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Charge dynamics and "in plane" magnetic field I: Rashba-Dresselhauss interaction, Majorana fermions and Aharonov-Casher theorems

The 2-dimensional charge transport with parallel (in plane) magnetic field is considered from the physical and mathematical point of view. To this end, we start with the magnetic field parallel to the plane of charge transport, in sharp contrast to the configuration described by the theorems of Aharonov and Casher where the magnetic field is perpendicular. We explicitly show that the specific form of the arising equation enforce the respective field solution to fulfil the Majorana condition. Consequently, as soon any physical system is represented by this equation, the rise of fields with Majorana type behaviour is immediately explained and predicted. In addition, there exists a quantized particular phase that removes the action of the vector potential producing interesting effects. Such new effects are able to explain due the geometrical framework introduced, several phenomenological results recently obtained in the area of spintronics and quantum electronic devices. The quantum ring as spin filter is worked out in this framework and also the case of the quantum Hall effect.