Graphene-based superlattice (SL) formed by a periodic gap modulation is studied theoretically using a Dirac-type Hamiltonian. Analyzing the dispersion relation we have found that new Dirac points arise in the electronic spectrum under certain conditions. As a result, the gap between conduction and valence minibands disappears. The expressions for the positions of these Dirac points in k-space and threshold value of the potential for their emergence were obtained. Also, the dispersion law and renormalized group velocities around the new Dirac points were calculated. At some parameters of the system, we have revealed interface states which form the top of the valence miniband.

In this work we study the effects of collapse and revival as well as Zitterbewegung (ZB) phenomenon, for the relativistic electron wave packets, which are a superposition of the states with quantum numbers sharply peaked around some Landau level n0 of the order of few tens. The probability densities as well as average velocities of the packet center and the average spin components were calculated analytically and visualized. Our computations demonstrate that due to dephasing of the states for times larger than the cyclotron period the initial wave packet (which includes the states with the positive energy only) loses the spatial localization so that the evolution can no longer be described classically. However, at the half-revival time t = TR/2 its reshaping takes place firstly. The behavior of the wave packet containing the states of both energy bands (with En > 0 and En < 0) is more complicated. At short times of a few classical periods such packet splits into two parts which rotate with cyclotron frequency in the opposite directions and meet each other every one-half of the cyclotron period. At these moments their wave functions have significant overlap that leads to ZB. At the time of fractional revival each of two sub-packets is decomposed into few packet-fractions. However, at t = TR each of the two sub-packets (with positive or negative energy) restores at various points of the cyclotron orbit, that makes it impossible reshaping of initial wave packet entirely unlike the wave packet which consists of states with energies En > 0 only. Obtained results can be useful for the description of electromagnetic radiation and absorption in relativistic plasma on astrophysics objects, where super high magnetic field has the value of the order 10 8 − 10 9 T, as well as for interpretation of experiments with trapped ions.

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The effects of collapse and revival of wave packets in monolayer and bilayer graphene in the presence of an external perpendicular magnetic field are investigated. An evolution of the electron wave packets, which are a superposition of states with quantum numbers n near the Landau level n0, are studied. The probability density and the average velocity of the center of packets were calculated analytically and visualized. The initial wave packet, which includes only states with positive energy, loses spatial localization, and splits into several sub-packets at the moments t = (m/n)TR, where TR is the revival time, m and n are co-primes. In addition, it is shown that a behavior of the wave packet, which consists of states of two energy bands (with En &gt; 0 and En &lt; 0) is more complex. Such a packet splits into two parts, which rotate with a cyclotron frequency in opposite directions. The nature of electromagnetic radiation of wave packets, undergoing periodic collapse and revival is discussed.