A microscopic model of indirect exchange interaction between transition metal impurities in dilute magnetic semiconductors (DMS) is proposed. The hybridization of the impurity delectrons with the heavy hole band states is largely responsible for the transfer of electrons between the impurities, whereas Hund rule for the electron occupation of the impurity d-shells makes the transfer spin selective. The model is applied to such systems as n−type GaN:Mn and p−type (Ga,Mn)As, p−type (Ga,Mn)P. In n−type DMS with Mn 2+/3+ impurities the exchange mechanisms is rather close to the kinematic exchange proposed by Zener for mixed-valence Mn ions. In p−type DMS ferromagnetism is governed by the kinematic mechanism involving the kinetic energy gain of heavy hole carriers caused by their hybridization with 3d electrons of Mn 2+ impurities. Using the molecular field approximation the Curie temperatures T C are calculated for several systems as functions of the impurity and hole concentrations. Comparison with the available experimental data shows a good agreement.

Analytical expressions for the Hall conductivity σ yx and the longitudinal resistivity ρ xx are derived in gapped, single-layer graphene using linear response theory. The gap 2 , described by a mass term, is induced by a substrate made of hexagonal boron nitride (h-BN) and produces two levels at ± . It is shown that σ yx has the same form as for a graphene sample supported by a common substrate without a mass term. The differences are a shift in the energy spectrum, which is not symmetric with respect to the Dirac point for either valley due to the gap, the absence of a zero-energy Landau level, and the nonequivalence of the K and K valleys. In addition, the dispersion of the energy levels, caused by electron scattering by impurities, modifies mostly plateaus due to the levels at ± . It is shown that the resistivity ρ xx exhibits an oscillatory dependence on the electron concentration. The main difference with the usual graphene samples, on SiO 2 substrates, occurs near zero concentration, as the energy spectra differ mostly near the Dirac point.

We study the influence of a finite Hall field E H on the Hall conductivity σ yx in graphene. Analytical expressions are derived for σ yx using the Kubo-Greenwood formula. For vanishing E H , we obtain the well-known expression σ yx = 4(N + 1/2)e 2 /h. The inclusion of the dispersion of the energy levels, previously not considered, and their width, due to scattering by impurities, produces the plateau of the n = 0 Landau level. Further, we evaluate the longitudinal resistivity ρ xx and show that it exhibits an oscillatory behavior with the electron concentration. The peak values of ρ xx depend strongly on the impurity concentration and their potential. For a finite E H , the result for σ yx is the same as that for E H = 0, provided E H is not strong, but the values and positions of the resistivity maxima are modified due to the E H-dependent dispersion of the energy levels.

An energy level diagram is constructed on the basis of a microscopic Hamiltonian proposed for a description of interacting manganese impurities in diluted magnetic semiconductors (DMS). It is shown that ferromagnetism in p-type III-V DMS is governed by the strong hybridization of Mn 2+ -electrons with the mobile holes and localized states near the top of the valence band. The Curie temperature estimated from the proposed kinematic exchange agrees with the experiments on GaAs : Mn. The model is also applicable to the GaP : Mn system.

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