Chapter 1: Introduction
1.4 Magnetic Orderings
where, S1
rand S2
r represent the spins of two interacting magnetic ions and D
r is DM vector.
Vector D
r vanishes when the crystal field has inversion symmetry with respect to the middle point connecting the two magnetic ions. Usually D
r lies either parallel or perpendicular to the line connecting the two spins, depending on the symmetry. The DM interaction tries to align
S1
r and S2 r
at right angle to each other in a plane perpendicular toD
r. Its effect is therefore very often to cant (i.e. slightly rotate) the spins by a small angle. It commonly occurs in antiferromagnets where the canted spins give rise to a small FM component of the moments perpendicular to the spin-axis of the antiferromagnet. Such DM interaction leads to the appearance of weak ferromagnetism or canted antiferromagnetism. Some of the AFM crystals like α-Fe2O3, Cr2O3, MnCO3, CoCO3, etc., exhibit such weak FM behavior [28,36].
The weak FM behavior in rare earth orthochromites arises mainly due to the DM interaction and it plays an important role in explaining some of the anomalous magnetic properties observed in orthochromites.
property exhibited by all materials. Some of the examples of diamagnetic materials are inert gases, metals like Zn, Cu, Au, Ag etc, most nonmetals like Si, C, Ge and many organic compounds [36].
1.4.2 Paramagnetic Materials
In paramagnetic materials, the atoms or ions have a net magnetic moment due to the presence of unpaired electron in partially filled orbitals. However, at room temperature, in the absence of any external magnetic field, the atomic moments are randomly oriented due to the thermal agitation and it gives rise to zero magnetization. The application of an external field prompts the magnetic moments to partially align along the direction of the applied field, resulting a net positive magnetic moment and positive susceptibility. In addition, the partial alignment along the external magnetic field is reduced with the increase in temperature due to randomization of spins. This indicates that the susceptibility of a paramagnetic material shows temperature dependence. Based on the semi-classical treatment the susceptibility of a paramagnetic material is governed by Curie’s law and is given as [28]
2 0
3 B N k T
µ µ
χ
= (1.3)where, N is the number of magnetic dipoles (magnetic ions) per unit volume each having a magnetic moment of µ. The above equation shows that the paramagnetic susceptibility follows an inverse relation with the temperature.
In quantum mechanical treatment, the paramagnetic susceptibility for low field can be described as,
( )
2 2
0 1
3
B B
N g J J k T
µ µ
χ
= + (1.4)where,
( ) ( )
( )
1 1
3
2 2 1
S S L L
g J J
+ − +
= + +
The constant g is known as Lande’s g factor and L, S and J represent orbital, spin and total angular momentum of the magnetic element respectively.
Some diatomic gases like O2, NO; ions of transition elements, rare earth metals, salts and oxides of rare earth are some of the examples of strong paramagnetic materials [28].
1.4.3 Ferromagnetic Materials
Ferromagnetic materials consist of ordered regions known as domains where the magnetic moments are oriented in a one direction giving rise to large finite magnetization even in the absence of a magnetic field. The parallel alignment of the magnetic moments is due to the presence of a molecular field which is strong enough to magnetize the material even in the absence of an external field. The phenomenon of ferromagnetism is observed below a critical temperature known as Curie temperature (TC) above which the ferromagnetic materials generally behave like a paramagnet, and their susceptibilities follow the Curie- Weiss law [28],
C C
χ
T=
θ
− (1.5) where, C and θC are Curie constant and Curie temperature, respectively. Here C =µoNg2µB2J (J+1)/3kB or C = µoNµ2/3kB. Typical θC values for common ferromagnetic materials like Fe, Ni and Co are 1043 K, 1394 K and 631 K, respectively [28].
1.4.4 Antiferromagnetic Materials
In antiferromagnetic materials the exchange interaction is negative as a result the molecular field is oriented in such a way that the nearest neighbour magnetic moments prefer to align antiparallel to each other giving rise to zero net magnetization. Antiferromagnetism is observed below a certain temperature known as Néel temperature (TN), and above which the material becomes normally a paramagnet. The magnetic susceptibility in the paramagnetic state (i.e. for temperatures, T > TN) for an antiferomagnetic material can be written as,
N
C T T
χ
=+ (1.6) Examples of AFM materials include metallic chromium, manganese, various transition metal oxides such as MnO, FeO, Cr2O3, peovskites such as BiFeO3, GdCrO3, SmCrO3 etc.
In antiferromagnetism, equal number of up and down spins on a lattice can be arranged in a large number of ways. Some selected arrangements of spins in perovskites are shown in Fig. 1.7. In A type AFM ordering, in each lattice plane (001) the magnetic spins are parallel to each other but they are opposite to adjacent planes leading to net AFM as shown in Fig. 1.7 (a). LaMnO3 falls in this category. In C type ordering the atoms in the (110) and (11'0) planes are ferromagnetically aligned such that net coupling is AFM. G type ordering is the most common type of ordering where each atom is coupled AFM to all its nearest neighbours. It is found in SmCrO3, GdCrO3, LaCrO3 and LaFeO3, etc [25]. The E type ordering represents a zigzag magnetic structure which does not contain any obvious planes of ferromagnetism.
Figure 1.7: Different types of antiferromagnetic order in a magnetic cell. The up and down arrows represent the orientation of the spin. Adapted from [37].
1.4.5 Ferrimagnetic Materials
Ferrimagnetism is a special case of antiferromagnetism which arises when the magnetic moment of the two sublattices are opposite in direction but not equal in magnitude leading to the presence of a net magnetization in the material. Due to the difference in the molecular field of each sublattice, the temperature dependences of spontaneous magnetization of the sublattices differ from each other. As a result, one sublattice magnetization dominates over other in certain temperature region and vice versa in some other temperature region. Such temperature dependence and the competition between the magnetization of two sublattices give rise to state where the net magnetization reduced to zero at a certain temperature known as the compensation temperature (Tcomp) below which the magnetization becomes negative. Examples of ferrimagnetic materials are Fe3O4, Fe2O3
CoFe O , NiFeO , Y Fe O , etc [28].