# Ferrimagnetism in Mean-Field Theory

Ferrimagnetic materials have complicated crystalline structures. There are often many sublattices of non equivalent spins interacting with each other via antiferromagnetic couplings. A well-known example is Fe_{2}0_{3} (ferrites) which has three sublattices: Fe^{3+} with f spin, Fe^{3+} with | spin and Fe^{2+} with t spin. The spin amplitude of Fe^{3+} is 5/2, that of Fe^{2+} is 2. As a consequence, the resulting magnetization comes from the spins of Fe^{2+}. For ferrimagnets, we should introduce several interaction parameters intra- and intersublattices. Ferrimagnets have very rich and complicated properties which are used in numerous applications such as recording devices, thanks to their very high critical temperatures of the order of 500- 800° C.

We introduce here a very simple model to illustrate some remarkable properties of ferrimagnets. We consider a system of Heisenberg spins which is composed of two sublattices, sublattice *A* containing t spins of amplitude *S _{A}* and sublattice

*В*containing j spins of amplitude Sb . The Hamiltonian is written as

where (/, /') and (m, m') indicate the sites of *A* and *B,* respectively. The interactions are /_{2}^{Л} between the neighbors belonging to the sublattice *A,* between neighbors belonging to the sublattice *B,* and *J* i between inter-sublattice nearest neighbors. We suppose /i>0 (antiferromagnetic). The signs of and /_{2}^{S} can be arbitrary. For simplicity, we assume /_{2}^{Л} = /_{2}^{8} = 0. We can start with equations (2.47) and (2.49) for two sublattices in zero applied field:

where *C* is the coordination number. Since the sublattices are not equivalent because S_{A} Ф Sb, we have to solve these two coupled equations by iteration.

Let us use the notations *M _{A}* =<

*S*and

^{Z}A >*M*=<

_{B}*S*>. At

^{Z}B*T*= 0, the expansion of the functions B

_{Sa}(• • •) and B

_{Sb}(• • •) gives

*M*=

_{A}*S*and M

_{A}_{B}= Sb (see Section 2.1). At low temperatures, we can obtain the solution for

*M*and

_{A}*M*by solving graphically Eqs. (2.73)-(2.74). However, it is more complicated to calculate the critical temperature. The high-temperature expansion similar to (2.20) gives two equations containing

_{B}*M*and

_{A}*M*of the form

_{B}*M*=

_{A}*a[S*+

_{A}, T)M_{B}*b[S*• • • and

_{A}, T)M^{3}B +*M*=

_{B}*c{S*

_{B}, T)M_{A}+*d[S _{B}, T)M_{A}* H----where я(5

_{л},

*T),*b(S,,,

*T*), c(S

_{B},

*T)*and d(S

_{B},

*T)*

are coefficients depending on *S _{A}, S_{B}* and

*T.*An explicit expression of the critical temperature

*T*can be obtained (see Problem 6 below). We have

_{N}

This result is equivalent to (2.52) for antiferromagnets if *S _{A}* =

*S*Let us give a qualitative argument. We suppose that

_{B}.*S*When

_{A}> S_{B}.*M*becomes very small

_{B}*M*is still large. It induces a local field on its

_{A}*В*neighbors, keeping them from going to zero. As long as

*M*is not zero,

_{A}*M*is maintained at a non zero value. However, fluctuations of

_{B}*M*affect in turn

_{B}*M*Therefore, the critical temperature is somewhere between the two critical temperatures of the sublattices when they are independent, namely

_{A}.

For *S _{A} = 2, S_{B} = 1* and

*C*= 8 (body-centered cubic lattice), we have

*k*= 16,

_{B}T_{A}/J_{x}*k*= 16/3 ~ 5.33, and

_{B}T_{B}/)i*k*i = 8^12/3 ~ 9.2376. The numerical solution of (2.73) and (2.74) for the above values of S^,

_{B}T_{N}/J*S*and

_{B}*C*is shown as a function of

*T*in Fig. 2.11.

Note that the sublattices have different characteristics: When we include /_{2}'^{4} and /_{2}^{B} for example, it can happen that *M _{A }*(>0) and

*M*(<0) cancel out at some

_{B}*T*below

*T*yielding a zero net magnetization. This temperature is called "magnetization compensation point” observed in some ferrimagnetic materials such as garnets and rare-earth transition-metal alloys. Ferrimagnets may also have an angular-momentum compensation point where the net angular momentum vanishes. This compensation point is a crucial point for achieving high speed magnetization reversal in magnetic memory device [326].

_{N},Figure 2.11 Numerical solution of (2.73) and (2.74) is shown as a function of Г for *S _{A} =2,S_{B} =* 1

*,C*= 8 and

*k*i = 1. See text for comments.

_{B}/)# Conclusion

In this chapter, we have presented the mean-field theory applied to ferromagnets, antiferromagnets and ferrimagnets. This theory is considered as the first approximation used to study behaviors of systems of interacting spins. Low-temperature properties and the phase transition are described with physical quantities such as the order parameter, the critical temperature, the susceptibility and the specific heat. The theoiy gives qualitative results for dimension 3, but becomes incorrect in lower dimensions. It can be improved by higher-order approximations such as the Landau-Ginzburg theory (see Section 5.3) and the cluster-variation method (CVM). The CVM consists in an exact treatment of a cluster of small size embedded in a crystal: The interaction of the cluster with the crystal is treated with the mean-field theory. The larger the cluster size, the better the results. However, the calculation is very complicated for large sizes. The method with smallest clusters is sometimes called "Bethe- Peierls-Weiss" approximation (see Problem 7). To obtain more precise properties of interacting spin systems, we can use other methods such as the spin-wave theory (Chapter 3), Green's function theory (Chapter 4), the renormalization group or the Monte Carlo simulation (Chapter 5).

# Problems

Problem 1. Define the order parameter of an antiferromagnetic lattice of Ising spins.

Problem 2. Consider the q-state Potts model defined by the Hamiltonian (1.66) on a square lattice.

- (a) Define the order parameter of the
*q*-state Potts model. - (b) Describe the ground state and its degeneracy when
*J >*0. - (c) If у <0, what is the ground state for
*q =*2 and*q =*3? For*q*= 3, find ways to construct some ground states and give comments. - (d) Show that the Potts model is equivalent to the Ising model when
*q*= 2.

Problem 3. Domain walls:

In magnetic materials, due to several reasons, we may have magnetic domains schematically illustrated in Fig. 2.5. The spins at the interface between two neighboring domains should arrange themselves in a smooth configuration in order to make a gradual change from one domain to the other. An example of such a "domain wall” is shown in that figure. Calculate the energy of a wall of thickness of *N* spins.

Problem 4. Bragg-Williams approximation:

The mean-field theory presented in this chapter can be demonstrated by the Bragg-Williams approximation described in this problem.

Consider a crystal of *N* sites each occupied by an Ising spin at a given temperature *T.* The coordination number (number of nearest neighbors) is *z.* One supposes the periodic boundary conditions. Let *N+* and AL be the number of up and down spins, respectively. The energy of a pair of parallel spins is —/ (/ > 0) and that of a pair of antiparallel spins is +/. Let *X* be defined by *N _{+}* = A/(l +

*X)/2.*One has then AL = Af(l -

*X)/2.*

- (a) Calculate the entropy
*S.* - (b) Calculate the probability to have an up spin at a lattice site. Deduce the numbers of up-up, down-down and up-down spin pairs, as functions of
*X.* - (c) Calculate the energy of the crystal as a functions of z,
*J*and*X.* - (d) Calculate the free energy
*F.*Deduce the expression of*X*at thermal equilibrium, namely at the minimum of*F.*Show that this expression is equivalent to the mean-field equation (2.12) with*S*= ±1. - (e) Give the mean-field solution for the critical temperature recalculate the entropy for
*T > T*_{c}.

Problem 5. Binary alloys by spin language:

Consider a lattice where there are two kinds of site such as the one shown in Fig. 2.12: sites of type / (white circles) and sites of type *II* (black circles). There are two kinds of atoms *A* and *В* occupying the lattice sites. The number of each atom type is *N/2.* The interaction energy between two neighbors of the same kind is e, that between two neighbors of different kinds is *ф.* One supposes e > *ф.*

In the disordered phase, half of *A* atoms are on the white sites and the other half on the black sites. The same situation is for *В* atoms. We can study the ordering structure of this binaiy alloy by mapping the problem into a spin language: an *A* atom is represented by an up Ising spin and a *В* atom by a down Ising spin. The *A — В* attractive interaction is replaced by an antiferromagnetic interaction.

Figure 2.12 Binary alloy (see Problem 5): white and black circles represent sites of type *I* and / /, respectively.

- (a) Describe the ground state.
- (b) The system energy is
*E.*Let*N^i*be the number of t-spins occupying sites of the type /.We define*x*by

• What is the value domain of *x* ? Which state does *x* = 0

correspond to? Calculate as a function of *x* the number of t-spins occupying sites of type *II* The same

question is for |-spins. One considers *x >* 0 hereafter.

- • Calculate the probabilities as functions of
*x*for a t-spin at a site of type*1*and at a site of type*11,*supposing that all probabilities are independent. The same question is for a t-spin. - • Let and j be the numbers of ft. 44 and

tl spin pairs, respectively. Calculate these quantities as functions ofx. Show that = *N[l-x ^{2})/2,* =

N[1-x* ^{2})/2, N_{bl} = N[1* + x

^{2}).

• Calculate *E* as functions of *x, e* and *ф.* Show that *E* can be written as

- • Calculate £?(£■) the number of microscopic states of energy
*E.*Deduce the entropy S. - • Calculate temperature
*T.*Show that

Show that *x* tends to 1 at low *T,* and that *x* = 0 when *T > Т _{с} = 2{е-ф)/к_{в}.*

Problem 6. Critical temperature of ferrimagnet:

Using the mean-field theoiy, calculate the critical temperature *T _{N}* of the simple model for a ferrimagnet described in Section 2.3.

Problem 7. Improvement of mean-field theory:

(a) Two-spin problem:

Consider the following Hamiltonian

where *J* (exchange interaction) and *D* (magnetic anisotropy) are positive constants and *В* magnitude of an applied magnetic field in the *z* direction. Find the eigenvalues and eigenvectors of *H* for spin one-half.

(b) Improved mean-field theory:

Consider the Heisenberg spin model:

In the first step, we treat exactly the interaction of two neighboring spins. In the second step, we use the mean- field theory to treat the interaction of the two-spin cluster embedded in the crystal. Explicitly, consider two spins S_{; }and S*j* embedded in a crystal. The Hamiltonian is given by

where *Z* is the coordination number. Show that the critical temperature *T _{c}* for 5 = 1/2 is given by

Problem 8. Interaction between next-nearest neighbors in mean- field treatment:

Consider a centered cubic lattice where each site is occupied by an Ising spin with values ±1. The Hamiltonian is given by

where *] _{2}* (>0) interaction between next-nearest neighbors. The first and second sums are made over pairs of corresponding neighbors.

- (a) Describe the magnetic ordering at temperature
*T =*0. - (b) Give briefly the hypothesis of the mean-field theory.
- (c) By a qualitative argument, show that the interaction between next-nearest neighbors,
*J*increases the critical temperature._{2},

- (d) Using the mean-field theory, calculate the partition function of a spin at a given temperature
*T.*Deduce an equation which allows us to calculate <*a*>, mean value of a spin, at*T.* - (e) Determine the critical temperature
*T*as functions of_{c}*J i*and*h-* - (П In the case where
*J*is negative (antiferromagnetic interaction), the above result is no more valid beyond a critical value of |/_{2}_{2}1. Determine that critical value /_{2}^{C}- What is the magnetic ordering when |/_{2}|*J*at_{2}*T*= 0?

Problem 9. Repeat Problem 7 in the case of an antiferromagnet.

Problem 10. Calculate the critical field *H _{c}* in the following cases

- (a) a simple cubic lattice of Ising spins with antiferromagnetic interaction between nearest neighbors
- (b) a square lattice of Ising spins with antiferromagnetic interaction
*J i*between nearest neighbors and ferromagnetic interaction*J*between next-nearest neighbors._{2}