RKKY interaction in semiconductors: Effects of magnetic field and screening C. H. Ziener, S. Glutsch, and F. Bechstedt Friedrich-Schiller-Universität Jena, Institut für Festkörpertheorie und Theoretische Optik, Max-Wien-Platz 1, 07743 Jena, Germany (Received 15 April 2003; revised manuscript received 5 March 2004; published 24 August 2004) We derive the Ruderman-Kittel-Kasuya-Yosida (RKKY) theory for semiconductors including band degen- eracy, modulation of the band edges, and external fields. Explicit expressions are given for a three- and two-dimensional semiconductor in a magnetic field. Screening effects are included by calculating the density correlation function. We show that the RKKY theory with screening is equivalent to the mean-field Zener model. DOI: 10.1103/PhysRevB.70.075205 PACS number(s): 75.30.Hx, 75.50.Pp, 75.75.+a I. INTRODUCTION Diluted magnetic semiconductors (DMS) have been ex- tensively studied for more than a decade. 1,2 The exchange coupling between magnetic ions and electrons leads to a gi- ant effective g factor. In recent years, ferromagnetism in Mn- doped III-V and II-VI semiconductors has been predicted theoretically 3–7 and measured experimentally; 8–14 for a re- view, see Ref. 15. In low-dimensional semiconductors, fer- romagnetism can be controlled through the modulation of the band edges, external fields, and carrier injection by optical pulses. 7,12,14 The goal is the development of semiconductor devices which use the spin degree of freedom for the storage and processing of information, known as spintronics. 16,17 The indirect exchange interaction of magnetic ions by electrons has been explained by the Ruderman-Kittel- Kasuya-Yosida (RKKY) theory, developed in the 1950s, 18–20 and by a mean-field Zener theory, which goes back to papers by Stoner from the 1930s. 21 The range function of the RKKY interaction has been calculated already in the early papers for interaction-free electrons in three, two, and one dimension, both in momentum 22 and in real space. 22–25 Dietl et al., 3 as- suming only the lowest subband to be occupied, expressed the RKKY interaction of a low-dimensional semiconductor in terms of the range function of the ideal three-, two-, and one-dimensional electron gas. However, realistic low-dimensional semiconductors are never perfectly two or one dimensional. For example, quan- tum wells and superlattices have an effective dimension be- tween two and three, quantum wires between one and two. As the translational symmetry is broken in the directions of confinement, the range function does no longer depend only on the difference of the coordinates. It is quite common to treat Bloch electrons like free electrons and few papers pay attention to the details of the band structure. 4,26–28 Moreover, the magnetic field itself leads to a quantization of the electron and hole motion and to reduction of the di- mensionality. For example, a bulk semiconductor in a mag- netic field has features both of a three- and of a one- dimensional semiconductor. To describe experiments in high magnetic fields, it is no longer justified to assume the RKKY interaction to be the same as in the field-free case and the influence of the magnetic field on the range function needs to be taken into account. An open problem is the effect of screening. It was argued on the basis of the diagram technique that screening plays no role in the absence of Zeeman splitting. 29 In semiconductors, Zeeman splitting is usually not negligible. The spin suscep- tibility, including screening, was calculated already in the late sixties, 30 but these results did not make their way into the theory of ferromagnetism in semiconductors. It is also not clear if the perturbation-theoretical character of the RKKY theory limits its validity and in which way RKKY theory and mean-field Zener theory are related to each other. In this paper, we present a rigorous derivation of the RKKY theory for semiconductors. First, in Sec. II, we derive the RKKY interaction by means of both perturbation theory and Kubo theory and show the equivalence. Then, in Sec. III, we derive the effective RKKY Hamiltonian for low- dimensional semiconductors, including the sample geometry, band degeneracy, and external fields. To illustrate the useful- ness of the theory, in Sec. IV we derive analytical results for a bulk semiconductor and an ideally two-dimensional semi- conductor in a magnetic field. The results are compared with the range functions for the three-, two-, and one-dimensional electron gas, and limiting cases are studied. In Sec. V, we calculate the range function in the presence of screening and show the equivalence of RKKY theory including screening and mean-field Zener theory. Summary and conclusions are given in Sec. VI. II. THE RKKY INTERACTION The derivation of the effective Hamiltonian by second- order perturbation theory in the original paper by Ruderman and Kittel is somewhat hand waving. 18 An alternative ap- proach to the effective interaction is based upon the magnetic susceptibility, which was pioneered by Wolff. 29,31 Here we give a rigorous derivation of the effective Hamiltonian by Löwdin’s perturbation theory and by linear response theory, which gives the same result. The effective interaction is re- lated to the density correlation function, which allows us to systematically study the role of many-particle effects. A. Degenerate perturbation theory We start with Löwdin’s degenerate perturbation theory. Suppose the Hamiltonian of a system is of the form PHYSICAL REVIEW B 70, 075205 (2004) 1098-0121/2004/70(7)/075205(14)/$22.50 ©2004 The American Physical Society 70 075205-1