Theory of electric dipole spin resonance in a parabolic quantum well

A theory of electric dipole spin resonance (EDSR), that is caused by various mechanisms of spin-orbit coupling, is developed as applied to free electrons in a parabolic quantum well. Choosing a parabolic shape of the well has allowed us to find explicit expressions for the EDSR intensity and its dependence on the magnetic field direction in terms of the basic parameters of the Hamiltonian. By using these expressions, we have investigated and compared the effect of specific mechanisms of spin orbit (SO) coupling and different polarizations of ac electric field on the intensity of EDSR. It is our basic assumption that the SO coupling energy is small compared with all different competing energies (the confinement energy, and the cyclotron and Zeeman energies) that allowed us to describe all SO coupling mechanisms in the framework of the same general approach. For this purpose, we have developed an operator formalism for calculating matrix elements of the transitions between different quantum levels. To make these calculations efficient enough and to derive explicit and concise expressions for the EDSR intensity, we have established a set of remarkable identities relating the eigenfrequencies and the angles defining the spatial orientation of the quantizing magnetic field $\mathbf{B}(\ensuremath{\theta},\ensuremath{\phi})$. Applicability of these identities is not restricted by EDSR and we expect them to be useful for the general theory of parabolic quantum wells. The angular dependences of the EDSR intensity, found for various SO coupling mechanisms, show a fine structure consisting of alternating up and down cusps originating from repopulating different quantum levels and their spin sublevels. Angular dependences of the EDSR intensity are indicative of the relative contributions of the competing mechanisms of SO coupling. Our results show that electrical manipulating electron spins in quantum wells is generally highly efficient, especially by an in-plane ac electric field.