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The spontaneous order of magnetic moments is a characteristic signature of so called cooperative or collective magnetism and is typically observed in ferro-magnetic, ferrimagnetic and antiferromagnetic materials. Such ordering takes place at temperatures below a critical temperature T^∗ which is material spe-cic. In the case of ferro and ferrimagnetic materials, the critical temperature T^∗ = T_C is known as the Curie Temperature, while in antiferromagnetic ma-terials, T^∗ = T_N and known as Neel temperature. In all solid state magnetic materials, the spontaneous magnetization is destroyed (vanishes) at tempera-tures above T^∗. Exchange interactions were discovered independently by Dirac and Heisenberg in 1926 to be primarily responsible for long range magnetic ordering (cooperative magnetism) in solid state materials [156]. The matrix elements of the exchange interaction constructed with completely antisymmet-ric wavefunctions, contain terms which are classically not understandable and correspond to an exchange of the indices of the identical particles (Fermions).

Nonetheless, as a precondition for collective magnetism in solid state materi-als, there must exist permanent magnetic moments in the material. In real systems direct and indirect exchange interactions are possible.

Direct Exchange

Consider a model consisting of only two electrons with position vectors r1 and r2 where the total wavefunction of the system is equivalent to the product

of single electron states ψa(r₁) and ψb(r₂). Electrons are fermions therefore the system wavefunction must be antisymmetric as a consequence of Pauli's exclusion principle on fermions. The electrons in the considered model are indistinguishable therefore the wavefunction squared must be invariant for the exchange of both electrons. In addition, due to spin of the electrons two possi-bilities arise: a symmetric spatial part ψ in combination with an antisymmetric spin part χ or an antisymmetric part in combination with a symmetric spin part. The rst case presents a singlet state with Stotal = 0 while the second case presents a triplet state with Stotal = 1. The corresponding total wave functions are given by:

ψ_S = 1

√2[ψ_a(r₁) ψ_b(r₂) + ψ_a(r₂) ψ_b(r₁)] · χS (4.18)

ψ_T = 1

√2[ψ_a(r₁) ψ_b(r₂) − ψ_a(r₂) ψ_b(r₁)] · χT (4.19)

The energies of the singlet and triplet states amount to:

E_S = Z

ψ_S^∗Hψ_SdV₁dV₂ (4.20)

E_T = Z

ψ_T^∗Hψ_TdV₁dV₂ (4.21) taking into account normalized spin components of the singlet and triplet wavefunctions, i.e.

s² = (S₁+ S₂)² = S₁²+ S₂²+ 2S₁ · S₂ (4.22) we thus obtain:

S₁· S₂ = 1

The eective Hamiltonian can be expressed as:

H = 1

4(ES+ 3ET) − (ES− ET) S1· S2 (4.26) The rst term is constant and often included in other energy contributions.

The second term is spin dependent and more important concerning ferromag-netic properties. The exchange constant or exchange integral J is given by:

J = ES− ET

2 =

Z

ψ^∗_a(r₁) ψ_b^∗(r₂) Hψ_a(r₂) ψ_b(r₁) dV₁dV₂ (4.27) Then the spin dependent term in the eective Hamiltonian can be written as:

H_spin = −2J S₁· S₂ (4.28)

If the exchange integral J is positive then ES > E_T, i.e. the triplet state with Stotal = 1 is energetically favored. If the exchange integral J is negative, then ES < E_T, i.e. singlet state with Stotal = 0 is energetically favoured. In the case of many electron atoms in magnetic systems, the Schrödinger equation cannot be solved without assumptions. The most important part of such an interaction like the exchange interaction mostly apply between neighbouring atoms. This consideration within the Heisenberg model leads to a term in the Hamiltonian of the form:

H = −X

ij

J_ijS_i· S_j (4.29)

with Jij being the exchange constant between spin i and spin j. The factor 2 is included in the double counting within the sum. Often a good approximation is given by:

Generally, J is positive for electrons at the same atom whereas it is often negative if both electrons belong to dierent atoms.

Indirect Exchange

In real systems, i.e. magnetic materials, magnetic nanostructures etc., the separation between two magnetic moments is generally large in comparison to average inter-electron distance presented in the direct exchange mechanism.

For this reason, direct exchange is frequently not acceptable as a coupling mechanism in magnetic materials. There are a number of indirect exchange mechanisms which within the framework of second-order perturbation theory lead to an eective Hamiltonian of the Heisenberg type. They are dierent from direct exchange because the direct exchange is a result of rst-order per-turbation theory. Nonetheless, the concept of indirect exchange is not uniquely dened, for this reason, three dierent types of indirect exchange mechanisms are briey discussed below [157].

• Superexchange: This type of indirect exchange occurs predominantly in ionic solids. The exchange interaction between non-neighboring magnetic ions is mediated by means of a non-magnetic ion which is located in-between. The distance between the magnetic ions is too large that a direct exchange can take place.

Figure 4.1: Graphical representation of the function F(x) dened in Eq.4.31 using x=2kFr. Positive values lead to a ferromagnetic coupling whereas negative values lead to an anti-ferromagnetic arrangement [157].

• RKKY exchange interaction: The RKKY exchange (Ruderman, Kittel, Kasuya, Yosida) occurs typically in metals with localized magnetic mo-ments. The exchange is mediated via the conduction electrons ("in-direct") where the coupling is characterized by a distance dependent exchange integral JRKKY (r) ∝ F (2k_Fr) with

F (x) = sin (x) − xcosx

x⁴ . (4.31)

This type of exchange coupling is long range and anisotropic which often results in complicated spin arrangements. In addition, the oscillating behaviour predictates a type of coupling (ferro- or anti-ferromagnetic nature) that is a function of the distance between the magnetic moments (see Fig.4.1).

• Double exchange: This type of exchange interaction occurs in some oxides where the magnetic ions exhibit mixed valencies, i.e. dierent oxidation

states which results in ferromagnetic arrangement, eg. magnetite (Fe3O4

which includes Fe²⁺ as well as Fe³⁺ ions.