Spin Impurities in Heisenberg Antiferromagnets

Abstract

Work on the effects produced by a single spin impurity placed in a uniform spin system which interacts through Heisenberg exchange and with anisotropy fields is considered. The problem is formulated in terms of a two-time Green's function, whose equation of motion is simplified by the rpa. An infinite set of equations arises which contains the spin magnetizations as parameters; these are to be found consistently in terms of the Green's functions using the Callen-Shtrikman algorithm. At low temperatures an approximation equivalent to that of spin-wave theory reduces the set of equations to nine in the bcc case. These may be solved and permit a discussion of the modes of excitation of the impurity host system. These may be localized, above or below the spin-wave band, or act as resonant modes within it. Explicit expressions for the spin deviation at any site may also be given. Exactly similar equations may be used at higher temperatures if it is assumed that in the equations all spins save the impurity have the same magnetization. The impurity magnetization itself may be found by solving iteratively the equations for the impurity Green's function and the magnetization. This model is very useful for general qualitative discussions. Magnetization vs. temperature curves have recently been obtained (with D. Hone and H. E. Callen) for various impurity situations, using a method previously applied by the same authors to ferromagnetic hosts. This model determines only the impurity magnetization, but the first shell of down-spins is allowed to depart from the host magnetization, although it is only treated approximately. Self-consistency is achieved, however, for the impurity. The convergence is excellent. Some vigilance has to be exercised in searching for localized modes and in doing integrations over the sharp peaks which may occur in the Green's functions. Examples of magnetization curves are given for systems with resonant modes, with localized modes above and below the spin wave band, with levels which change from resonant to localized as the temperature changes and so forth. In the absence of anisotropy, some of the features of the zero-point deviation of the impurity which appear may be predicted by the use of second-order perturbation theory treating the transverse exchange as a perturbation.