Using the well-known Kubo formula, we evaluate magnetotransport quantities, such as the collisional and Hall conductivities of the α-T model. The collisional conductivity exhibits a series of peaks at a strong magnetic field. Each of the conductivity peaks for [Formula: see text] (graphene) splits into two in the presence of a finite α. This splitting occurs due to a finite phase difference between the contributions coming from the two valleys. The density of states is also calculated to explore the origin of the splitting of conductivity peaks. As α approaches 1, the right split part of a conductivity peak comes closer to the left split part of the next conductivity peak. At [Formula: see text], they merge with each other to produce a new series of the conductivity peaks. On the other hand, the Hall conductivity undergoes a smooth transition from [Formula: see text] to [Formula: see text] with n  =  0,1,2,... as we tune α from 0-1. For intermediate α, we obtain the Hall plateaus at values 0,2,4,6,8,... in units of e/h.

We consider the $\alpha$-$T_3$ model which provides a smooth crossover between the honeycomb lattice with pseudospin $1/2$ and the dice lattice with pseudospin $1$ through the variation of a parameter $\alpha$. We study the dynamics of a wave packet representing a quasiparticle in the $\alpha$-T$_3$ model with zero and finite transverse magnetic field. For zero field, it is shown that the wave packet undergoes a transient $zitterbewegung$ (ZB). Various features of ZB depending on the initial pseudospin polarization of the wave packet have been revealed. For an intermediate value of the parameter $\alpha$ i.e. for $0<\alpha<1$ the resulting ZB consists of two distinct frequencies when the wave packet was located initially in $rim$ site. However, the wave packet exhibits single frequency ZB for $\alpha=0$ and $\alpha=1$. It is also unveiled that the frequency of ZB corresponding to $\alpha=1$ gets exactly half of that corresponding to the $\alpha=0$ case. On the other hand, when the initial wave packet was in $hub$ site, the ZB consists of only one frequency for all values of $\alpha$. Using stationary phase approximation we find analytical expression of velocity average which can be used to extract the associated timescale over which the transient nature of ZB persists. On the contrary the wave packet undergoes permanent ZB in presence of a transverse magnetic field. Due to the presence of large number of Landau energy levels the oscillations in ZB appear to be much more complicated. The oscillation pattern depends significantly on the initial pseudospin polarization of the wave packet. Furthermore, it is revealed that the number of the frequency components involved in ZB depends on the parameter $\alpha$.

We study the effect of an in-plane magnetic field on the zitterbewegung (ZB) of electrons in a semiconductor quantum well (QW) and in a quantum dot (QD) with the Rashba and Dresselhaus spin-orbit interactions (SOIs). We obtain a general expression of the time-evolution of the position vector and current of the electron in a semiconductor QW. The amplitude of the oscillatory motion is directly related to the Berry connection in momentum space. We find that in presence of the magnetic field the ZB in a QW does not vanish when the strengths of the Rashba and Dresselhaus SOIs are equal. The in-plane magnetic field helps to sustain the ZB in QWs even at a low value of k(0)d (where d is the width of the Gaussian wavepacket and k(0) is the initial wavevector). The trembling motion of an electron in a semiconductor QW with high Landé g-factor (e.g. InSb) is sustained over a long time, even at a low value of k(0)d. Further, we study the ZB of an electron in QDs within the two sub-band model numerically. The trembling motion persists in time even when the magnetic field is absent as well as when the strengths of the SOI are equal. The ZB in QDs is due to the superposition of oscillatory motions corresponding to all possible differences of the energy eigenvalues of the system. This is an another example of multi-frequency ZB phenomenon.

A theory of hot electron cooling power due to polar optical phonons P is developed in 3D Dirac semimetal (3DDS) CdAs taking account of hot phonon effect. Hot phonon distribution N and P are investigated as a function of electron temperature T , electron density n, and phonon relaxation time [Formula: see text]. It is found that P increases rapidly (slowly) with T at lower (higher) temperature regime. Whereas, P is weakly decreasing with increasing n. The results are compared with those for three-dimensional electron gas (3DEG) in CdAs semiconductor. Hot phonon effect is found to reduce P considerably and it is stronger in 3DDS CdAs than in CdAs semiconductor. P is also compared with the hot electron cooling power due to acoustic phonons P. We find that a crossover takes place from P dominated cooling at low T to P dominated cooling at higher T. The temperature at which this crossover occurs shifts towards higher values with the increase of n . Also, hot electron energy relaxation time [Formula: see text] is discussed. It is suggested that [Formula: see text] can be tuned to achieve faster or slower energy loss for suitable applications of CdAs.