Fundamentals of the Physics of Solids / 14-Magnetically Ordered Systems

only on the value of the correlation length, and that it did not make any di erence whether it arose as the result of changing the magnetic field or the temperature. Within the correlation length, the length scale could be chosen at will. If instead of the natural scale of the lattice constant a its s-fold multiple is chosen as the unit, then the correlation length will be correspondingly smaller: ξ = ξ/s. It is as if the scaled system were farther from the critical point – that is, if instead of the reduced temperature t and magnetic field H it was characterized by a more distant temperature t and stronger field H . The parameters yt and yH , defined by

are called the scaling dimensions of temperature and magnetic field. In general, the scaling dimension of any physical quantity A is determined by the relation between A , the quantity obtained by changing the length scale, and A:

It follows from the relation ξ t−ν that yt = 1/ν. As for any extensive quantity, the scaling dimension for the free energy is the same as the dimension of space, therefore Kadano ’s assumption implies

This is just (14.5.39) – the equation asserting that free energy is a generalized homogeneous function – with the exponents

Expressing at with the exponent of the specific heat, the Josephson inequality is also found to be replaced by an equality:

In perfect analogy, it is possible to obtain a scaling formula for the correlation function, too. In addition to t and H, the distance r is also scaled now by the straightforward transformation r = r/s:

Here η characterizes the deviation from the mean-field theory; this is why it is called anomalous dimension. It then follows that

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The exponent η is related to the other critical exponents by the equalities

14.5.5 Elimination of Fluctuations and the Renormalization Group

Based on the Kadano theory, a physical picture was obtained for the scaling behavior of thermodynamic quantities and correlation functions close to the critical point. This allowed us to derive relations among the critical exponents. It was also understood that the deviation of experimental results from the predictions of the Landau theory are due to fluctuations. The correlation between these fluctuations becomes long-ranged in the vicinity of the critical point, and infinitely long-ranged in the critical point itself. The divergence of the correlation length gives divergent contributions to the susceptibility and other physical quantities. However, the scaling hypothesis in itself does not specify either the critical exponents or the scaling functions. It has to be complemented by K. G. Wilson’s10 formulation (1971) of a method widely used in field theory, the renormalization-group method. Below we shall present only the main ideas.

Since the correlation length gets macroscopically large around the critical point, fluctuations of any wavelength become important, and so the approximation that the order parameter can be determined from the minimum of the free-energy density is no longer su cient. Characterizing the system with the e ective Hamiltonian H, we shall examine the total free energy and the partition function

where the sum of the diagonal matrix elements is taken over all possible states of the system. Unless the critical point is at zero temperature, quantum e ects are unimportant in its vicinity, so one has to deal only with thermal fluctuations that can be treated classically.11 To this end, we shall write K = H/kBT in a simple form in terms of the Fourier components ψk of the fluctuations of the order parameter. Starting with the form

10Kenneth Geddes Wilson (1936–) was awarded the Nobel prize in 1982 “for his theory for critical phenomena in connection with phase transitions”.

11For an appropriate description of phase transitions at zero temperature (called quantum phase transitions) at a quantum critical point – where the transition is not driven by temperature but by the change of the coupling constants of the Hamiltonian – quantum fluctuations must also be taken into account.

which bears some resemblance to the Landau expansion of the free-energy density (14.5.17). In terms of ψk we have

The critical point is specified by r0 = 0, while u0 is the interaction strength of fluctuations. Both of them will be treated as phenomenological parameters. One of the most striking features of the theory is the universality of critical behavior: the critical exponents depend only on the dimensionality of the system and the order parameter, as well as the symmetries of the system – however, they are independent of the initial values of the parameters.

Taking the fluctuations into account means taking an average over fluctuations of all possible wavelengths:

∞

?

Assuming that the most important contribution to the critical behavior comes from long-wavelength fluctuations, an average may be taken over shortwavelength (large-wave-number) fluctuations as a first step. Replacing the Brillouin zone by a sphere of radius Λ, this averaging procedure corresponds to an integration over the wavelengths between Λ = Λ/s and Λ. This way a system of fewer degrees of freedom is obtained. Then a Hamiltonian H can be constructed in the smaller Brillouin zone, along with the corresponding K defined as

Obviously, the partition function of this system obtained by averaging over the still allowed fluctuations must be the same as the original partition function:

For purposes of comparing the parameters and coupling constants of the new and the original systems, the distance has to be scaled from r to r = r/s – which corresponds to scaling the wave numbers from k to k = sk. Elimination of a part of the degrees of freedom through the above transformation is called renormalization, and the new coupling constant is called the renormalized coupling.

By continuing the procedure and eliminating further degrees of freedom

– which explains the origin of the name renormalization group, although the group character of the transformations is not apparent in this description – the

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flow of the coupling constants can be determined.12 Things are highly simplified when the coupling constant of an interaction gets weaker and eventually vanishes as fluctuations are gradually eliminated and their e ect is absorbed into the new, renormalized value of the coupling constant. Such interactions are not relevant from the viewpoint of critical behavior. However, this is generally not the case; instead the renormalized coupling tends to a finite or infinite fixed point, and often additional many-particle interactions appear as a result of renormalization. In such cases the renormalization-group transformation can be treated only numerically. The exponents that characterize the critical properties can be determined from the fixed point obtained at the end of the iteration procedure and the behavior of the system in its vicinity.

Using expressions (14.5.56) and (14.5.57) – which contain the gradient term – for the free-energy density, this renormalization program can be carried out. It turns out that in d > 4 dimensional systems the contribution of fluctuations is not important, and the results of the mean-field theory are recovered. In d = 4 dimensions the role of fluctuations is marginal, which means that the critical exponents are the same as in the mean-field approximation, however logarithmic corrections appear in the correlation functions. In the physically important case of d < 4 dimensions fluctuations are relevant, and so critical exponents depend on the dimensionality of space and of the order parameter. Table 14.8 shows the critical exponents determined for the three-dimensional n = 1, 2, 3-component Ising, XY, and isotropic Heisenberg models, compared to the values derived from the mean-field theory and the exact solution of the two-dimensional Ising model.

Table 14.8. Calculated critical exponents of the three-dimensional n = 1, 2, 3- component Ising, XY, and isotropic Heisenberg models, compared to the values obtained from the mean-field theory and for the two-dimensional Ising model

Comparison with the data listed in Table 14.7 shows that the agreement between theoretical and experimental results is fairly good for the isotropic Heisenberg model.

12In connection with another problem, an explicit example of the renormalization procedure will be presented in the appendix of Volume 3.

14.6 High-Temperature Expansion

On account of its initial hypotheses, the Landau theory and its improvement that takes the e ects of fluctuations into consideration can describe the behavior of a magnetic system only in the vicinity of the phase transition point. We have seen that fluctuations give significant corrections to the Landau theory. Since the latter is equivalent to the mean-field theory, the question naturally arises: to what extent is the mean-field approximation justified at low and high temperatures? We shall devote the next chapter to the study of lowtemperature behavior, that is the quantum mechanics of ordered magnetic structures. In the present section we shall discuss the method used in the high-temperature region.

The free energy of a system characterized by the Hamiltonian H can be derived from the partition function

as F = −kBT ln Z, where Tr stands for the trace operator, that is, the sum of the diagonal matrix elements over a complete set of basis functions. The thermal average of a physical quantity characterized by the operator A is

At high temperatures the exponential in the partition function can be

where Tr(1) is the number of possible states in the complete system of functions, and B 0 = Tr B/ Tr(1) stands for the thermodynamic average at infinitely large temperature. By taking the logarithm of the series, free energy can be written in terms of the so-called cumulants:

To determine the susceptibility, the contribution of the external magnetic field has to be included in the Hamiltonian, and the terms proportional to the field have to be collected. Using the same steps as in the derivation of (3.2.49), the susceptibility formula generalized to the case at hand is

i

is the α component of the magnetic moment density. In the high-temperature region, where no spontaneous magnetization is present and the spin components of di erent directions are uncorrelated, the susceptibility tensor is diagonal:

Determining the thermal average in the previous formula for the Heisenberg model as above,

Using the mean-field value for the Curie temperature,

which gives the first two terms in the expansion of the Curie–Weiss susceptibility. Deviations from the mean-field theory appear in the next terms of the high-temperature expansion. By continuing the expansion to su ciently high orders, one may try to sum up the expansion formula, that is, to fit a scaling function – in such a way that the result should be valid even in the vicinity of the transition point. The values obtained in this way for the critical temperature and the critical exponent of susceptibility in the Heisenberg model are in fair agreement with experimental data.

14.7 Magnetic Anisotropy, Domains

We have seen that the exchange interaction determines the orientation of magnetic moments only relative to each other. In crystals the orientation relative to the crystallographic axes is determined by the much weaker anisotropic terms due to relativistic spin–orbit interactions. A particularly interesting feature of ferromagnets is that in large, macroscopic samples the local magnetization does not point in the same crystallographic direction over the whole of the sample: instead, as the system starts to become ordered below the transition point, magnetization points in one of the equivalent easy axis directions around each nucleation center. The sample is thus made up of a large number of domains with di erent magnetization directions, which are separated by domain walls. To describe them, we shall first introduce a continuum model, and then determine the characteristic dimensions of domains and the domain-wall energy.

14.7.1 A Continuum Model of Magnetic Systems

We used a continuous order parameter in the Landau theory, and determined its magnitude from the minimum of free energy. The continuous order parameter can be kept even far from the transition point, at low temperatures

– however in this region we are not concerned primarily with its magnitude (since magnetization is almost saturated here) but rather with its orientation and possible rotation.

The continuous magnetization density function is defined in terms of the magnetization density operator

i

Even when the operator is replaced by its expectation value, the resulting expression still contains a sum of delta peaks. To obtain a coarse-grained (smoothed-out) magnetization density, the expectation value has to be averaged around r over a volume v whose radius is much larger than atomic dimensions but still small on the characteristic scale of the spatial variations of atomic magnetic moments. The expression

A

is indeed a classical quantity that varies slowly in space.

To express the energy with M (r), we have to start with the Heisenberg

i,j

is recovered. It is then plausible to assume that in terms of the density M (r) the magnetic energy can be written in the Hamiltonian-like form

Just like the exchange integral J(r), J(r) is also short-ranged, therefore the expansion of the position-dependent magnetization M (r ) in the integrand around r = r gives

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M (r ) = M (r) + ∂M (r) (rμ − rμ)

μ ∂rμ

+ 1 ∂2M (r) (rμ − rμ)(rν − rν ) + . . . .

(14.7.6)

2 μν ∂rμ∂rν

Substituting this into the integrand, the linear term vanishes on account of the inversion symmetry of the crystal lattice. The leading term of the magnetic energy is then

Note that this expression bears strong resemblance to the Landau expansion of free energy; however we are now interested only in the slow spatial variations in the direction of M (r) therefore the quartic term M 4 can be ignored.

The presence of M 2 in the first term is the consequence of having started with the isotropic Heisenberg model. In the more general case the expression for the energy density must be invariant under the symmetry operations of the crystal. It follows from symmetry considerations that in a uniaxial crystal the combination

has to appear. This determines the orientation of magnetization with respect to the crystallographic axes. The di erence of the coe cients K and K is therefore related to anisotropy. In cubic crystals higher-order terms have to be taken into account, since the second-order expression of cubic symmetry

possesses full spherical symmetry, hence it does not single out a preferred direction with respect to the crystallographic axes. In the fourth order, in addition to the spherically symmetric term M 4 there are two other terms that show cubic symmetry:

Mx4 + My4 + Mz4 and Mx2My2 + Mx2Mz2 + My2Mz2 . (14.7.12)

These are not independent of each other, since a suitably chosen linear combination gives M 4. When one is not concerned with the magnitude of the moment only with its orientation, it is enough to keep either of them. Using the direction cosines α1, α2, α3, the energy contribution associated with magnetic anisotropy in a cubic crystal is

which contains contributions up to the sixth order. The orientation of the moment relative to the crystallographic axes is determined by the sign of the two anisotropy constants. By minimizing the energy it is straightforward to show that for K1 > 0 the magnetic moment is along one of the directions100 , while for K1 < 0 along one of the directions 111 .

The anisotropy constants of ferromagnetic iron and nickel measured at room temperature are given in Table 14.9. These constants depend sensitively on temperature, however even at low temperatures K1 is positive for iron and negative for nickel. Therefore magnetization is along the edges of the cubic primitive cell in iron, and along the space diagonal in nickel.

Table 14.9. Phenomenological anisotropy constants of iron and nickel, measured at room temperature (in units of 103 J m−3)

Needless to say, the term arising from the slow spatial variations of magnetization in the energy density expression (14.7.9) also depends on the symmetries of the system. The formula obtained in the Landau expansion is recovered only in the isotropic case and for cubic crystals, where the three Cartesian coordinates are equivalent, Jμν = Jδμν , and hence the derivative term is

In uniaxial crystals, where one axis is inequivalent to the two others the contribution is

A similar expression is found for the energy density in antiferromagnets provided the sublattice magnetization is considered as a classical vector that varies slowly in space:

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14.7.2 Magnetic Domains

In the previous calculations we ignored the energy of the magnetic field around the finite-sized magnetic sample, however this cannot be neglected in the energy balance. For homogeneous ferromagnets of finite size this field energy can be significant. As shown in Fig. 14.18, the magnetic field outside the sample – and along with it, the field energy – is reduced substantially when the sample is not homogeneously magnetized but is made up of oppositely polarized domains.

Fig. 14.18. Magnetic structures with one and two domains, and the lines of the induced magnetic field

At the boundary of the two domains there is a region in which spins are not aligned properly, therefore the formation of a wall between the domains entails an increase in magnetic energy. The latter is proportional to the surface area of the wall, while the decrease in field energy is proportional to the volume of the sample. In large samples this decrease may become dominant, in which case it is energetically more favorable to have a domain structure. However, small samples may contain a single domain.

Inside a domain the direction of magnetization is determined by anisotropy. The antiparallel orientation of the moments on the two sides of the wall is equally favorable from the viewpoint of anisotropy, however, there is a significant increase in the exchange energy. With a slow rotation over a longer distance the increase in the exchange energy can be reduced – however, this would give rise to an increase in the anisotropy energy. The competition of the two contributions determine the details of the reversal of the moment across the domain wall. Assuming uniaxial anisotropy, where the upward or