⊙ Theories of ordered magnetism (What types of ordered magnetic structures exist and how do they differ?) •Different types of magnetic order in solids including ferromagnetism antiferromagnetism ferrimagnetism helimagnetism • Some materials (heavy rare earth), exhibit more than one ordered magnetic states. • Ferromagnetism T > Curie T paramagnetic
Antiferromagnetism T > Neel T paramagnetic
•Some solids( Tb, Dy, Ho) have Curie T Neel T
⊙Ferromagnetism (What cause the transition from paramagnetic to ferromagnetism?)
•In ferromagnetic solid at T < Tc, the magnetic moments within domains are aligned parallel. (This can be explained by the Weiss interaction field.) •Transition metal ferromagnets o o o Fe, Tc= 770 C, Ni, Tc= 358 C, Co, Tc= 1131 C •Rare earth metals
Gd, Tc=293K, Dy, Tc=85K, Tb, Tc=219K, Ho, Tc=19K
Er, Tc=19.5K, Tm, Tc=32K
1 • The alignment of magnetic moments in various ordered ferromagnetic solid.
• At a critical T, the randomizing effect of thermal energy overcomes the aligning effect of the interaction energy, and above this T, the magnetic state becomes disordered.
2 Ni
Fe
Co
3 ⊙ Weiss theory of ferromagnetism (How can the Weiss interaction be used to explain magnetic order in ferromagnets? ) • If the unpaired electronic magnetic moments which are responsible for the magnetic properties are localized on the atomic sites. • The interaction between the unpaired moments, leads to the existence of a critical T be low which the thermal energy of the electronic moments is insufficient to cause random paramagnetic alignment.
• The effective field He can be used to explain the alignment of magnetic moments within domains for T < Tc. • Theme for the interatomic interaction (exchange field): (1) the mean-field approximation -> used for the paramagnetic region (2) a nearest-neighbor interaction-> used for the ferromagnetic region
• Suppose that any atomic magnetic moment mi experiences an effective field Heij due to another moment mj. ( If we assume that this field is also in the direction of mj) Jij m Heij j
• The total exchange interaction field at the moment mi will be the vector sum of all interaction with other moments. Hei Jijm j all j
4 ⊙Mean-field approximation (Is there a simple explanation of the Weiss interaction?)
• If the interactions between all moments are identical and hence
independent of displacement between the moments, then all of the Jij are equal. He m j all j
• within a domain: He α(Ms mi )
≒ αMs
• The interaction energy of the moment under these conditions:
Ee 0mi He 0αmi Ms the original formulation of the Weiss theory
5 • If we consider the case of a zero external field, then the only field operating within a domain will be the Weiss field
Htot = He
• If we apply the mean-field model, the interaction field will be
proportional to the spontaneous magnetization Ms within a domain.
• Following an analogous argument to that given by Langevin for paramagnetism
M mM k T s coth( 0 s ) B M0 kBT 0mMs
• As T ↑, the spontaneous M within a domain ↓.
• The energy of a moment within a domain can be generalized to include the effect of a magnetic field H as follows. E 0m(H Ms )
6 • Magnetization within a domain
M m(H M ) k T s coth[ 0 s ] B M0 kBT [0m(H Ms )]
• This eq. is not encountered very often, because αMs >>H in 6 8 ferromagnet (in iron, Ms = 1.7x10 A/m, so αMs can up to 6.8x10 A/m, while H will rarely exceed 2x106 A/m)
• Consequently within the body of a domain, the action of the H field is not very significant when compared with the interaction field.
• Moderate magnetic field (H ≒8x103 A/m) can cause significant changes in the bulk magnetization M in ferromagnet.
• These changes occur principally at the domain boundaries where the exchange interaction is competing with the anisotropy energy to give an energy balance.
• Under these conditions, the additional field energy can just tip the balance and result in change in the direction of magnetic moments within the domain wall. ↓ domain wall motion
• The magnetic moment in the domain wall do not couple to the spontaneous magnetization of the domain.
7 ◎ Nearest – neighbor interactions ( Can the Weiss model be interpreted on the basis of localized interaction only? ) • In the nearest – neighbor approximation, the electronic moments interact only with those of its Z nearest neighbors.
• For a simple cubic lattice, Z = 6 body – centered cubic, Z = 8 face – centered cubic, Z = 12 hexagonal lattice, Z = 12
• The nearest – neighbor approach is particularly useful for considering magnetic moment in the domain wall.
• In this case, the moments do not couple to the magnetization within the body of the domain simply because they lie between domains with different magnetic directions and the direction of magnetization changes within the wall.
8 • In this approximation, the exchange interaction field
H e Jij m j neares neighbor
• We assume that each nearest – neighbor interaction is identical and equal to J.
• J = 0, non – interacting J ≠ 0, each moment interacts equally with each of its nearest neighbors.
He J m j J m j nearest mearest neighbor neighbor
9 Corresponds to J > 0 ferromagnetic alignment J < 0 antiferromagnetic alignment
• Interaction energy of magnetic moment Ee μom J m j nearest neighbor • Summing over the Z nearest neighbor
2 Ee μo Z J m
• Weiss interaction it is possible to provide a description of ferromagnets which is similar to the Langevin model of Paramagnetism.
• Weiss interaction model is only correct for ferromagnets in which the moments are localized on the atomic cores.
• Thus it applies to La series, because 4f electrons are tightly bound to the nuclei. The model also works reasonably well for Ni, which obeys the curie – Weiss law.
10 ⊙ Weiss mean field theory (What is the underlying cause of the alignment of atomic magnetic moments?)
• In the original Weiss theory, the mean field was proportional to the
bulk magnetization M so that He = αM (α: the mean field constant)
• This is assumed that each atomic moment interacts equally with every other atomic moment within the solid.
• This was found to be a viable assumption in the paramagnetic phase because due to the homogeneous distribution of magnetic moment directions.
• However in the ferromagnetic phase the magnetization is locally inhomogenous on a scale larger than the domain size due to the variation in the direction of magnetization from domain to domain.
• Therefore, the idea of a Weiss mean field is applied only within a domain, arguing that the interaction between the atomic moments decayed with distance.
• It is generally considered that the Weiss field is good approximation to the real situation within a given domain because within the
domain, the magnetization is homogeneous and has a value Ms.
• The interaction field which is responsible for the ordering of
moments within domains can be expressed He= αMs
Ms: the spontaneous magnetization within the domain
11 • Ms ≈ the saturation magnetization at 0K, but decreases as the T↑ • Ising model applied to ferromagnets, based on interaction fields only between the nearest neighbor.
• When α > 0, the ordering of moments within a domain is parallel, leading to ferromagnetism.
• When α < 0, the ordering is antiparallel leading to antiferromagnetism.
• A number of different types of magnetic order are possible depending on the nature of the interaction parameter α.
12 • Suppose the field experienced by any magnetic moment mi within a domain due to its interaction with any other moment mj is
H e ij m j • The interaction with all moments is the sum over the moments within the domain,
He ij m j • The energy of moment
E 0 mi H e
0 mi ij m j
• If the interactions with all moments are equal then all the αij are equal. Let these be α
He m j • The vector sum over all the moments within a domain gives the
spontaneous magnetization Ms
He αMs mean field 6 Ex: For iron, M0 =1.7x10 A/m (saturation magnetization), at R.T. Ms(spontaneous magnetization) ≒ M0 (saturation magnetization)
if α = 400, He = αMs = (400)(1.7x106) A/m = 6.8x108 A/m
13 ⊙energy states of different arrangements of moments
• Energy of configuration of (a) < Energy of configuration of (b)
• If we consider the exchange energy of the six moment system
the energy of any moment mi, Ei = -μ0miΣαijmj
with the mean field approximation, Ei = -μ0αmiΣmj
the total energy, Ei = -μ0αΣmiΣmj
(a) When all moments are parallel, E = -μ0α(6m)(5m) 2 = -30m μ0α
(b) When all moments are antiparallel, E = -μ0α(5m4m-m5m) 2 = -15m μ0α • The positive exchange interaction, the energy is lower when all moments are aligned parallel within the domain and hence the aligned state is preferred.
14 ◎ Energy considerations and domain patterns
• The existence of domains is a consequence of energy minimization.
• A single domain specimen has associated with it a large magnetostatic energy, but the breakup of the magnetization into localized regions ( domains ), providing for flux closure at the ends of the specimen, reduces the magnetostatic energy.
• The decrease in magnetostatic energy is greater than the energy needed to form magnetic domain walls then multi-domain specimens will arise.
Magnetostatic energy of single-domain specimens • The energy per unit volume of a dipole magnetization M in a magnetic field H is given by E -μ HdM 0
• When it is subjected only to its own demagnetizing field Hd, which is generated by M
( Hd = -NdM, Nd : the demagnetizing factor ) E μ N MdM 0 d 0 N M 2 2 d
• If M can be reduced by the emergence of domains, the magnetostatic energy will be reduced.
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