Patterson, Bailey - Solid State Physics Introduction to theory

408 7 Magnetism, Magnons, and Magnetic Resonance

Since the Hamiltonian is a many-electron Hamiltonian, it is not exactly solvable for a general lattice. We solve it in the mean-field approximation and thus replace

2I ∑α,σ nασ nα,−σ ,

with

Itinerant Ferromagnetism and the Stoner Model (B)

The mean-field approximation has been criticized on the basis that it builds in the possibility of an ordered ferromagnetic ground state regardless of whether the Hubbard Hamiltonian exact solution for a given lattice would predict this. Nevertheless, we continue, as we are more interested in the model we will eventually reach (the Stoner model) than in whether the theoretical underpinnings from the Hubbard model are physical. The mean-field approximation to the Hubbard model gives

Actually, in the mean-field approximation, the band picture is more convenient to use. Since we can show

The average number of electrons per site n is less than or equal to 2 and n = n+ + n−, while the magnetization per site n is M = (n+ − n−)μB, where μB is the Bohr magneton.

Note: In order not to introduce another “−” sign, we will say “spin up” for now. This really means “moment up” or spin down, since the electron has a negative charge.

Note n + (M/μB) = 2n+ and n − (M/μB) = 2n−. Thus, up to an additive constant

Although these equations are easy to write down, it is not easy to obtain simple convenient solutions from them. As already noted, the Stoner model contains two basic assumptions: (1) The electronic energy band in the metal is described by a known εk. By standard means, one can then derive a density of states. For free electrons, N(E) (E)1/2. (2) A molecular field approximately describes the effects of the interactions and we assume Fermi–Dirac statistics can be used for the spinup and spin-down states. Much of the detail and even standard notation has been presented by Wohlfarth [7.69]. See also references to Stoner’s work in the works by Wohlfarth.

The only consistent way to determine εk and, hence, N(E) is to derive it from the Hubbard Hamiltonian. However, following the usual Stoner model we will just use an N(E) for free electrons.

The maximum saturation magnetization (moment per site) is M0 = μBn and the actual magnetization is M = μB(n↑ − n↓). For the Stoner model, a relative magnetization is defined below:

Using (7.212) and (7.213), we have

410 7 Magnetism, Magnons, and Magnetic Resonance

It is also convenient to define a temperature θ′, which measures the strength of the exchange interaction

We now suppose that the exchange energy is strong enough to cause an imbalance in the number of spin-up and spin-down electrons. We can picture the situation with constant Fermi energy μ = EF (at T = 0) and a rigid shifting of the up N+ and the down N− density states as shown in Fig. 7.15.

The ↑ represents the “spin-up” (moment up actually) band and the ↓ the “spindown” band. The shading represents states filled with electrons. The exchange energy causes the splitting of the two bands. We have pictured the density of states by a curve that goes to zero at the top and bottom of the band unlike a freeelectron density of states that goes to zero only at the bottom.

Fig. 7.15. Density states imbalanced by exchange energy

At T = 0, we have

This can be easily worked out for free electrons if E = 0 at the bottom of both bands,

We now derive conditions for which the magnetized state is stable at T = 0. If we just use a single-electron picture and add up the single-electron energies, we find, with the (–) band shifted up by and the (+) band shifted down by , for the energy per site

However, just as in the Hartree–Fock analysis, this exchange term has double counted the interaction energies (once as a source of the field and once as interaction with the field). Putting in a factor of 1/2, we finally have for the total energy

Differentiating (d/dξ) (7.216) and (7.218) and combining the results, we can show

2kθ′ξ = (EF+ − EF− ) = 2μB Heff = 2 .

Note if ξ = 0 (paramagnetism) and dE/dξ = 0, while d2E/dξ2 < 0 the paramagnetism is unstable with respect to ferromagnetism. ξ = 0, dE/dξ = 0 implies EF+ = EF- and N(EF-) = N(EF+) = N(EF). So by (7.220) with d2E/dξ2 ≤ 0 we have

For a parabolic band with N(E) E1/2, this implies

412 7 Magnetism, Magnons, and Magnetic Resonance

We now calculate the relative magnetization (ξ0) at absolute zero for a parabolic band where N(E) = K(E)1/2 where K is a constant. From (7.216)

Eliminating K and using EF+ − EF− = 2kθ′ξ0, we have

which is valid for 0 ≤ ξ0 ≤ 1. The maximum ξ0 can be is 1 for which kθ′/EF = 2 , and at the threshold for ferromagnetism ξ0 is 0. So, kθ′/EF = 2/3 as already predicted by the Stoner criterion.

Summary of Results at Absolute Zero

We have three ranges:

The middle range, where 0 < ξ0 < 1 is special to Stoner ferromagnetism and not to be found in the Weiss theory. This middle range is called “unstructured” or “weak” ferromagnetism. It corresponds to having electrons in both ↑ and ↓ bands. For very low, but not zero, temperatures, one can show for weak ferromagnetism that

where C is a constant. This is particularly easy to show for very weak ferromagnetism, where ξ0 << 1 and is left as an exercise for the reader.

414 7 Magnetism, Magnons, and Magnetic Resonance

Thus, as long as T θ, we have a Curie–Weiss-like law:

At very high temperatures, one can also show that an ordinary Curie–Weiss-like law is obtained:

Summary Comments About the Stoner Model

1.The low-temperature results need to be augmented with spin waves. Although in this book we only derive the results of spin waves for the localized model, it turns out that spin waves can also be derived within the context of the itinerant electron model.

2.Results near the Curie temperature are never qualitatively good in a mean-field approximation because the mean-field approximation does not properly treat fluctuations.

3.The Stoner model gives a simple explanation of why one can have a fractional number of electrons contributing to the magnetization (the case of weak ferromagnetism where ξ0 = MT=0/nμB is between 0 and 1).

4.To apply these results to real materials, one usually needs to consider that there are overlapping bands (e.g. both s and d bands), and not all bands necessarily split into subbands. However, the Stoner model does seem to work for ZrZn2.

7.2.5 Magnetic Phase Transitions (A)

Simple ideas about spin waves break down as Tc is approached. We indicate here one way of viewing magnetic phenomena near the T = Tc region. In this Section we will discuss magnetic phase transitions in which the magnetization (for ferromagnets with H = 0) goes continuously to zero as the critical temperature is approached from below. Thus at the critical temperature (Curie temperature for a ferromagnet) the ordered (ferromagnetic) phase goes over to the disordered (paramagnetic) phase. This “smooth” transition from one phase (or more than one phase in more general cases) to another is characteristic of the behavior of many substances near their critical temperature. In such continuous phase transitions there is no latent heat and these phase transitions are called second-order phase transitions. All second-order phase transitions show many similarities. We shall consider only phase transitions in which there is no latent heat.

No complete explanation of the equilibrium properties of ferromagnets near the magnetic critical temperature (Tc) has yet been given, although the renormalization technique, referred to later, comes close. At temperatures well below Tc we know that the method of spin waves often yields good results for describing the

416 7 Magnetism, Magnons, and Magnetic Resonance

Critical Exponents and Failures of Mean-Field Theory (B)

Although mean-field theory has been extraordinarily useful and in fact, is still the “workhorse” of theories of magnetism (as well as theories of the thermodynamics behavior of other types of systems that show phase transitions), it does suffer from several problems. Some of these problems have become better understood in recent years through studies of critical phenomena, particularly in magnetic materials, although the studies of “critical exponents” relates to a much broader set of materials than just magnets as referred to above. It is helpful now to define some quantities and to introduce some concepts.

A sensitive test of mean-field theory is in predicting critical exponents, which define the nature of the singularities of thermodynamic variables at critical points of second-order phase transitions. For example,

for T < Tc, where β, ν are critical exponents, φ is the order parameter, which for ferromagnets is the average magnetization M and ξ is the correlation length. In magnetic systems, the correlation length measures the characteristic length over which the spins are ordered, and we note that it diverges as the Curie temperature Tc is approached. In general, the order parameter φ is just some quantity whose value changes from disordered phases (where it may be zero) to ordered phases (where it is nonzero). Note for ferromagnets that φ is zero in the disordered paramagnetic phase and nonzero in the ordered ferromagnetic situation.

Mean-field theory can be quite good above an upper critical (spatial) dimension where by definition it gives the correct value of the critical exponents. Below the upper critical dimension (UCD), thermodynamic fluctuations become very important, and mean-field theory has problems. In particular, it gives incorrect critical exponents. There also exists a lower critical dimension (LCD) for which these fluctuations become so important that the system does not even order (by definition of the LCD). Here, mean-field theory can give qualitatively incorrect results by predicting the existence of an ordered phase. The lower critical dimension is the largest dimension for which long-range order is not possible. In connection with these ideas, the notion of a universality class has also been recognized. Systems with the same spatial dimension d and the same dimension of the order parameter D are usually in the same universality class. Range and symmetry of the interaction potential can also play a role in determining the universality class. Quite dissimilar systems in the same universality class will, by definition, exhibit the same critical exponents. Of course, the order parameter itself as well as the critical temperature Tc, may be quite different for systems in the same universality class. In this connection, one also needs to discuss concepts like the renormalization group, but this would take us too far afield. Reference can be made to excellent statistical mechanics books like the one by Huang.21

21 See Huang [7.32, p441ff].