PHYSICAL REVIE% B VOLUME 25, NUMBER 1 1 JANUARY 1982 Indirect exchange interaction in lead salts L. Liu' and G. Bastard Groupe de Physique des Solides de l'Ecole ¹rmale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05, France (Received 27 July 1981) The indirect exchange interaction between magnetic ions embedded in a lead salt host lattice, mediated by the spin-dependent polarization effects, is theoretically studied. It is found that the interaction is ferromagnetic in sign and greatly enhanced in strength because of the multivalley band structure of the host material. The interaction decays exponentially with the interspin dis- tance, and the decay constant is determined by the energy gap and the effective masses of the electron and the hole. A recent experiment in Pb~ „Mn„Te is also briefly discussed. Recently there has been some interest in the in- teraction between magnetic ions dilutely embedded in a nonmagnetic semiconductor host. In such a sys- tem, the magnetic interaction mediated by the spin- dependent polarization of the valence-band electrons may become important with a proper set of band parameters. This so called indirect exchange interac- tion has been theoretically studied by several au- thors. ' For a finite-gap semiconductor with isotro- pic energy bands, the interaction was found to decay exponentially with the interspin distance like e "". The decay constant ) is determined by the energy gap and the effective masses of the electron and the hole. For narrow-gap semiconductors with small electron and hole masses, the interaction range can be rela- tively long, extending over several lattice spacings. For zero-gap semiconductors" like Hg~ „Mn„Te, the interaction drops off like R with 5 ~ v ~4. Since the spin-dependent polarization process in a semicon- ductor involves interband electron-hole excitations, one may expect that the indirect exchange interaction should be greatly enhanced in a semiconductor host with a multivalley band structure which provides for more electron-hole excitation channels. The family of lead salts like PbTe, in addition to having a narrow energy gap and small electron and hole effective masses, has precisely such a multielectron valley and multi-hole-valley structure, and hence should show strong polarization effects. We thus decided to make a model calculation of the indirect exchange interac- tion in lead salts with a band structure similar to that of PbTe in this paper. Apart from the complications arising from the anisotropy of the energy bands, the calculation is very similar to that of Ref. 5. The in- teraction is shown to be ferromagnetic and decay ex- ponentially with the interspin distance. We shall also give a brief discussion on the recent susceptibility data in Pb~ „Mn„Te obtained by Andrianov et al. 6 The semiconductor PbTe crystallizes in the NaC1 structure. Both the conduction- and the valence- band edges occur at the L point of the Brillouin zone. In the immediate vicinity of the L points, the energy bands are described by , , (k„~)'+(k;)' (k;)' C g mph' me (k„") +(ky")2 (k, ") tl mar mw where the energy is measured from the top of the valence band, and the wave vector from the p, th L point (L„,p, = 1,2, 3,4). The z direction is chosen to be along the I -L„axis. If we neglect the spin-orbit coupling, the indirect exchange interaction between two spins S; and S~ lo- calized at the magnetic ion sites is of the Heisenberg type': and H = — HgS( Sg, (2) H„= —, g (f„-„— f, , ) 1 n k ~/ nn kk „ IJ(nk, n'k )I xpe[i(k — k ) K] E, (k ) -E„(k) where f„-„ is the Fermi distribution function for the k state in the nth band, and R is the lattice vector joining S; and S&. The exchange integral J, as usual, involves the localized magnetic orbitals and the Bloch orbitals. ' In this calculation J is taken to be a con- stant. The summation over k and k is confined within the first Brillouin zone. To calculate H& in Eq. (3), we assume two bands, a conduction band which is completely empty and a valence band which is fully occupied at T =0. We break the k summation over the Brillouin zone into that over different electron and hole valleys. As we shall neglect the band nonparabolicity, we simply use 487 19S2 The American Physical Society