We review the physical properties of diluted magnetic semiconductors (DMS) of the type AII1−xMnxBVI (e.g., Cd1−xMnxSe, Hg1−xMnxTe). Crystallographic properties are discussed first, with emphasis on the common structural features which these materials have as a result of tetrahedral bonding. We then describe the band structure of the AII1−xMnxBVI alloys in the absence of an external magnetic field, stressing the close relationship of the sp electron bands in these materials to the band structure of the nonmagnetic AIIBVI ‘‘parent’’ semiconductors. In addition, the characteristics of the narrow (nearly localized) band arising from the half‐filled Mn 3d5 shells are described, along with their profound effect on the optical properties of DMS. We then describe our present understanding of the magnetic properties of the AII1−xMnxBVI alloys. In particular, we discuss the mechanism of the Mn++‐Mn++ exchange, which underlies the magnetism of these materials; we present an analytic formulation for the magnetic susc...

Diluted magnetic semiconductors

The electronic properties and stereochemistry of mono-nuclear complexes of the copper(II) ion

A Mathematical Model for the Narrowing of Spectral Lines by Exchange or Motion

When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum‐mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.

Stochastic liouville equations

A theory of ESR linewidths for substances in which the magnetic anisotropy is small and for which the orbital magnetism has been essentially quenched is developed. Nuclear quadrupole moments, zero field splittings, anisotropic Zeeman terms, and intramolecular electron‐nuclear dipolar interactions, as well as motional and exchange effects are considered in the strong field case. The theory is developed in a manner that is particularly adaptable to the study of liquids but it is also applied to crystals since the resulting equations, though only applicable for small anisotropies, are particularly simple. The theory provides an extension of previous theories on ESR spectra in liquids. Several applications of the theory are discussed with particular emphasis on V4+ chelates. The theory of exchange in liquids, a phenomenon which is complicated by the noncommutivity of the motional and exchange Hamiltonians is considered in special detail. It is shown that this theory can be used to explain Hausser's results—th...

Theory of ESR Linewidths of Free Radicals

In this paper the problem of the line shape in paramagnetic resonance when large exchange interaction is present is discussed from the standpoint of a simplified mathematical model. The mathematical model can be called the model of "random frequency modulation": It is assumed that the atom absorbs a single frequency, which varies over a distribution determined by the dipolar local fields, but that this frequency varies randomly in time at a rate determined by the exchange interactions.The predicted line shape in the case in which exchange is large is of resonance type in the observable center of the line, but falls off more rapidly in the wings. This line shape has been verified experimentally in a number of cases. This conclusion seems quite independent of any assumption about the type of random frequency modulation, etc. The quantitative conclusions are reached in the following way: It is suspected, since the exchange motion is the superposition of the effects of a number of neighbors which is not particularly small, that a good approximation to the modulation function is Gaussian noise with a Gaussian spectrum. This, of course, is what would result from the superposition of a large number of rather small effects. Under this assumption both the second moment (which is independent of exchange) and the fourth moment of the line shape can be calculated. This kind of modulation is the simplest one which does give a finite fourth moment; a Markoffian, or "jump," type of modulation, which might seem more reasonable at first, does not. These moments are then compared with the moments computed by Van Vleck [Phys. Rev. 74, 1168 (1948)] to fix the two adjustable parameters, mean square frequency, and average rate of change of frequency, of the theory.The result as to line breadth, which is essentially $\ensuremath{\Delta}\ensuremath{\cong}\frac{{〈(\ensuremath{\Delta}{\ensuremath{\omega}}^{2})〉}_{\mathrm{Av}} \mathrm{d}\mathrm{i}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}{\frac{J}{\ensuremath{\hbar}}},$ if $J$ is the exchange integral, can be compared with observed line breadths by estimating $J$ from Curie-Weiss constants for a number of materials. The results are quite satisfactory if the theory is extended in two ways: (a) When the exchange frequency is larger than the resonance frequency, it can be shown that the off-diagonal elements of the dipolar interaction must be included, leading to a line-width larger by a factor of roughly 10/3; (b) in a number of cases hyperfine and Stark splitting is contributing importantly to the width.The good agreement with experiment in the cases we have investigated leads us to believe that a quantitative approach to the paramagnetic resonance line breadth problem, using only the already known concepts of dipolar interaction, exchange narrowing, and fine structure splitting, will probably explain all the observed phenomena.

P. R. Weiss

Papers

Exchange Narrowing in Paramagnetic Resonance