Ferromagnetism

Chapter 2 Fundamental Aspects and Theoretical Modeling

2.2. Origin of magnetism

2.2.3. Ferromagnetism

Certain materials possess a permanent magnetic moment resulting in from strong interaction between the magnetic moments even in the absence of an external field. This dominates over the thermal energy and reveals an alignment of magnetization in a particular direction.

Such behaviors are displayed by the transition metals and some of the rare earth metals. In

ferromagnetic material, there are two distinct characteristics: (1) their spontaneous

magnetization and (2) existence of magnetic ordering temperature. Spontaneous

magnetization is the net magnetization that exists inside a uniformly magnetized

microscopic volume even in the absence of external magnetic field. The magnitude of this magnetization at absolute temperature depends on the spin magnetic moments of electrons.

The atomic moments in ferromagnetic materials align either a parallel or an antiparallel arrangements showing very strong interactions, which are produced by electronic exchange forces. As a result, a large net magnetization even after removing the external applied magnetic field exists in ferromagnetic materials.

Figure 2.02: Magnetic hysteresis loop of a ferromagnetic material.

All ferromagnetic materials exhibit magnetic hysteresis loop (M-H loop) under the

application of magnetic field as displayed in Figure 2.02. By studying its hysteresis loop we

can get information about the magnetic properties of a ferromagnetic material. The loop is

generated by measuring the magnetic flux of a ferromagnetic material while the magnetizing

field is changed continuously. Ferromagnetic materials in virgin states follow the dashed

line (starting from the origin ‘o’) as applied field is increased and reach the point 'a' where

almost all of the magnetic domains are aligned to field direction and an additional increase

in the magnetizing field produces a very little or no increase in magnetic flux. The

magnetization obtained at this point is called saturation magnetization (M

S

). When the field

is reduced to zero, the curve moves from point ‘a’ to ‘b’. At this point, some magnetic flux

remains in the material even at zero magnetic field. This is called as retentivity and indicates

the remanence or level of residual magnetism in the material. As the magnetic field is

reversed, the curve moves to point ‘c’, where the magnetization reaches to zero. This point

is called as coercivity (H

C

). On further increasing the field in the negative direction,

materials become magnetically saturated but in the opposite direction (point ‘d’). Reducing

the field to zero brings the curve to point ‘e’. At this point, the level of residual magnetism

is almost equal to that achieved in the other direction (point ‘b’). Increasing the field back in the positive direction returns the magnetization to zero. Subsequently, the curve takes a different path from point 'f' back to the saturation point (point ‘a’) and thereby completing the loop. From the

M-H loop, the following magnetic parameters can be determined: (i)

Retentivity: the material's ability to retain a certain amount of magnetization when the magnetizing field is removed after achieving saturation, (ii) Coercivity: The magnitude of reverse magnetic field is required to make the magnetization to zero; (iii) Permeability: A property of a material that describes the ease with which a magnetic flux is established in the component. These hysteresis parameters are not solely intrinsic properties but are dependent on various parameters such as grain size, domain state, internal stresses and temperature. Since the hysteresis parameters are dependent on grain size, they are useful for magnetic grain sizing of natural samples. The elements Fe, Ni, and Co and their alloys are typical examples of ferromagnetic materials. Ferromagnetic materials are mainly divided into two groups (see Figure 2.03): (a) hard ferromagnetic materials which exhibit very high

HC

(> 1000 Oe). These materials are mainly used as permanent magnets and recording media for data storage and (b) soft ferromagnetic materials with low H

C

(< 100 Oe) are used for transformer core, read head and magnetic sensor applications.

Figure 2.03: Typical magnetic hysteresis loops of soft and hard ferromagnetic materials.

Another important parameter is the magnetic induction [B = μ

0

(H + M)], where μ

0

is

the permeability of free space], which is the total flux of magnetic field lines through a unit

cross sectional area of the material. From the initial magnetization curve, the initial magnetic

permeability µ

I

(= B/H), for very small applied magnetic field and maximum permeability

µmax

[=(B/H)

max

] can be obtained. These parameters indicate the amount of induction generated by the material in a given magnetic field and are useful in characterizing magnetic materials.

µI

and

HC

have a reciprocal relationship. So, materials exhibiting low

HC

necessarily have a high

µI

. When increasing temperature, a transition from ferromagnetic state to paramagnetic state occurs at a temperature called Curie temperature (T

C

) that is due to thermal energy eventually overcomes the exchange energy and produces a randomizing effect leading to paramagnetism. The phenomenon of ferromagnetic can often be described by mean field or molecular field model. The molecular field model simply assumes that all the interactions from the neighboring magnetic species can be described in terms of an effective internal or molecular field B

m

, which is proportional to magnetization (B

m

= λM, where

λ is Weiss molecular field constant). So total magnetic field experienced by each

dipole is sum of applied field B and the molecular field B

m

. The expression for magnetization can be re-written by following eqn.(2.08) as

𝑀 = 𝑁𝑔²𝜇_𝐵²𝐽(𝐽 + 1)

3𝑘𝑇 (𝐵_𝑎+ 𝜆𝑀)

(2.11)

Figure 2.04: Different type of antiferromagnetic arrangement in a unit cell.