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Mean-Field Theory of Magnetic Materials

The mean-field theory, or molecular-field approximation, is considered as the first-order approximation to treat a system of interacting spins. This chapter shows how this theory is applied to ferromagnets, antiferromagnets and ferrimagnets. As a first- order approximation, its results give a quick look at the system’s properties. We will discuss the validity of the mean-field theory below and show how to improve it in Chapter 5.

Mean-Field Theory of Ferromagnets

We consider the Heisenberg model for a ferromagnet with the following Hamiltonian

where H0 is a magnetic field applied in the z direction, g the Lande factor and ц в the Bohr magneton. The first sum is performed over spin pairs (S,, Sy) occupying lattice sites / and j. For simplicity, we suppose in the following only the interaction between nearest

Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com neighbors is not zero. Note that this hypothesis is not a hypothesis of the mean-field theory because the mean-field theoiy can be applied to systems including far-neighbor interactions as seen in Problem 8.

We consider the spin at the siteThe interaction energy with its nearest neighbors and with the magnetic field are written as

where p are vectors connecting the site i to its nearest neighbors and J denotes the exchange integral between S, with its nearest neighbors.

Mean-Field Equation

The only assumption of the mean-field theoiy is to suppose that all neighboring spins have the same average value, namely < S,+p >= < Sz > for all (/ + p). This value is to be computed in the following. We choose the z axis as the spin quantization axis. The average values of the x and у spin components are then zero since the spin precesses circularly around the z axis:

For the z component, we have for all neighbors

where < Sz > is the average value in the absence of the magnetic field, and < Д52 > is the variation of < Sz > induced by the field. Equation (2.2) is rewritten as

where C is the coordination number (number of nearest neighbors). We can express 'Hi as

where

H is called "molecular field" acting on the spin S, .

Let us suppose H0 = 0 for the moment. In that case < ASZ >= 0 in Eq. (2.7). We have

The average value < Sz > is calculated using the canonical description (see Appendix A) as follows:

where f) = and Z, the partition function defined by [see Eq. (A.9)]

where 5 = |S, |. We obtain from which one gets

В six') is the Brillouin function defined by where

Equation (2.12) is called "mean-field equation.” Since the argument* of Bs[x) contains < Sz >, (2.12) is therefore an implicit equation of < Sz > which depends on the temperature. In the case of spin one-half, S = the Brillouin function is

In the case where S —> oo, we have from Eq. (2.13)

Now, suppose that H0 is not zero but very weak. We use H of Eq. (2.7) with < ASZ > being very small. The mean-field equation is

where

Expanding the Brillouin function near* = x0 = /32CJ S < Sz > and identifying the second terms of the two sides of (2.17), we have

where ^(xo) is the derivative of Bs(x) with respect to x taken at x0.

Mean-Field Critical Temperature

Let us study the mean-field equation with respect to T.

At high T, p < Sz >« 1, we obtain from (2.13)

Equation (2.12) becomes

This equation has a solution < Sz 0 only if namely

Once this condition is satisfied, < Sz > is given by

Tc is called "critical temperature.” When T > Tc, the solution of (2.21) is < Sz >= 0.

At low temperatures, 2CJ S < Sz > is much larger than kBT, the expansion of (2.13) gives

which leads to

If Г =0, we have < Sz >= S.

Graphical Solution

In general, we solve (2.12) by a graphical method: We look for the intersection of the two curves andy2 = Bs(x)

which represent the two sides of (2.12). The first curve yx versus x is a straight line with a slope proportional to the temperature. For a given value of T, there are two symmetric intersections at ±M as shown in Fig. 2.1. It is obvious that if the slope of yx is larger than the slope ofy2 at x = 0, there is no intersection other than the one at x = 0. The solution is then < Sz >= 0. The slope ofy2 at x = 0 thus determines the critical temperature Tc, namely

which is identical to Tc given by Eq. (2.23).

Graphical solutions of Eq. (2.12)

Figure 2.1 Graphical solutions of Eq. (2.12).

Thermal average < S > versus T

Figure 2.2 Thermal average < Sz > versus T.

We display the positive solution of < Sz > as a function of T in Fig. 2.2.

Specific Heat

The average energy of a spin when H0 = 0 is calculated by [see Eq. (A. 10)]

The total ferromagnetic energy of the crystal is

where the factor is added in order to count each interaction just once. The specific heat is

At low temperatures, < Sz >= S - e-2Cls/kBT |-see (2.26)], we have When T 0, we have Cy — 0.

CV calculated by the mean-field theory versus T

Figure 2.3 CV calculated by the mean-field theory versus T.

For T > Tc, we have E = 0; therefore Cv = 0. Let us calculate Cv when T -*■ T~. We have from (2.24)

so that

The discontinuity of Cv at Tc is thus

This discontinuity is an artifact of the mean-field theory resulting from the fact that critical fluctuations near Tc have been neglected by replacing all spins by a uniform average. When fluctuations around the average values of spins are taken into account, Cv diverges at Tc when we approach Tc from both sides. Some more details on this point are given in Chapter 5. We show in Fig. 2.3 Cv calculated by the mean-field theory as a function of T.

Susceptibility

The susceptibility is defined by

where N is the total number of spins (M = Ngu в < Sz > is the total magnetic moment). From Eq. (2.19), we have

therefore,

where x = 2C/f<*>.

кцТ

When T > Tc, we have < Sz >= 0 and B((0) = ^. We get

When T < Tc, we have < Sz >-*■ 0. Expanding Bj(x) with respect to < Sz >, we obtain

It is noted that the coefficient in this case is twice smaller than that in (2.38). When T —> 0, /ц —» 0 because M -» constant. The inverse of the susceptibility is schematically shown as a function of T in Fig. 2.4.

In reality, a ferromagnetic crystal can have several ferromagnetic domains with spins pointing in different directions. This is due to the presence of defects, dislocations and imperfections during the formation of the crystal. The region between two magnetic domains is called "domain wall” in which the matching of two spin orientations is progressively realized. We show schematically magnetic domains and a domain-wall spin configuration in Fig. 2.5. The presence of domain walls makes it difficult to compare calculated and experimental susceptibilities.

Validity of Mean-Field Theory

The mean-field theory assumes that all spins have the same value, meaning that it neglects instantaneous fluctuations of each spin. Fluctuations favor disorder, so when taken into account, fluctuations cause a transition at a temperature lower than Tc given by (2.23).

Inverse of the susceptibility obtained by mean-field theory versus T

Figure 2.4 Inverse of the susceptibility obtained by mean-field theory versus T.

(a) Ferromagnetic domains in an imperfect crystal (b) Example

Figure 2.5 (a) Ferromagnetic domains in an imperfect crystal (b) Example

of a spin structure in a domain wall.

Due to the approximation of uniform spins, the mean-field theory thus overestimates the critical temperature Tc. This point is studied in Section 5.3 with the Landau-Ginzburg theory.

Another artifact of the mean-field theory is that it results in a phase transition at a finite temperature in spin systems in any space dimension: Tc given by Eq. (2.23) is not zero even for one dimension (C = 2). This is not correct because we know that in dimensions d = 1 and d = 2, fluctuations are so strong that they destroy magnetic long-range order at any finite temperature in many systems. The mean-field theory, however, becomes exact for dimension d > 4 (see Chapter 5 for more details).

Antiferromagnetism in Mean-Field Theory

In Section 1.5.2, we have seen that depending on the sign of the exchange interaction a spin system can have an antiferromagnetic order at zero and low temperatures. We study here some properties of antiferromagnets by the mean-field theory.

We consider a system of Heisenberg spins interacting with each other via the Hamiltonian

where g and цв are the Lande factor and the Bohr magneton, respectively. H0 is a magnetic field applied along the z axis. To simplify the presentation, we suppose that the exchange interaction Jij is limited to the nearest neighbors with /,y = ]. We have

Note that we have defined the exchange terms in the Hamiltonian with a positive sign so that the antiferromagnetic interaction corresponds to / > 0. In zero applied field, the neighboring spins are antiparallel, except in geometrically frustrated systems (see Chapter 5). A few antiferromagnetic systems are displayed in Fig. 2.6.

Mean-Field Theory

In the case of non frustrated lattice, the antiferromagnetic ordering has two sublattices (see Fig. 2.6): sublattice of f spins and sublattice

Antiferromagnetic ordering

Figure 2.6 Antiferromagnetic ordering: Black and white circles denote 1 and | spins, respectively.

of 4- spins, indicated hereafter by indices / and m, respectively. For simplicity, we treat the case of weak field Ho ] so that the antiferromagnetic ordering remains.

The mean-field theoiy is applied to an antiferromagnet as follows. We write the following mean-field energies of spins / and m:

where C is the coordination number, < Sf >=< 5+ > + < AS+ > denotes the average value of Sf, and < AS+ > the spin variation induced by the applied field. Using Hi, we calculate < Sf > as follows:

where #s(x) is the Brillouin function given by with

For weak fields, we expand the function Ss(x) around We then obtain

therefore

B's[x0) being the derivative of Bs(x0) with respect to x taken at x0. In the same manner, we obtain for down sublattice spin < Sz >

with Xq = -ffC] S < Sz+ >.

If the two sublattices are symmetric, namely < S+ > = — < Si >=< Sz >, then Eqs. (2.47) and (2.49) are equivalent because the Brillouin function is an odd function. We then have only one implicit equation for < Sz > to solve

This mean-field equation for a sublattice spin is the same as that for ferromagnets, Eq. (2.12) [note that there is no factor 2 in Eq. (2.51) because we did not use the factor 2 for the exchange interaction in the Hamiltonian (2.40)]. We have thus the same result on the temperature dependence of < Sz > and on the critical temperature. Therefore, the critical temperature for antiferromagnets, called "Neel temperature" and denoted by TN, is given by

We calculate < AS± >. Since H0 induces a positive amount of the z component for both sublattices, and by symmetry, we have < AS+ >=< AS_ >=< AS >. Note that £^(x) is an even function of x; therefore from Eqs. (2.48) and (2.50), we have

The susceptibility is given by

When T —> 0, B's(- ■ ■) tends to 0 faster than T. We deduce that Xu = 0. On the contrary, for T >TN, B^(- • •) — we 8et

Susceptibility xn and xj. of an antiferromagnet versus T

Figure 2.7 Susceptibility xn and xj. of an antiferromagnet versus T.

where we notice the + sign in front of TN, in contrast to the ferromagnetic case. There is thus no divergence of the susceptibility at the phase transition for an antiferromagnet.

In the case where the applied field is also weak but perpendicular to the z axis, for example H0 || Ox, we modify (2.42) and (2.43) to obtain

and

We show in Fig. 2.7 xn and versus T.

In materials which have magnetic domains or in powdered systems, experimental susceptibility at Г < TN is an average with spatial weight coefficients 1/3 and 2/3:

Spin Orientation in a Strong Applied Magnetic Field

The results shown above have been calculated with the assumption of weak field. When H0 is sufficiently strong, the results will be different as seen below.

We suppose that H0 is parallel to the z axis. The f spins have their energy lowered by the Zeeman effect —g/uBS?H0 while the i spins have their energy increased by —giuBSfnH0>0[S^ < 0). Contrary to the weak field case where the spins remain approximately antiparallel because of the dominant J, in the case of strong field the competition between the Zeeman effect and the exchange interaction determines the stable spin configuration as seen below.

We consider the general case where we add a uniaxial anisotropy term to the Hamiltonian (2.41) to fix the easy-magnetization axis. We suppose that H0 is applied in the direction which forms an angle <■(<■€ [0, л]) with respect to the easy-magnetization axis. The competition between the Zeeman effect and J gives rise to a configuration of the two sublattices shown in Fig. 2.8 where в is the angle of H0 with respect to the +z axis.

The exchange energy is written as

where

with M being the magnetization modulus.

Spin orientation with respect to the direction of H

Figure 2.8 Spin orientation with respect to the direction of H0.

The anisotropy energy is written for sublattices ML and M_ (see Fig. 2.8) as

where К is the anisotropy constant.

The Zeeman energy is

The energy induced by the variation of M under the applied field is

because, by definition,

The total energy is thus

By minimizing E of (2.66) with respect to (p and £ we obtain therefore,

Since, by definition, = 2^ss^, we get

Before minimizing (2.66) with respect to £, we consider the regime where Л » K, H0. In this case ~ 0 (see Fig. 2.8), so that cos(f — ~ cos(?r — t; - (p). Replacing this and (2.67)-(2.68) in (2.66), we arrive at

The minimization with respect to c leads to

We examine a particular case where 0 = 0 (H0 || Oz). The solutions are

  • • ? = 0 if H0 < HCl
  • • < ~ n/2 if H0 > Hc where

The spin configurations corresponding to these two solutions are displayed in Fig. 2.9. The transition between these phases when H0 = Hc is called "spin-flop transition”: the spins are approximately perpendicular to H0 for H0 > Hc. Hc est called "critical field.”

This result has been obtained with the hypothesis Л » K, H0. In the case where H0 is larger than Hc and larger than the local exchange field acting on a spin, all spins will turn into the direction of H0.

Spin orientation with respect to H when H < H (left) and H > H (right)

Figure 2.9 Spin orientation with respect to H0 when H0 < Hc (left) and H0 > Hc (right).

Left: Phase diagram of an antiferromagnet with Ising spins under an applied field of amplitude H. The line separates the antiferromagnetic and paramagnetic phases under field. Right

Figure 2.10 Left: Phase diagram of an antiferromagnet with Ising spins under an applied field of amplitude H0. The line separates the antiferromagnetic and paramagnetic phases under field. Right: Phase diagram in the case of Heisenberg spins, there is a spin-flop phase.

Phase Transition in an Applied Magnetic Field

The results shown above were obtained at Г = 0. We discuss now the effect of T in an antiferromagnet under a strong applied field.

To simplify the description, let us consider the Ising spin model. The field H0 is supposed to be parallel to Oz. If H0 < Hc where Hc is the critical field which is to be determined for the Ising model (see Problem 10 below), the spins remain antiparallel between them. If H0 > Hc, all spins are parallel to H0: There is no spin-flop phase for Ising spins.

In the case of ferromagnets in a field, the magnetization is never zero, so a phase transition is impossible at any temperature. In the case of antiferromagnets, when H0 < Hc there is a possibility that the antiferromagnetic order is broken with increasing T: At high temperatures, spins excited by the temperature finish by turning themselves parallel to the field at a temperature Tc. Of course, Tc depends on H0. We display schematically a phase diagram in Fig. 2.10. More details on the phase transition are given in Chapter 5.

 
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