Crystallization of dense magnetic polarons in degenerate magnetic semiconductors

Crystallization of dense magnetic polarons (CDM) is studied on the basis of the density-functional method with parametrized variational electron densities at zero and finite temperatures. In the present paper, the case for antiferromagnetic semiconductors is studied in detail, in relation to heavily doped EuTe and ${\mathrm{Gd}}_{3\mathrm{-}\mathrm{x}}$ ${\mathrm{\square }}_{\mathrm{x}}$ ${\mathrm{S}}_4$ (x∼(1/3), where □ stands for vacancies. For relatively low electron concentrations, dense magnetic polarons may be crystallized; in this case, the electron densities are extremely concentrated in the central part of each crystallization unit cell so that the ferromagnetic domain with (nearly) saturated magnetization, that is, the magnetic polaron, is established in the central part of the cell, while in the remaining part of the cell, the electron densities are extremely small with a nearly antiferromagnetic ordering of localized spins. By this process, the interaction energy between electrons and localized spins is almost fully gained at the cost of the kinetic energy, the Coulomb interaction energy of electrons, and the energy of localized spins. When this state is energetically favorable compared with the homogeneous state studied previously by Umehara and Kasuya [J. Phys. Soc. Jpn. 40, 13 (1976)], CDM is stabilized. For larger electron concentrations, a different type of crystallization occurs: In the central part of the cell, the electron densities become extremely small with nearly antiferromagnetic ordering of localized spins, while in the remaining part of the cell, the electron densities are larger than the average density with ferromagnetic ordering of localized spins, which is the reverse CDM and is called the crystallization of dense bubbles (CDB). With further increasing electron concentration, CDB melts into the homogeneous state. The temperature dependence of both crystallization states is very weak except in the vicinity of the critical temperature for the transition to the homogeneous state. The result obtained from the present study is compared with experimental data, which shows that the present mechanism is a convincing one for degenerate EuTe and ${\mathrm{Gd}}_{3\mathrm{-}\mathrm{x}}$ ${\mathrm{\square }}_{\mathrm{x}}$ ${\mathrm{S}}_4$, at least at low temperatures.