Theory of Magnetic Exchange Interactions: Exchange in Insulators and Semiconductors

PHILIP. W . ANDERSON
Bell Telephone Laboratories, Incorporated, Murray HiU, New Jersey

“The three central themes of the theory of magnetism in insulators were stated early in the 1930’s: the Heisenberg-Dirac Hami1tonian;- the concept of antiferromagnetism,’ll and the idea of superexchange.12 To these one might add the ideas, of basic importance in understanding the field,
of the crystal field theory.

The Heisenberg paper pointed out that ordinary Coulomb interactions among electrons, together with the Pauli principle, could lead to an exchange effect strongly coupling their spins, which might give rise to ferromagnetism.

The hypothetically ferromagnetic interaction (11.1) formed the basis for Heisenberg's theory of ferromagnetism. Clearly, the lowest state of such an interaction is that in which all spins are parallel, or ferromagnetic. While it is now generally accepted that magnetic interactions result from
some such combination of Coulomb effects with the Pauli principle, it is highly probable-and well accepted in the field-that this specific mechanism is very seldom the sole cause of ferromagnetism. This may be said because in insulators, the only case in which the Heisenberg Hamiltonian is correct, the interaction is almost universally antiferromagnetic.

Née1 first expressed the possibility that the negative sign of interaction, if present, could lead to a state in which different subsets or sublattices of the spins in a crystal could align themselves antiparallel. This is the state later" called antiferromagnetism. Néel, however, did not discover the
thermal transition point analogous to the Curie point, now called the Néel point; Landaulo stressed the phenomenological similarity of ferro- and antiferromagnetism and suggested the existence of the Néel point. Néel's theory also suffered by being at first intended for metals, where antiferromagnetism is a much more complicated and subtle phenomenon than in insulators.

Hulthén'' proposed a spin-wave theory of antiferromagnetism of the semiclassical Heller-Kramers type, but in his thesisI8 he pointed out the still serious questions of principle about such a theory, adapting earlier work of Bethels to calculate the exact ground state for a one-dimensional
antiferromagnet. He suggested that it was not in fact antiferromagnetically ordered. The dilemma was essentially that one can prove that the lowest energy state of an antiferromagnet must be a singlet state, and thus have no directional character.

Meanwhile Néel,Bitter,and especially Van Vleck” were developing the simpler molecular field type of theory for antiferromagnetism.

The idea of superexchange first appeared when Kramers12 tried to understand the early results of adiabatic demagnetization, which indicated that small exchange couplings existed even between ions separated by one or several diamagnetic groups (halide ions, waters of crystallization, etc.) . He pointed out that the magnetic ions could cause spin-dependent perturbations in the wave functions of intervening ions, thereby transmitting the exchange effect over large distances; but ‘no specific mechanisms were discussed.