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Home , Ferrimagnetism

PHYSICS 2800 – 2nd TERM Outline Notes (continued)

Section 4. Magnetic Properties (see also textbook chapter 16)

The properties to be considered are  “weak” magnetic materials (paramagnets, diamagnets);  “strong” magnetic materials (ferromagnets, antiferromagnets, ferrimagnets);  applications of strong (e.g., in magnetic recording, computer memory storage, fast switching, etc.);  relation to (the topic of section 5).

4.1 Basic definitions and concepts (see also textbook section 16.2)

In electricity we have electric charges (electric monopoles) q, and an electric dipole can be formed from + q and −q close together. In we have magnetic dipoles , but no magnetic monopoles under normal conditions. A magnetic dipole is equivalent to a small current loop (it produces an equivalent distribution of lines).

Magnetic field lines of force around a current loop and a bar .

We can define two different magnetic field vectors H and B as follows:-

(a) The magnetic field H generated by a cylindrical coil depends on the current I, the number of turns N, and the coil length l. The magnetic induction B0 = µ0 H in a vacuum is shown.

(b) The magnetic induction B inside a is defined by B = µ H.

40 H is usually defined first in terms of an electric current flowing along a wire. For example, if we take a long solenoid of length l , N turns, and current I, the field inside the solenoid is NI H = l This is just a special case of Ampere’s law, which says that “The line integral (or sum) of H round a closed path = The total current enclosed” Therefore H l = NI , provided we assume no field outside the solenoid and ignore end effects.

The units of H are Amp/m. It is called the magnetic field strength (or intensity ). It is the same for whatever medium is filling the solenoid.

We define another field quantity B (called the magnetic induction ) to represent the resultant internal field within a material that is subjected to a H field. In general we write B = µ H where µ is a property of the material called the permeability . The units of B are Tesla (T), basically kg/s-C, and so the units of µ can be deduced.

In the case of a vacuum we write B0 = µ0 H -7 2 where µ0 = permeability of vacuum = 4 π × 10 kg-m/C

It is convenient to define a few other quantities from the above definitions:- The difference between B0 (in vacuum) and B (when a material is present) is also used to define the M in the material: B = B0 + µ0 M = µ0 (H + M) M is the per unit volume in the material. It has the same dimensions (and units) as H.

Some other relations are to write: M = χm H where χm is called the and B = µ H = µ0 µr H where µr is called the relative permeability Both χm and µr are dimensionless numbers. Substituting into B = µ0 (H + M ) gives µ0 µr H = µ0 (H + χm H) , which simplifies to µr = (1 + χm)

What is the origin of the magnetic moments in materials? In other words, what makes B different from the vacuum B0?

Basically, it comes from more current loops (but this time on an atomic scale or smaller):- (a) Electrons may have an orbital motion around the nucleus (orbital angular momentum → like a small current loop → magnetic moment) (b) Electrons also have an intrinsic spin (spin angular momentum) → also gives a magnetic moment We add these together to get the total contribution from each electron, and then add again for all the electrons in the . Usually this is nonzero (except in a few cases where there is cancellation).

41 The basic unit on the atomic scale is taken as that due to the electron spin: it is called the Bohr magneton and denoted by µB. -24 2 µB = 9.27 × 10 A-m (from QM, it is e h /2m , where m = electron mass).

In “ weak ” magnetic materials, χm << 1 in magnitude, but can be either positive or negative (so µr is close to 1). In “strong ” magnetic materials, χm is of order unity or more (sometimes much more), so µr >> 1 typically.

4.2 Weak magnetic materials (see also textbook section 16.3)

There are two cases: χm is small and negative ( µr is slightly less than 1) → χm is small and positive ( µr is slightly greater than 1) →

Diamagnetism This is observed in materials when: the magnetic moment of each atom = 0

This occurs if the contributions from all the different electrons add up (vectorially) to zero, e.g., because of complete shell (as in He, Ne, etc) or just because of cancellation (as in Si, Hg, etc).

When a field H ≠ 0 is applied, the electron orbits distort slightly because electrons experience a Lorentz force. This changes the current loops and induces a small magnetic moment (in the opposite direction to H):-

The atomic dipole configuration for a diamagnetic material with and without a magnetic field H. Without H, no dipoles exist; with H, dipoles are induced and are aligned opposite to the field.

∴ If H > 0, it implies that induced M < 0, and since B = µ0 (H + M ) it implies B < µ0 H. −5 Hence χm < 0 (typically around −10 ) and µr < 1 (but only slightly).

In principle, the diamagnetic effect occurs in all materials, but it is so small that it is only observed when other magnetic effects are zero (i.e., when the atomic magnetic moments are zero).

Paramagnetism This occurs when  The magnetic dipole moment of each atom is nonzero, and  There are negligible magnetic interactions between neighbouring .

When H = 0 the dipole moments are pointing randomly in all directions and cancel out, so M = 0.

42 When H ≠ 0 the dipole moments preferentially line up in the direction of H (in order to reduce their potential energy, which is given by −M⋅⋅⋅H). ∴ M > 0 (with magnitude depending on the strength of H).

The atomic dipole configuration for a paramagnetic material with and without a magnetic field H.

−5 −2 Hence χm > 0 (typically 10 to 10 ) and µr > 1 (slightly).

Some examples of room-temperature susceptibilities for diamagnetic and paramagnetic materials:

Schematic representation of B vs H for diamagnetic, paramagnetic and ferromagnetic materials.

4.3 Strong magnetic materials (see also textbook section 16.3)

These occur when • The magnetic dipole moment of each atom is nonzero, and • There are strong magnetic interactions between neighbouring atoms (by contrast with paramagnetism).

The interactions are QM in origin (and are called exchange interactions ):

43 • They depend on the overlap of wave functions on neighbouring atoms • They give a potential energy contribution that makes neighbouring magnetic moments align either parallel (like ↑ ↑) or antiparallel (like ↑ ↓)

There are three important types of strong magnetic behaviour:-

Ferromagnetism : this is when all atoms have the same magnitude of magnetic moment and the interactions favour parallel alignment (even when H = 0) → Large M ≠ 0 : same as above but the interactions favour antiparallel alignment (even when H = 0) → M = 0 (by cancellation) Ferrimagnetism : this occurs in compounds where there are two (or more) types of atoms having different magnetic moments, and the interactions favour antiparallel alignment of some of these (even when H = 0) → M ≠ 0 (only partial cancellation), but not as large as in .

Ferromagnetism

6 M is large, so that typically M >> H, i.e. χm >> 1 (can be up to about 10 ), and so B = µ0 (H + M ) ≈ µ0 M This can occur even when H = 0 (unlike both diamagnetism and paramagnetism).

Schematic illustration of the mutual alignment of atomic dipoles for a ferromagnetic material, which will exist even when H = 0.

The max value of M in any material occurs when all the atomic dipoles in the solid line up parallel − this is called the magnetization Ms The corresponding value of B is denoted by Bs (≈ µ0 Ms).

Example : Calculate Bs and Ms for Ni given that the magnetic moment of each atom is 0.6 µB and the density of Ni is 8.9 g/cm 3.

Answer : Ms = max magnetic moment/unit volume = 0.6 µBN where N = no. of atoms/unit volume (m 3) But N = ρNi NA /ANi , where ρNi = density, NA = Avogadro’s number, and ANi = atomic weight (8.9× 106 g/m 3 )(6 × 10 23 atoms/mol) ∴ N = = 9.1 × 10 28 atoms/m 3 58.7 g/mol -24 28 5 so Ms = 0.6 × 9.27 × 10 × 9.1 × 10 A/m = 5.1 × 10 A/m -7 5 Then Bs = µ0Ms = 4 π × 10 × 5.1 × 10 Tesla (T) = 0.64 T

Other common ferromagnets are Fe and Co (and metallic alloys).

44 Antiferromagnetism

These materials have a high degree of magnetic order (but antiparallel arrangement of nearest neighbours, because the interaction has the opposite sign) but M = 0 due to cancellation.

Occurs in many insulating compounds and ceramics. An example is MnO where the Mn 2+ have canceling magnetic moments and the O 2− ions have zero magnetic moment:

Schematic representation of antiparallel alignment of spin magnetic moments for antiferromagnetic MnO.

Ferrimagnetism

Again there are antiparallel alignment of some magnetic moments in this case, but the cancellation effects are only partial and so M is nonzero.

Some of the most common ferrimagnets are: Cubic ferrites M Fe 2O4 , where M is a metal e.g. Fe 3O4 (when M = Fe), which is M3Fe 5O12 , where M is a rare earth such as Sa, Eu, Gd, Y, etc e.g. Y 3Fe 5O12 , which is yttrium iron (or YIG). For example, the behaviour in Fe 3O4 is like:

Schematic diagram showing the spin magnetic moment configuration for the Fe 2+ 3+ and Fe ions in Fe 3O4.

To summarize, the different types of strong magnetic materials have magnetic moments aligned schematically as:-

Ferromagnetism Antiferromagnetism Ferrimagnetism

45 4.4 Temperature dependence and domains (see also textbook sections 16.4 − 16.8)

Temperature dependence

In ferromagnets and ferrimagnets, M decreases as T is increased, because thermal effects reduce the magnetic order caused by the exchange interactions. The M value starts at Ms at low T (with T ≈ 0) and steadily decreases and becomes zero (if H = 0) at a transition temperature TC (called the ). For T > T C the material becomes paramagnetic.

Plot of saturation magnetization Ms vs . T for Fe and Fe 3O4.

In antiferromagnets, M is always zero because there is the good antiparallel ordering at low T. This ordering is reduced by thermal fluctuations as T is increased. Again it disappears at a transition temperature TN (called the Neel temperature). For T > T N the material becomes paramagnetic.

Domains

These are small regions of uniform magnetic alignment (in ferromagnets and ferrimagnets). Their size depends very much on the type of material (including whether it is in a field H of given strength) and on the temperature − sizes might often be of ~ µm or mm − domains are separated by small boundary regions (domain walls) where there is a transition from one direction of M to another.

Schematic pattern of domains in a ferromagnetic (or ferrimagnetic) material, with arrows representing the atomic magnetic dipoles. Within each domain all the dipoles are aligned parallel, but this direction varies between different domains.

46 Example of a domain wall showing a gradual change of magnetic dipole orientation in the wall region.

Magnetization process and hysteresis

Suppose we start from an initially unmagnetized piece of ferromagnet (i.e., one where all the domains are random and cancel out, so B =0 when H = 0).

If H is gradually increased, the domains pointing roughly in the direction of H tend to grow in size. Those in the “wrong” direction shrink. So B increases as H increases, until there is saturation ( B = B s) when H is large enough to make all domains parallel.

The B vs. H behavior for a ferromagnetic (or ferrimagnetic) material that was initially unmagnetized. Domain configurations during several stages of magnetization are shown.

If H is now gradually reduced, B becomes smaller but does not necessarily follow the original magnetization curve (because the domains do not form again exactly as before) − this effect is known as hysteresis .

By cycling H from large positive to large negative values, the B versus H plot traces out a closed loop − known as the hysteresis loop .

47 B vs. H for a ferromagnetic material that is taken through forward and reverse saturations (points S and S’). The hysteresis loop is represented by the solid curve, while the dashed line represents the initial magnetization process.

Note the definitions of Br (the value of B when H is reduced to zero) and Hc (the magnitude of H required to make B = 0).

Hard and soft magnetic materials

“Hard” − large area hysteresis loop, large Br and Hc. “Soft” − small area hysteresis loop, small Hc.

Hard materials are suitable for applications as permanent magnets (e.g., tungsten steel).

Soft materials are suitable for applications where the magnetic state is constantly being changed, as in high-frequency magnetic devices, switches, transformer cores, etc. (e.g. soft iron, Si-iron, Fe/Ni alloys, etc).

It can be shown that the work needed to take a material through a complete hysteresis cycle is proportional to the area of the hysteresis loop. ∴ A small area loop is an advantage in soft materials, as is small Hc.

Magnetic information storage

The domain structure can be used for information storage (e.g. in computer memories or magnetic recording). The basic idea is that some magnetic materials (particularly when grown as a thin film) have just two stable directions for the domain (say ← and →). These can be used to represent “0” and “1” in the binary code. An external circuit (a “recording head” can be used to read or write the information). Schematically:-

More recently it has become advantageous to use films magnetized in the perpendicular direction (rather than the parallel orientation shown here).

Arrays of artificially produced magnetic nanometer-sized elements (such as nanowires) are being studied as a way to achieve higher- density magnetic memories.

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