In this section, we describe the models that are most commonly used to describe the magnetic interactions in diluted magnetic semiconductors.
(I) Origin of exchange [57]
Consider a simple model with just two electrons which have spatial coordinate r1
and r2 respectively. The wave function for the joint state can be written as a product of single electron states, so that if the first electron is in state ψa (r1) and the second electron is in state ψb (r2), then the joint wave function can be written as ψa (r1)ψb (r2).
For electrons the overall wave function must be antisymmetric so the spin part of the wave function must either be an antisymmetric singlet state χS (S = 0) in the case of a symmetric spatial state or a symmetric triplet state χT (S = 1) in the case of an antisymmetric spatial state. Therefore we can write the wave function for the singlet case ΨS and the triplet c eas ΨT as
where both the spatial and spin parts of the wave function are included. The energies t o e states are normalized. The difference between the two energies is
E E
T2
ψa ψbΗ
ψa ψb dr1dr2For a singlet state S1 · S2 = -3/4 while for a triplet state S1 ·S2 = 1/4. Hence the Hamiltonian can be written in the form of an “effective Hamiltonian”
Η 1
4 E
3E
TE E
TS · S
This is the sum of a constant term and a term which depends on spin. The constant can be absorbed into other constant energy terms, but the second term is more interesting. The exchange constant (or e hxc ang integral , J i de ined bye ) s f
J E E
T2
ψa ψbΗ
ψa ψb dr1dr2 and hence the spin-dependent term in the effective Hamiltonian can be writtenH 2JS · S
If J > 0, ES > ET, the triplet state S = 1 is favored. On the other hand, if J < 0, ES < ET, the singlet state S = 0 is favored. This equation is relatively simple to derive for two electrons, but generalizing to a many-body system is far from trivial. This motivates the Hamiltonian of the Heisenberg model:
Η S · S
where Jij is the exchange constant between the ith and jth spins.
(II) Direct exchange
If the electrons on neighboring magnetic atoms interact via an exchange interaction, this is known as direct exchange. This is because the exchange interaction proceeds directly without the need for an intermediary. Though this seems to be the most obvious route for the exchange interaction to take, the reality in physical situations is rarely that simple. Very often direct exchange cannot be an important mechanism in controlling the magnetic properties because there is insufficient direct overlap
between the orbitals of the neighboring magnetic ions.
(III) Superexchange
Superexchange can be defined as an indirect exchange interaction between non-neighboring magnetic ions which is mediated by a non-magnetic ion which is placed in between the magnetic ions. It arises because there is a kinetic energy advantage for antiferromagnetism, which can be understood by referring to Fig. 6(a) which shows two transition metal ions separated by an oxygen ion. For simplicity we will assume that the magnetic moment on the transition metal ion is due to a single unpaired electron (more complicated cases can be dealt with in analogous ways).
Hence if this system were perfectly ionic, each metal ion would have a single unpaired electron in each d orbitand the oxygen would have two p electrons in its outermost occupied states. The figure demonstrates that antifcrromagnetic coupling lowers the energy of the system by allowing these electrons to become delocalized over the whole structure, thus lowering the kinetic energy.
Fig. 6 Depict the hoping processes of the superexchange.
(IV) Double exchange
In some oxides, it is possible to have a ferromagnetic exchange interaction which
more than one oxidation state (Fig. 6(b)). Examples of this include compounds containing the Mn ion which can exist in oxidation state 3 or 4, i.e. as Mn3+ or Mn4+
(Fig. 7).The eg electron on a Mn3+ ion can hop to a neighboring site only if there is a vacancy there of the same spin (since hopping proceeds without spin-flip of the hopping electron). If the neighbor is a Mn4+ which has no electrons in its eg shell, this should present
no problem. However, there is a strong single-centre exchange interaction between the eg electron and the three electrons in the t2g level which wants to keep them all aligned. Thus it is not energetically favourable for an eg electron to hop to a neighboring ion in which the t2g spins will be antiparallel to the eg electron (Fig. 7(b)).
Ferromagnetic alignment of neighboring ions is therefore required to maintain the high-spin arrangement on both the donating and receiving ion. Because the ability to hop gives a kinetic energy saving, allowing the hopping process shown in Fig. 7(a) reduces the overall energy. Thus the system favors to align ferromagnetically and saves energy.
The physics of this mechanism is applied to mixed-valence DMSs, i.e.
semiconductors in which one can find coexisting magnetic ions of the same chemical nature but with different charge states. The magnetic ions with different charge states were also observed in III–V DMS [58]. Although it is believed that in GaAs and InAs the Mn ions are in the high spin 2+ charge state [59, 60], the precise nature of the charge state related to Mn ions is not finally determined in these materials. Recently, the coexistence of Mn2+ and Mn3+ ions leading to double exchange induced ferromagnetism was also suggested in ZnO [61].
Fig. 7 Schematics that depict the hoping processes of the double exchange.