7.3 Magnetic Domains and Magnetic Materials (B) 427
M
M
Fig. 7.21. Magnetic-domain splitting within a material
45º
Fig. 7.22. Formation of magnetic domains of closure
Fig. 7.23. Formation of more magnetic domains of closure
Hysteresis, Remanence, and Coercive Force (B)
Consider an unmagnetized ferromagnet well below its Curie temperature. We can understand the material being unmagnetized if it consists of a large number of domains, each of which is spontaneously magnetized, but that have different directions of magnetization so the net magnetization averages to zero.
The magnetization changes from one domain to another through thin but finitewidth domain walls. Typically, domain walls are of thickness of about 10−7 meters or some hundreds of atomic spacings, while the sides of the domains are a few micrometers and larger.
The hysteresis loop can be visualized by plotting M vs. H or B = μ0(H + M) (in SI) = H + 4πM (in Gaussian units) (see Fig. 7.24). The virgin curve is obtained by starting in an ideal demagnetized state in which one is at the absolute minimum of energy.
When an external field is turned on, “favorable” domains have lower energy than “unfavorable” ones, and thus the favorable ones grow at the expense of the unfavorable ones.
Imperfections determine the properties of the hysteresis loop. Moving a domain wall generally increases the energy of a ferromagnetic material due to a complex combination of interactions of the domain wall with dislocations, grain boundaries, or other kinds of defects. Generally the first part of the virgin curve is reversible, but as the walls sweep past defects one enters an irreversible region, then
428 7 Magnetism, Magnons, and Magnetic Resonance
in the final approach to saturation, one generally has some rotation of domains. As H is reduced to zero, one is left with a remanent magnetization (in a metastable state with a “local” rather than absolute minimum of energy) at H = 0 and B only goes to zero at −Hc, the coercive “force”.25 For permanent magnetic materials, MR and Hc should be as large as possible. On the other hand, soft magnets will have very low coercivity. The hysteresis and domain properties of magnetic materials are of vast technological importance, but a detailed discussion would take us too far afield. See Cullity [7.16].
Fig. 7.24. Magnetic hysteresis loop identifying the virgin curve Hc ≡ coercive “force”.
Figure 7.25 provides a convenient way to distinguish Bloch and Néel walls. Bloch walls have φ = 0, while Néel walls have φ = π/2. Néel walls occur in thin films of materials such as permalloy in order to reduce surface magnetostatic energy as suggested by Fig. 7.26. There are many other complexities involved in domainwall structures. See, e.g., Malozemoff and Slonczewski [7.44].
25 Some authors define Hc as the field that reduces M to zero.
7.3 Magnetic Domains and Magnetic Materials (B) 429
z
z
M
z
M
θ
y
ϕ
M
x
Fig. 7.25. Bloch wall: φ = 0; Néel wall: φ = π/2
wall
Fig. 7.26. Néel wall in thin film
Methods of Observing Domains (EE, MS)
We briefly summarize five methods.
1.Bitter patterns–a colloidal suspension of particles of magnetite is placed on a polished surface of the magnetic material to be examined. The particles are attracted to regions of nonuniform magnetization (the walls) and hence the walls are readily seen by a microscope.
2.Faraday and Kerr effects–these involve rotation of the plane of polarization on transmission and reflection (respectively) from magnetic substances.
3.Neutrons–since neutrons have magnetic moments they experience interaction with the internal magnetization and its direction, see Bacon GE, “Neutron Diffraction,” Oxford 1962 (p355ff).
4.Transmission electron microscopy (TEM)–Moving electrons are influenced by forces due to internal magnetic fields.
5.Scanning electron microscopy (SEM)–Moving secondary electrons sample internal magnetic fields.
430 7 Magnetism, Magnons, and Magnetic Resonance
7.3.2 Magnetic Materials (EE, MS)
Some Representative Magnetic Materials (EE, MS)
Table 7.4. Ferromagnets
Ferromagnets
Tc (K)
Ms (T = 0 K, Gauss)
Fe
1043
1752
Ni
631
510
Co
1394
1446
EuO
77
1910
Gd
293
1980
From Parker SP (ed), Solid State Physics Sourcebook, McGraw-Hill Book Co., New York, 1987, p. 225.
Table 7.5. Antiferromagnets
Antiferromagnets
TN (K)
MnO
122
NiO
523
CoO
293
From Cullity BD, Introduction to Magnetic Materials, AddisonWesley Publ Co, Reading, Mass, 1972, p. 157.
Table 7.6. Ferrimagnets
Ferrimagnets
Tc (K)
Ms (T = 0 K, Gauss)
YIG (Y3Fe5O12)
560
195
a garnet
Magnetite (Fe3O4)
858
510
a spinel
(From Solid State Physics Sourcebook, op cit p. 225)
We should emphasize that these classes do not exhaust the types of magnetic order that one can find. At suitably low temperatures the heavy rare earths, may show helical or conical order. and there are other types of order, as for example, spin glass order. Amorphous ferromagnets show many kinds of order such as speromagnetic and asperomagnetic. (See, e.g., Solid State Physics Source Book, op cit p 89.)
Ferrites are perhaps the most common type of ferrimagnets. Magnetite, the oldest magnetic material that is known, is a ferrite also called lodestone. In general, ferrites are double oxides of iron and another metal such as Ni or Ba (e.g. nickel ferrite: NiOFe2O3 and barium ferrite: BaO·6Fe2O3). Many ferrites find application in high-frequency devices because they have high resistivity and hence do not have appreciable eddy currents. They are used in microwave devices, radar, etc. Barium
7.3 Magnetic Domains and Magnetic Materials (B) 431
ferrite, a hard magnet, is one of the materials used for magnetic recording that is a very large area of application of magnets (see, e.g., Craik [7.15 p. 379]).
Hard and Soft Magnetic Materials (EE, MS) The clearest way to distinguish between hard and soft magnetic materials is by a hysteresis loop (see Fig. 7.27). Hard permanent magnets are hard to magnetize and demagnetize, and their coercive forces can be of the order of 106 A/m or larger. For a soft magnetic material, the coercive force can be of order 1 A/m or smaller. For conversions: 1 A/m is 4π × 10−3 Oersted, 1 kJ/m3 converts to MGOe (mega Gauss Oersted) if we multiply by 0.04π, 1Tesla = 104 G.
B
Hard
Soft
Hchard
Hcsoft
H
Fig. 7.27. Hard and soft magnetic material hysteresis loops (schematic)
Permanent Magnets (EE, MS) There are many examples of permanent magnetic materials. The largest class of magnets used for applications are permanent magnets. They are used in electric motors, in speakers for audio systems, as wiggler magnets in synchrotrons, etc. We tabulate here only two examples that have among the highest energy products (BH)max.
Table 7.7. Permanent Magnets
Tc (K)
Ms (kA m−1)
Hc (kA m−1)
(BH)max (kJ m−3)
(1) SmCo5
997
768
700–800
183
(2) Nd2Fe14B
~583
―
~880
~290
(1)Craik [7.15 pp. 385, 387]. Sm2Co17 is in some respects better, see [7.15 p. 388].
(2)Solid State Physics Source Book op cit p 232. Many other hard magnetic materials are mentioned here such as the AlNiCos, barium ferrite, etc. See also Herbst [7.29].
432 7 Magnetism, Magnons, and Magnetic Resonance
Soft Magnetic Materials (EE, MS) There are also many kinds of soft magnetic materials. They find application in communication materials, motors, generators, transformers, etc. Permalloys form a very common class of soft magnets. These are Ni-Fe alloys with sometimes small additions of other elements. 78 Permalloy means, e.g., 78% Ni and 22% Fe.
Table 7.8. Soft Magnet
Tc (K)
Hc (A m−1)
Bs (T)
78 Permalloy
873
4
1.08
See Solid State Physics Source Book op cit, p. 231. There are several other examples such as high-purity iron.
7.4 Magnetic Resonance and Crystal Field Theory
7.4.1 Simple Ideas About Magnetic Resonance (B)
This Section is the first of several that discuss magnetic resonance. For further details on magnetic resonance than we will present, see Slichter [91]. The technique of magnetic resonance can be used to investigate very small energy differences between individual energy levels in magnetic systems. The energy levels of interest arise from the orientation of magnetic moments of the system in, for example, an external magnetic field. The magnetic moments can arise from either electrons or nuclei.
Consider a particle with magnetic moment μ and total angular momentum J and assume that the two are proportional so that we can write
μ = γJ ,
(7.252)
where the proportionality constant γ is called the gyromagnetic ratio and equals −gμB/ (for electrons, it would be + for protons) in previous notation. We will then suppose that we apply a magnetic induction B in the z direction so that the Hamiltonian of the particle with magnetic moment becomes
H 0 = −γμ0HJ z ,
(7.253)
where we have used (7.252), and B = μ0H, where H is the magnetic field. If we define j (which are either integers or half-integers) so that the eigenvalues of J2 are j(j + 1) 2, then we know that the eigenvalues of H0 are
Em = −γ μ0Hm ,
(7.254)
where –j≤ m ≤ j.
From (7.254) we see that the difference between adjacent energy levels is determined by the magnetic field and the gyromagnetic ratio. We can induce transitions
7.4 Magnetic Resonance and Crystal Field Theory
433
between these energy levels by applying an alternating magnetic field (perpendicular to the z direction) of frequency ω, where
ω = | γ | μ0H or ω = | γ | μ0H .
(7.255)
These results follow directly from energy conservation and they will be discussed further in the next section. It is worthwhile to estimate typical frequencies that are involved in resonance experiments for a convenient size magnetic field. For an electron with charge e and mass m, if the gyromagnetic ratio γ is defined as the ratio of magnetic moment to orbital angular momentum, it is given by
γ = e / 2m , for e < 0 .
(7.256)
For an electron with spin but no orbital angular momentum, the ratio of magnetic moment to spin angular momentum is 2γ = e/m. For an electron with both orbital and spin angular momentum, the contributions to the magnetic moment are as described and are additive. If we use (7.255) and (7.256) with magnetic fields of order 8000 G, we find that the resonance frequency for electrons is in the microwave part of the spectrum. Since nuclei have much greater mass, the resonance frequency for nuclei lies in the radio frequency part of the spectrum. This change in frequency results in a considerable change in the type of equipment that is used in observing electron or nuclear resonance.
Abbreviations that are often used are NMR for nuclear magnetic resonance and EPR or ESR for electron paramagnetic resonance or electron spin resonance.
7.4.2 A Classical Picture of Resonance (B)
Except for the concepts of spin-lattice and spin-spin relaxation times (to be discussed in the Section on the Bloch equations) we have already introduced many of the most basic ideas connected with magnetic resonance. It is useful to present a classical description of magnetic resonance [7.39]. This description is more pictorial than the quantum description. Further, it is true (with a suitable definition of the time derivative of the magnetic moment operator) that the classical magnetic moment in an external magnetic field obeys the same equations of motion as the magnetic moment operator. We shall not prove this theorem here, but it is because of it that the classical picture of resonance has considerable use. The simplest way of presenting the classical picture of resonance is by use of the concept of the rotating coordinate system. It also should be pointed out that we will leave out of our discussion any relaxation phenomena until we get to the Section on the Bloch equations.
As before, let a magnetic system have angular momentum J and magnetic moment μ, where μ = γJ. By classical mechanics, we know that the time rate of change of angular momentum equals the external torque. Therefore we can write for a magnetic moment in an external field H,
dJ
= μ× μ0 H .
(7.257)
dt
434 7 Magnetism, Magnons, and Magnetic Resonance
Since μ = γJ (γ < 0 for electrons), we can write
dμ
= μ× (γμ0 )H .
(7.258)
dt
This is the general equation for the motion of the magnetic moment in an external magnetic field.
To obtain the solution to (7.258) and especially in order to picture this solution, it is convenient to use the concept of the rotating coordinate system. Let
= ˆ + ˆ + ˆ
A iAx jAy kAz
be any vector, and let ˆi, ˆj, kˆ be unit vectors in a rotating coordinate system. If Ω is the angular velocity of the rotating coordinate system relative to a fixed coordinate system, then relative to a fixed coordinate system we can show that
diˆ
= Ω× iˆ .
(7.259)
dt
This implies that
dA
=
δA
+ Ω× A ,
(7.260)
dt
δt
where δA/δt is the rate of change of A relative to the rotating coordinate system and dA/dt is the rate of change of A relative to the fixed coordinate system.
By using (7.260), we can write (7.258) in a rotating coordinate system. The result is
δμ
= μ× (Ω +γμ0 H ) .
(7.261)
δt
Equation (7.261) is the same as (7.258). The only difference is that in the rotating coordinate system the effective magnetic field is
Heff = H +
Ω
.
(7.262)
γμ0
If H is constant and Ω is chosen to have the constant value Ω = −γμ0H, then δμ/δt = 0. This means that the spin precesses about H with angular velocity γμ0H. Note that this is the same as the frequency for magnetic resonance absorption. We will return to this point below.
It is convenient to get a little closer to the magnetic resonance experiment by supposing that we have a static magnetic field H0 along the z direction and an alternating magnetic field Hx(t) = 2H′cos(ωt) along the x-axis. We can resolve the alternating field into two rotating magnetic fields (one clockwise, one counterclockwise) as shown in Fig. 7.28. Simple vector addition shows that the two fields add up to Hx(t) along the x-axis.
With the static magnetic field along the z direction, the magnetic moment will precess about the z-axis. The moment will precess in the same sense as one of the rotating magnetic fields. Now that we have both constant and alternating magnetic
7.4 Magnetic Resonance and Crystal Field Theory
435
y
|H1| = H′
|H2| = H′
H1
Hx(t)
ωt
x
H2
ωt
Fig. 7.28. Decomposition of an alternating magnetic field into two rotating magnetic fields
fields, something interesting begins to happen. The component of the alternating magnetic field that rotates in the same direction as the magnetic moment is the important component [91]. Near resonance, the magnetic moment and one of the circularly polarized components of the alternating magnetic field rotate with almost the same angular velocity. In this situation the rotating magnetic field exerts an almost constant torque on the magnetic moment and tends to tip it over. Physically, this is what happens in resonance absorption.
Let us be a little more quantitative about this problem. If we include only one component of the rotating magnetic field and if we assume that Ω is the cyclic frequency of the alternating magnetic field, then we can write
δμ
ˆ
ˆ
= μ×[Ω+γμ0
(iH
′+ kH0 )] .
δt
This can be further written as
δμ
= μ× Heff
,
δt
where now
ˆ
Ω
ˆ
H
eff
≡ k
H
0
+
+ iH ′.
γμ0
(7.263)
(7.264)
Since in the rotating coordinate system μ precesses about Heff, we have the picture shown in Fig. 7.29. If we adjust the static magnetic field so that
H0 = − Ω ,
γμ0
then we have satisfied the conditions of resonance. In this situation Heff is along the x-axis (in the rotating coordinate system) and the magnetic moment flops up and down with frequency γμ0H′.
436 7 Magnetism, Magnons, and Magnetic Resonance
Ω
ˆ
+
k H0
γμ0
z
μ
Heff
x
H′
Fig. 7.29. Precession of the magnetic moment μ about the effective magnetic field Heff in a coordinate system rotating with angular velocity Ω about the z-axis
Similar quantum-mechanical calculations can be done in a rotating coordinate system, but we shall not do them as they do not add much that is new. What we have done so far is useful in forming a pictorial image of magnetic resonance, but it is not easy to see how to put in spin-lattice interactions, or other important interactions. In order to make progress in interpreting experiments, it is necessary to generalize our formalism somewhat.
7.4.3 The Bloch Equations and Magnetic Resonance (B)
These equations are used for a qualitative and phenomenological discussion of NMR and EPR. In general, however, it is easier to describe NMR than EPR. This is because the nuclei do not interact nearly so strongly with their surroundings as do the electrons. We shall later devote a Section to discussing how the electrons interact with their surroundings.
The Bloch equations are equations that describe precessing magnetic moments, and various relaxation mechanisms. They are almost purely phenomenological, but they do provide us a means of calculating the power absorbed versus the frequency. Without the interactions responsible for the relaxation times, this plot would be a delta function. Such a situation would not be very interesting. It is the relaxation times that give us information about what is going on in the solid.26
Definition of Bloch Equations and Relaxation Times (B)
The theory of the resonance of free spins in a magnetic field is simple but it holds little inherent interest. To relate to more physically interesting phenomena it is necessary to include the interactions of the spins with their environment. The Bloch equations include these interactions in a phenomenological way.
y
