# Mean-Field Theory of Magnetic Materials

- Mean-Field Theory of Ferromagnets
- Mean-Field Equation
- Mean-Field Critical Temperature
- Graphical Solution
- Specific Heat
- Susceptibility
- Validity of Mean-Field Theory
- Antiferromagnetism in Mean-Field Theory
- Mean-Field Theory
- Spin Orientation in a Strong Applied Magnetic Field
- Phase Transition in an Applied Magnetic Field

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 H_{0} 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,, S_{y}) 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 >= < *S ^{z}* > 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 < *S ^{z} >* is the average value in the absence of the magnetic field, and < Д5

^{2}> is the variation of <

*S*induced by the field. Equation (2.2) is rewritten as

^{z}>

where *C* is the coordination number (number of nearest neighbors). We can express *'H _{i}* as

where

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

Let us suppose *H _{0}* = 0 for the moment. In that case <

*AS*0 in Eq. (2.7). We have

^{Z}>=

The average value < *S ^{z}* > 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 *B _{s}[x)* contains <

*S*(2.12) is therefore an implicit equation of <

^{z}>,*S*which depends on the temperature. In the case of spin one-half,

^{z}>*S =*the Brillouin function is

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

Now, suppose that *H _{0}* is not zero but very weak. We use

*H*of Eq. (2.7) with <

*AS*being very small. The mean-field equation is

^{Z}>

where

Expanding the Brillouin function near* = x_{0} = /32*CJ S < S ^{z} >* and identifying the second terms of the two sides of (2.17), we have

where ^(xo) is the derivative of B_{s}(x) with respect to x taken at x_{0}.

### Mean-Field Critical Temperature

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

At high *T, p* < S^{z} >« 1, we obtain from (2.13)

Equation (2.12) becomes

This equation has a solution < *S ^{z} >ф* 0 only if
namely

Once this condition is satisfied, < *S ^{z} >* is given by

*T _{c}* is called "critical temperature.” When

*T > T*the solution of (2.21) is <

_{c},*S*0.

^{z}>=At low temperatures, *2CJ S < S ^{z} >* is much larger than

*k*the expansion of (2.13) gives

_{B}T,

which leads to

If Г =0, we have < *S ^{z} >= S.*

### Graphical Solution

In general, we solve (2.12) by a graphical method: We look for the intersection of the two curves andy_{2} = Bs(x)

which represent the two sides of (2.12). The first curve *y _{x}* 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

*y*is larger than the slope ofy

_{x}_{2}at x = 0, there is no intersection other than the one at x = 0. The solution is then <

*S*0. The slope ofy

^{z}>=_{2}at x = 0 thus determines the critical temperature

*T*namely

_{c},

which is identical to *T _{c}* given by Eq. (2.23).

**Figure 2.1 **Graphical solutions of Eq. (2.12).

Figure 2.2 Thermal average *< S ^{z} >* versus

*T.*

We display the positive solution of < *S ^{z} >* as a function of

*T*in Fig. 2.2.

### Specific Heat

The average energy of a spin when *H _{0}* = 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, < *S ^{z}* >= S -

*e-*|-

^{2C}ls/k_{B}T_{see}(2.26)], we have When

*T*0, we have

*Cy —*0.

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

For *T > T _{c},* we have

*E*= 0; therefore

*Cv*= 0. Let us calculate

*C*when

_{v}*T -*■ T~.*We have from (2.24)

so that

The discontinuity of *C _{v}* at

*T*is thus

_{c}

This discontinuity is an artifact of the mean-field theory resulting from the fact that critical fluctuations near *T _{c}* have been neglected by replacing all spins by a uniform average. When fluctuations around the average values of spins are taken into account,

*C*diverges at

_{v}*T*when we approach

_{c}*T*from both sides. Some more details on this point are given in Chapter 5. We show in Fig. 2.3

_{c}*C*calculated by the mean-field theory as a function of

_{v}*T.*

### Susceptibility

The susceptibility is defined by

where *N* is the total number of spins (M = *Ngu* в < S^{z} > is the total magnetic moment). From Eq. (2.19), we have

therefore,

where x = ^{2C}/f<*>.

*кцТ*

When *T > T _{c},* we have <

*S*>= 0 and B((0) = ^. We get

^{z}

When *T* < *T _{c},* we have <

*S*0. Expanding Bj(x) with respect to <

^{z}>-*■*S*>, we obtain

^{z}

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 *T _{c}* given by (2.23).

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

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 *T _{c}.* 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: *T _{c}* 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. **H _{0} **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

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

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 S_{s}(x) around
We then obtain

therefore

*B' _{s}[x*

_{0}) being the derivative of B

_{s}(x

_{0}) with respect to

*x*taken at x

_{0}. In the same manner, we obtain for down sublattice spin < S

^{z}>

with Xq = *-ffC] S < S ^{z}+ >.*

If the two sublattices are symmetric, namely < S+ > = — < *Si >=< S ^{z}* >, 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 <

*S*> to solve

^{z}

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 < S^{z} > and on the critical temperature. Therefore, the critical temperature for antiferromagnets, called "Neel temperature" and denoted by *T _{N},* is given by

We calculate < *AS± >.* Since *H _{0}* 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 >T*B^(- • •) —

_{N},^{we}8

^{et }

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

where we notice the + sign in front of *T _{N},* 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 H_{0} || Ox, we modify (2.42) and (2.43) to obtain

and

We show in Fig. 2.7 xn and *x±* versus *T.*

In materials which have magnetic domains or in powdered systems, experimental susceptibility at Г < *T _{N}* 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 H_{0} is sufficiently strong, the results will be different as seen below.

We suppose that H_{0} is parallel to the *z* axis. The f spins have their energy lowered by the Zeeman effect *—g/u _{B}S?H_{0}* while the

*i*spins have their energy increased by

*—giu*0). Contrary to the weak field case where the spins remain approximately antiparallel because of the dominant

_{B}Sf_{n}H_{0}>0[S^ <*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 H_{0} 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 H_{0} with respect to the *+z* axis.

The exchange energy is written as

where

with *M* being the magnetization modulus.

Figure 2.8 Spin orientation with respect to the direction of H_{0}.

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}^_{s}^{s}^, we get

Before minimizing (2.66) with respect to £, we consider the regime where *Л* » *K, H _{0}.* 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 (H_{0} || Oz). The solutions are

- • ? = 0 if
*H*_{0}< H_{Cl} - • < ~
*n/2*if*H*where_{0}> H_{c}

The spin configurations corresponding to these two solutions are displayed in Fig. 2.9. The transition between these phases when *H _{0} = H_{c}* is called "spin-flop transition”: the spins are approximately perpendicular to H

_{0}for

*H*est called "critical field.”

_{0}> H_{c}. H_{c}This result has been obtained with the hypothesis Л » *K, H _{0}. *In the case where

*H*is larger than

_{0}*H*and larger than the local exchange field acting on a spin, all spins will turn into the direction of H

_{c}_{0}.

Figure 2.9 Spin orientation with respect to H_{0} when *H _{0} < H_{c}* (left) and

*H*(right).

_{0}> H_{c}Figure 2.10 Left: Phase diagram of an antiferromagnet with Ising spins under an applied field of amplitude *H _{0}.* 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 H_{0} is supposed to be parallel to Oz. If *H*_{0}* < H _{c}* where

*H*is the critical field which is to be determined for the Ising model (see Problem 10 below), the spins remain antiparallel between them. If

_{c}*H*

_{0}*> H*all spins are parallel to H

_{c},_{0}: 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 *H*_{0}* < H _{c}* 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

*T*Of course,

_{c}.*T*depends on

_{c}*H*We display schematically a phase diagram in Fig. 2.10. More details on the phase transition are given in Chapter 5.

_{0}.