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A new two-band polaronic Hamiltonian is derived from a generalized periodic Anderson model. This is achieved by making use of a new generalized Lang–Firsov transformation on the periodic Anderson model completed with new terms representing the interactions between local phonons and both single electrons and pairs of them with opposite spins residing on the same lattice site. The novelty consists in the appearance of three- and four-fermion terms in the resulting Hamiltonian. The interaction between phonons and pairs of electrons with opposite spins was introduced at first time by Hirsch in order to describe the weakening of the Coulomb repulsion acting between two electrons with opposite spins occupying the same atomic orbital observed in experiments. The three-fermion terms represent the local interactions between a single polarons from one band and a pair of them with opposite spins from the other one. Whether it is attractive or repulsive depends on the signs of couplings of phonons to electron pairs. The four-fermion term stands for the local attraction between pairs of electrons with opposite spins from two bands. In the case of presence of such an interaction, the formation of local fermion quartets is possible. A simplified version of the generalized Hamiltonian is investigated in the end of the paper. The influence of the three- and the four-fermion terms on the properties of the two-band fermion system with two order parameters is examined. Expressions for the order parameters at zero temperatures and finite ones along with the critical temperatures for two-electron subsystems have been derived. This was achieved in the atomic limit by making use of the mean field method.

The large nonlinear Hall effect was found in (FeCo)${}_{0.67}$Ge${}_{0.33}$/Ge heterojunctions formed by sputtering amorphous $p$-type (FeCo)${}_{0.67}$Ge${}_{0.33}$ magnetic semiconductor films on near intrinsic n-type Ge substrate. It is very interesting that the mechanisms of the large nonlinear Hall effect in (FeCo)${}_{0.67}$Ge${}_{0.33}$/Ge heterojunctions are different at different temperature ranges. Below 10 K, the Hall resistance of (FeCo)${}_{0.67}$Ge${}_{0.33}$/Ge heterojunctions is almost the same as the anomalous Hall effect of (FeCo)${}_{0.67}$Ge${}_{0.33}$ ferromagnetic films. While the temperature increased from 10 to 60 K, the nonlinear Hall resistance, longitudinal conductance, and magnetoresistance all increased quickly and reached the maximum at T=60 K. In this case, thermally excited conducting carriers can tunnel through the interfacial potential barrier in (FeCo)${}_{0.67}$Ge${}_{0.33}$/Ge heterojunctions. Thus, in the range of 10–60 K, the enhanced nonlinear Hall resistance can be attributed to the anomalous Hall effect which was further enhanced by interfacial Rashba spin–orbit coupling effect. When the temperature further increased from 60 to 250 K, the interfacial potential barrier weakened gradually, and the Hall resistance and magnetoresistance decreased due to the shunting of the Ge substrate. In this case, the nonlinear Hall effect of (FeCo)${}_{0.67}$Ge${}_{0.33}$/Ge heterojunctions can be explained very well by the two-band model of nonlinear Hall effect.

We report on tunneling and Andreev-reflection conductance spectra of 39 K superconducting magnesium diboride, obtained with Pb and Au counter-electrodes. Two distinct steps at close to 2.7 and 7.1 meV appear in a low-resistance metallic-type Au–MgB₂ junction characteristic, whereas a tunneling-like spectrum measured for the same junction, annealed by the application of DC current, exhibits only a rounded contribution of the larger gap. Junctions with a superconducting lead counter-electrode pressed into a bulk MgB₂ sample reveal two conductance peaks that are interpreted as the result of the formation of a highly-transmitting break junctions inside the magnesium diboride ceramic. Our results strongly support the two-band model with two different gap values on quasi-two-dimensional σ (7.1 meV) and three-dimensional π (2.7 meV) Fermi surface sheets of MgB₂.

We describe recent research on Zitterbewegung (ZB, trembling motion) of electrons in semiconductors. A brief history of the subject is presented. Zitterbewegung of charge carriers in monolayer and bilayer graphene is elaborated in some detail. Effects of an external magnetic field on ZB are described using monolayer graphene as an example. Nature of electron ZB in crystalline solids is explained. It is proposed to trigger the coherent trembling motion of many electrons by exciting them from the valence to the conduction band by laser pulses and detecting the resulting ZB by nonlinear optical effects. Simulations of the trembling motion in vacuum and in semiconductors are reviewed. A recent experimental observation of the trembling motion in semiconducting material InGaAs is mentioned.