Magneto-Quantum-Electric Effect

Abstract

In this chapter we study the motion of the center of the cyclotron orbit when the electron in the orbit absorbs photons or phonons, and we also discuss related phenomena such as the magneto-quantum-electric effect and phonon amplification. As is discussed in detail by Zawadzki in Chapter 13 of this volume, the center of the cyclotron orbit is given by X 0 = -P x /mω c , Y 0 = y + (P x /mω c ) if we choose a gauge A = [0, H 0 x,0]. In this gauge the x coordinate of the center of the cyclotron orbit depends only on the “good quantum number” P y . Therefore we should be able to shift the cyclotron orbit in the x direction by giving a momentum ΔP y to an electron in cyclotron orbit. However, if we choose another gauge A’ = [—H 0 y, 0, 0], the center of cyclotron orbit is given by X 0 = x + (P y /mω c ), Y 0 = P x /mω c . In this case P y is not a good quantum number, and it is not easy to see what will happen when the electron absorbs a momentum ΔP y . To see what really happens to the cyclotron electron in absorbing a momentum from phonons or photons, we must carry out a gauge-independent calculation of a physical quantity such as the current associated with this shift of centers of orbits. This calculation is done in the next section. It turns out that the lateral displacement of the cyclotron orbits occurs when the momenta ZIP transfered to the electrons have components in the plane of the orbit in real space.