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The effect of confined optical phonons and laser radiation are taken into account to study the Hall effect in a compositional semiconductor superlattice (CSSL). The new theoretical expressions of the Hall conductivity, the Hall coefficient and the magnetoresistance are obtained as functions of the external fields (the magnetic field B , the laser frequency Ω and amplitude E ), the CSSL parameters (the well length d I , the barrier length d II and height U ) and the quantum number m characterizing the phonon confinement effect. The magneto-confined phonon resonance condition in a CSSL is also carried out to explain the numerical evaluation results for the GaAs / Al x Ga 1 − x As CSSL. When optical phonons are confined, the Hall conductivity has more resonance peaks in comparison with that for bulk phonons. In addition, these sub-peaks are more separate from the main peaks as the CSSL layers thickness decreases. The tighter CSSL period is, the more forcefully the phonon confinement makes. When the CSSL period d is smaller than 30 nm , the contribution of confined optical phonons is significant and could not be neglected. The magnetophonon resonance for bulk semiconductor could be achieved in the limit the CSSL period goes to infinity. Furthermore, the Hall conductivity increases and the Hall coefficient, the magnetoresistance decrease due to the laser radiation and the effect of confined optical phonons. Especially, the near-linear dependence of the magnetoresistance on temperature is in good agreement with the experimental result.

The influence of confined optical phonons on the Hall Coefficient (HC) in a Cylindrycal Quantum Wire (CQW) with an infinite potential (for electron – confined optical phonons scattering). Consider a case where CQW is placed in a perpendicular magnetic field , a constant - electric field and an intense electromagnetic wave . By using the quantum kinetic equation for electrons interacting with Confined Optical Phonon (COP), we obtain analytical expressions for (HC), which are different from in comparison to those obtained for the HC in the case of normal bulk semiconductor and in the case of cylindrycal quantum wire with electron – unconfined phonons scattering mechanism. Numerical calculations are also applied for AlGaAs/GaAs/AlGaAs cylindrycal quantum wire, we see the HC depends on magnetic field B, temperature T, frequency Ω and amplitude E0 of laser radiation and especially quantum index m1 and m2 characterizing the phonon confinement. This influence is due to the quantum index m1 and m2, which makes an increase of Hall coefficient by 2,3 times in comparition with the case of unconfined phonons. When the quantum number m1 and m2 goes to zero, the result is the same as in the case of unconfined phonons.

Based on the quantum kinetic equation method, the Hall effect in doped semiconductor superlattices (DSSL) has been theoretically studied under the influence of confined LO-phonons and the laser radiation. The analytical expression of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient of a GaAs:Si/GaAs:Be DSSL is obtained in terms of the external fields, lattice period and doping concentration. The quantum numbers N, n, m were varied in order to characterize the effect of electron and LO-phonon confinement. Numerical evaluations showed that LO-phonon confinement enhanced the probability of electron scattering, thus increasing the number of resonance peaks in the Hall conductivity tensor and decreasing the magnitude of the magnetoresistance as well as the Hall coefficient when compared to the case of bulk phonons. The nearly linear increase of the magnetoresistance with temperature was found to be in good agreement with experiment.

The influence of confined optical phonons on the Hall effect in a quantum well under the present of a perpendicular magnetic field and an in-plane intense electromagnetic wave, has been theoretical studied. The analytical expression of the Hall coefficient for electron — confined optical phonon interaction is obtained base on the quantum kinetic equation method. It is found that the Hall coefficient depends on the frequency and amplitude of the electromagnetic wave, the magnetic induction, the quantum well width, the temperature of the system and especially the quantum number m characterizing the phonon confinement. Numerical calculations are also applied for AlGaAs/GaAs/AlGaAs quantum well to show the nonlinear dependence of the Hall coefficient on the electromagnetic wave amplitude and the remarkable contribution of confined optical phonons on the magnitude of the Hall coefficient as well as the magneto-phonon resonance condition. All results are compared with that in case of unconfined phonons to see differences.