Patterson, Bailey - Solid State Physics Introduction to theory

418 7 Magnetism, Magnons, and Magnetic Resonance

Huang’s excellent text can be consulted.21 Tables 7.2 and 7.3 summarize some of the results.

Table 7.2. Summary of mean-field theory

Table 7.3. Critical exponents (calculated)

Adapted with permission from Chaikin PM and Lubensky TC,

Principles of Condensed Matter Physics, Cambridge University

Press, 1995, p. 231.

Two-Dimensional Structures (A)

Lower-dimensional structures are no longer of purely theoretical interest. One way to realize two dimensions is with thin films. Suppose the thin film is of thickness t and suppose the correlation length of the quantity of interest is c. When the thickness is much less than the correlation length (t << c), the film will behave two dimensionally and when t >> c the film will behave as a bulk threedimensional material. If there is a critical point, since c grows without bound as the critical point is approached, a thin film will behave two-dimensionally near the two-dimensional critical point. Another way to have two-dimensional behavior is in layered magnetic materials in which the coupling between magnetic layers, of spacing d, is weak. Then when c << d, all coupling between the layers can

be neglected and one sees 2D behavior, whereas if c >> d, then interlayer coupling can no longer be neglected. This means with magnetic layers, a twodimensional critical point will be modified by 3D behavior near the critical temperature.

In this chapter we are mainly concerned with materials for which the threedimensional isotropic systems are a fairly good or at least qualitative model. However, it is interesting that two-dimensional isotropic Heisenberg systems can be shown to have no spontaneous (sublattice − for antiferromagnets) magnetization [7.49]. On the other hand, it can be shown [7.26] that the highly anisotropic

two-dimensional Ising ferromagnet (defined by the Hamiltonian H ∑i,j(nn.)σizσjz, where the σs refer to Pauli spin matrices, the i and j refer to lattice sites) must show spontaneous magnetization.

We have just mentioned the two-dimensional Heisenberg model in connection with the Mermin–Wagner theorem. The planar Heisenberg model is in some ways even more interesting. It serves as a model for superfluid helium films and predicts the long-range order is destroyed by formation of vortices [7.40].

Another common way to produce two-dimensional behavior is in an electronic inversion layer in a semiconductor. This is important in semiconductor devices.

Spontaneously Broken Symmetry (A)

A Heisenberg Hamiltonian is invariant under rotations, so the ensemble average of the magnetization is zero. For every M there is a −M of the same energy. Physically this answer is not correct since magnets do magnetize. The symmetry is spontaneously broken when the ground state does not have the same symmetry as the Hamiltonian, The symmetry is recovered by having degenerate ground states whose totality recovers the rotational symmetry. Once the magnet magnetizes, however, it does not go to another degenerate state because all the magnets would have to rotate spontaneously by the same amount. The probability for this to happen is negligible for a realistic system. Quantum mechanically in the infinite limit, each ground state generates a separate Hilbert space and transitions between them are forbidden—a super selection rule. Because of the symmetry there are excited states that are wave-like in the sense that the local ground state changes slowly over space (as in a wave). These are the Goldstone excitations and they are orthogonal to any ground state. Actually each of the (infinite) number of ground states is orthogonal to each other: The concept of spontaneously broken symmetry is much more general than just for magnets. For ferromagnets the rotational symmetry is broken and spin waves or magnons appear. Other examples include crystals (translation symmetry is broken and phonons appear), and superconductors (local gauge symmetry is broken and a Higgs mode appears—this is related to the Meissner effect – see Chap. 8).22

22 See Weinberg [7.67].

420 7 Magnetism, Magnons, and Magnetic Resonance

7.3 Magnetic Domains and Magnetic Materials (B)

7.3.1 Origin of Domains and General Comments23 (B)

Because of their great practical importance, a short discussion of domains is merited even though we are primarily interested in what happens in a single domain.

We want to address the following questions: What are the domains? Why do they form? Why are they important? What are domain walls? How can we analyze the structure of domains, and domain walls? Is there more than one kind of domain wall?

Magnetic domains are small regions in which the atomic magnetic moments are lined up. For a given temperature, the magnetization is saturated in a single domain, but ferromagnets are normally divided into regions with different domains magnetized in different directions.

When a ferromagnet splits into domains, it does so in order to lower its free energy. However, the free energy and the internal energy differ by TS and if T is well below the Curie temperature, TS is small since also the entropy S is small because the order is high. Here we will neglect the difference between the internal energy and the free energy. There are several contributions to the internal energy that we will discuss presently.

Magnetic domains can explain why the overall magnetization can vanish even if we are well below the Curie temperature Tc. In a single domain the M vs. T curve looks somewhat like Fig. 7.16.

M

MS

H = 0

Tc T

Fig. 7.16. M vs. T curve for a single magnetic domain

For reference, the Curie temperature of iron is 1043 K and its saturation magnetization MS is 1707 G. But when there are several domains, they can point in different directions so the overall magnetization can attain any value from zero up to saturation magnetization. In a magnetic field, the domains can change in size (with those that are energetically preferred growing). Thus the phenomena of hysteresis, which we sketch in Fig. 7.17 starting from the ideal demagnetized state, can be understood (see Section Hysteresis, Remanence, and Coercive Force).

23 More details can be found in Morrish [68] and Chikazumi [7.11].

422 7 Magnetism, Magnons, and Magnetic Resonance

where the spin Si for atom i when averaged over many neighboring spins gives us the local magnetization. We now make a classical continuum approximation. For the interaction energy of two spins we write:

Assuming ui is a unit vector in the direction of Si we have since Si = Sui:

If rji is the vector connecting spins i and j, then

treating u as a continuous function r, u = u(r). Then since

we have, neglecting an additive constant that is independent of the directions of ui and uj,

Uij = +JS 2 (u j − ui )2 .

where we have inserted a 1/2 so as not to count bonds twice. If u =α1i +α2 j +α3k ,

where the αi are the direction cosines, for rji = ai, for example:

For a simple cubic lattice where we must also include neighbors at rji = ±aj and

±ak, we have:24

24An alternative derivation is based on writing U ∑μiBi, where μi is the magnetic moment Si and Bi is the effective exchange field ∑j(nn) JijSj, treating the Sj in a continuum spatial approximation and expanding Sj in a Taylor series (Sj = Si + a∂Si /∂x + etc. to 2nd order). See (7.275) and following.

424 7 Magnetism, Magnons, and Magnetic Resonance

Also, for proper choice of K1, this may describe hexagonal crystals, e.g. cobalt (hcp) where θ is the angle between M and the hexagonal axis. Figure 7.19 shows some data related to anisotropy. Note Fe with a bcc structure has easy directions in 100 and Ni with fcc has easy directions in 111 .

Fig. 7.19. Magnetization curves showing anisotropy for single crystals of iron with 3.85% silicon [Reprinted with permission from Williams HJ, Phys Rev 52, 1 (1937). Copyright 1937 by the American Physical Society.]

Wall Energy (B)

The wall energy is an additive combination of exchange and anisotropy energy, which are independent. Exchange favors parallel moments and a wide wall. Anisotropy prefers moments along an easy direction and a narrow wall. Minimizing the sum of the two determines the width of the wall. Consider a uniaxial ferromagnet with the magnetization varying only in the y direction. If the energy per unit volume is (using spherical coordinates, see, e.g., (7.242) and Fig. 7.25)

and α1, κ1 differ for different crystal structures, but both are approximately unity. For simplicity in what follows we will set α1 and κ1 equal to one.

7.3 Magnetic Domains and Magnetic Materials (B) 425

Using δ∫wdy = 0 we get two Euler–Lagrange equations. Inserting (7.245) in the Euler–Lagrange equations, we get the results indicated by the arrows.

For Bloch walls by definition, φ = 0, which is a possible solution. The first equation (7.247) has a first integral of

The effective wall width is obtained by approximating dθ/dy by its value at the midpoint of the wall, where θ = π/2.

One can also show the wall width per unit area (perpendicular to the y-axis in Fig. 7.25) is 4(AK1)1/2. For Iron, the wall energy per unit area is of order 1

erg/cm2, and the wall width is of order 500 Å.

Magnetostrictive Energy (B)

Magnetostriction is the variation of size of a magnetic material when its magnetization varies. Magnetostriction implies a coupling between elastic and magnetic effects caused by the interaction of atomic magnetic moments and the lattice. The magnetostrictive coefficient λ is δl/l, where δl is the change in length associated with the magnetization change. In general λ can be either sign and is typically of the order of 10−5 or so. There may also be a change in volume due to changing magnetization. In any case the deformation is caused by a lowering of the energy.

426 7 Magnetism, Magnons, and Magnetic Resonance

Magnetostriction is a very complex matter and a detailed description is really outside the scope of this book. We needed to mention it because it has a bearing on domains. See, e.g., Gibbs [7.24].

Formation of Magnetic Domains (B)

We now give a qualitative account of the formation of domains. Consider a cubic material, originally magnetized along an easy direction as shown in Fig. 7.20. Because the magnetization M and demagnetizing fields have opposite directions (7.136), this configuration has large magnetostatic energy. The magnetostatic energy can be reduced if the material splits into domains as shown in Fig. 7.21

Fig. 7.20. Magnetic domain formation within a material

where nM2 is a unit vector pointing from region 1 to region 2. Thus when M · n is continuous, there are no demagnetizing fields (assuming also M is uniform in the interior). Thus (for typical magnetic materials with cubic symmetry) the magnetostatic energy can be further reduced by forming domains of closure, as shown in Fig. 7.22. The overall magnetostrictive and strain energy can be reduced by the formation of more domains of closure (see Fig. 7.23). That is, this splitting into smaller domains reduces the extra energy caused by the internal strain brought about by the spontaneous strain in the direction of magnetization. This process will not continue forever because of the increase in the wall energy (due to exchange and anisotropy). An actual material will of course have many imperfections as well as other complications that will cause irregularities in the domain structure.