Lecture note on solid state physics Superexchange interaction

In preparing this note, we have examined many textbooks of magnetism where the mechanism of the superexchange interaction is discussed. We realize that it may be difficult for readers (in particular graduate students and undergraduate students studying on the magnetism) to understand the physics on the superexchange interactions from these textbooks, partly because of the limited space of the textbooks and the requirement of the amount of knowledge in quantum mechanics. The present note is based on the lecture note of the Solid State Physics which one of the authors (MS) has prepared since 1986. The note has been revised many times. The Mathematica programs are used for calculations of the eigenvalue problems and plotting the electronic density of the wavefunctions. The use of the Mathematica will be helpful for students to understand the mechanism of the superexchange interactions visually. As a supplementary, one can see our lecture note on the spin Hamiltonian and the crystal field of transition metal ions. In this note, we discuss the development of various interactions between magnetic ions; such as direct exchange interaction and superexchange interaction. Direct exchange involves an overlap of electron wavefunctions from the two sites and Coulomb electrostatic interaction repulsion. The Pauli exclusion principle keeps the electrons with parallel spin away from each other, thereby reducing the Coulomb repulsion. Originally superexchange acquired its name because of the relatively large distances, occupied by normally diamagnetic ions, radicals, or molecules. Small exchange coupling existed even between 3d ions separated by one negative ion. Anderson (1959) considered a molecular orbitals formed of the admixture of the localized 3d orbitals and p orbitals of the intervening negative ion. The bonding orbital is mainly occupied by a negative ion, while the antibonding orbital is partially occupied by 3d electrons, leading to the magnetism of the system. Thus the wavefunction of localized d spins extends over the neighboring negative ion. There is a probability of transferring from one 3d orbital of the magnetic ion to the neighboring 3d orbitals, leading to the exchange interaction. A considerably more satisfactory system of semi-empirical rules was developed over a period of years mainly by Goodenough and Kanamori. These rules have the important features of taking into account the occupation of the various d levels as dictated by ligand field theory. They are related to the prescriptions of Anderson’s paper about the sign of superexchange. The main features of the superexchange interactions are usually explained in terms of the so-called Goodenough-Kanamori-Anderson rules. According to these rules, a 180o superexchange (the magnetic ion-ligand-magnetic ion angle is 180o) of two magnetic ions with partially filled d shells is strongly antiferromagnetic, whereas a 90o superexchange interaction is ferromagnetic and much weaker.