Oliver Thewalt

    Oliver Thewalt

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    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.