energy. The most stable structure is attained when overall potential energy of materials is minimum. The total energy is sum of contributions from Magnetostatic energy Exchange energy Magnetocrystalline anisotropy energy Domain wall energy Magnetostrictive energy Types of Energy

31. Tendency for neighbouring atomic dipoles to line up parallel or anti-parallel to each other is called exchange. Potential energy within domain is minimized when all atomic dipoles align in one direction, giving a small exchange energy. However, even though potential energy within domain is minimized, external potential energy is increased by formation of external magnetic field. MagnetostaticEnergy Exchange Energy • Magnetostatic energy is the potential energy stored in an external magnetic field. • It takes external work to establish a magnetic field, and this energy is said to be stored in the magnetic field. • Magnetostatic energyof a ferromagnetic material is energy produced by its external field. • This potential energy can be minimized by domain formation, as in eg discussed. • Multiple domain formation reduces the magnetostatic energy of a unit volume of material.

32. Magnetocrystalline anisotropy refers to a preferred direction of magnetization when no fields are applied to a magnetic material. The direction of magnetization points in certain directions called easy axes. Magnetocrystalline anisotropy energy refers to atomic dipoles preferring to point in certain crystallographic directions in the crystal lattice. Examples: in Fe, magnetocrystalline anisotropy energyis lowest when the magnetization points along a [100] direction; while in Ni the easy axes lie along the [111] direction as Ni is fcc. However, in polycrystalline material magnetocrystalline anisotropy energy is independent of direction of magnetization. Magnetocrystalline Anisotropy Energy

33. A domain wallis the boundary between two domains whose overall magnetic moments are at different orientations. Analogous to grain boundary where crystal orientation changes from one grain to another. However, grains change orientation abruptly over distances of a few atoms wide but domain wall does not as this would mean two neighbouring spins at large angles to each other and possessing excessive exchange interaction. A domain changes orientation within a domain boundary about several hundreds atoms wide, of the order of 0.1 μm. The following figure shows a schematic drawing of a domain boundary of 180º change in magnetic moment dir that takes place gradually across the boundary. Domain Wall Energy

34. Large width of domain wall is due to balance in exchange energy and magnetocrystalline anisotropy energy. A wide wall will allow a gradual reorientation of the spin moments with small difference in orientation between dipoles  exchange forces between dipoles minimized and exchange energy reduced. Thus exchange energy tends to widen domain wall. However, wider wall means greater number of dipoles forced to lie in direction different from those of easy magnetization and magnetocrystalline anisotropy energy will increase. Trade-off between these two effects gives a equilibrium domain wall which is achieved when the sum of exchange and magnetocrystalline anisotropy energies is a minimum. Domain wall energy is the excess energy in the domain wall as a result of the neighboring spin magnetic moments of atoms through the wall region.

35. When a sample is small (<10 nm for Fe), it is no longer energetically favorable for domain walls to form. This is because: Magnetostatic energy in a uniform field is proportional to volume of sample, while the domain wall energy varies with cross-section area. As particle size decreases, the surface-to-volume ratio increases rapidly and hence it is energetically unfavorable to form domain walls and sample exists as single domain and hence always magnetized. Critical dimension below which single-domain behavior occurs is of the order of a domain wall width. The dominant industrial and commercial applications of ferromagnetism are in magnetic recording devices, where the magnetic material is in the form of single domain particles or regions. An ideal single domain particle is a fine particle, usually elongated, that has its magnetic moment directed towards one end or the other end, as 0 and 1 in digital recording patterned media Single Domain Particles Co single domain animation

36. If we apply stress to a ferromagnetic crystal along a certain direction, we change the interatomic spacing not only along this direction but also in other directions and hence change the exchange interactions between atomic spins. Conversely, magnetization of crystal generates strains or changes in physical dimensions of crystal. (cf: piezo electric effect.) Longitudinal strain (Δl) along direction of magnetization, length l is called magnetostrictive constant: May be +ve (extension) or –ve (contraction) The crystal lattice strain energy associated with magnetostriction is called magnetostrictive energy. Magnetostrictive Energy

37. Assume each magnetic dipole has the same strength, then the north pole of one dipole would cancel out the south pole of the dipole above it. At the top (bottom) surface, there will be north poles that are not canceled out since there are no dipoles above (below) it. Note that poles are not created on surfaces parallel to M. Hence magnetic poles are created on the surface of the magnet by an abrupt change in magnetization at top and bottom surfaces. Uncompensated poles at top and bottom of magnet will produce the magnetic field shown in the center fig, opposite to the direction of magnetization. This demagnetizing(demag) field exists both inside and outside the magnet. Since the magnetization is upward, the atomic dipoles are pointing upwards and would like to rotate in the H field to the downward direction, trying to demagnetize the magnet, hence its name. Demagnetizing Field/Energy Net B inside magnet still upward

38. Note that the demag field is not uniformwithin the magnet. Uniform demag field exists in an ellipsoidal shape magnet, being the only shape which produces a uniform demag field. However, most geometries that occur in practice are close to ellipsoidal in shape: spheres, long rods, flat plates are all special cases of an ellipsoid. • For a uniform demag field Hd, the field is proportional to and opposite in direction to the magnetization: • where N is the demagnetizing factor. The ‘-‘ sign denotes that the field is opposite to M.

39. For the given ellipsoidal shape, demag field depends upon direction of magnetization. Assume magnetization along x, y, z axes to be Mx, My, Mz. The corresponding demagnetizing factors Nx, Ny, Nz along these directions are shown in fig 3.18. Eg. the demag field in x direction is –NxMx. • The demag factors sum to unity as • Imagine the ellipsoid to be stretched in the x direction to make a long thin rod, then in the limit the demag factors are Nx=0, Ny=0.5, Nz=0.5 (Ny=Nzby symmetry). In this case since the magnetic charge is only at the end of the rod and the ends are very far away from most of the rod, the demag field in the x direction must be zero in the limit. Lack of a demag field in this direction makes it easier to magnetize the ellipsoid in this direction (for H applied in the positive x direction). • Shape anisotropy --Creation of demag field expends energy since there is an energy associated with any magnetic field. Hence shape of sample is creating a magnetic anisotropy. Lowest energy direction for magnetization is called easy axis of magnetization. • Eg. for the ellipsoid, the easy axis due to shape anisotropy will lie in the horizontal direction since that is the direction that minimizes the demag field and demag field energy.

40. Summary: • The domain configuration of magnetic sample depends on the interplay of many terms mentioned in this section such as crystal orientation, size/shape (shape anisotropy), even on deposition procedure (field induced anisotropy). • The majority of the magnetic materials are polycrystalline and consist of many grains of various sizes and orientations depending on the preparation and thermal history of the component. • Domain structure in each grain will depend on the size and shape of grain and, to some extent, on the magnetization in neighboring grains. • Overall, the structure will possess no net magnetization, provided that it was not previously subjected to an applied magnetic field and assume that the sample was heated to a temperature above Curie point and then allowed to cool to room temperature without an applied field.