الفهرس 
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Abstract
The study of the spin orbit interactions has gained a great attention in low dimensional structures, The three main types of spin orbit interactions are conventional, Dresselhaus and Rashba interactions. The conventional interactions result naturally due to the motion of the electron in its orbit. This motion, in turn, causes a coupling between the electron spin and its angular momentum, The Dresselhaus interactions arise in crystals which possess a lake of symmetry under reflection in a plane that involves at least one lattice point, The Rashba interactions are the most important type of spin orbit interactions that occur in low dimensional structures. They occur in two dimensional geometries. Their coupling coefficient is much higher than that of Dresselhaus interactions. They mainly depend on the inversion symmetry breaking in the direction perpendicular to the two dimensional structure.
The present work is mainly concerned with studying the Rashba spin orbit effect in different two dimensional geometries subject to a variety of confining potentials and external fields. The geometries which have been taken into consideration are a two dimensional discs subject to a parabolic confining potential and a perpendicular magnetic field. Also, an infinite quantum well wire subject to a finite confining potential and in the absence and presence of an axial magnetic field. Finally, the role of spin orbit interactions has been investigated in a spherical quantum dot with a radial parabolic confinement potential and in the presence of external magnetic and electrical fields. The solution of the Schrödinger equation in the presence of the Rashba interactions has been derived by applying a new approach that differs from the one used in an earlier treatment. The wave function in the presence of these interactions has been expanded in terms of the eigenfunctions of the Hamiltonian in the absence of them.
The first problem considered is the case of a two dimensional disc subject to a parabolic confining potential and a perpendicular magnetic field. In this respect, we have criticized the work which has been performed by M. Kumar and his coworkers (Refs. [17], [24], [26]). It has been shown that an obvious contradiction exists between the form of the Rashba spin orbit Hamiltonian and the form of the energy eigenvalues and eigenfunctions they have been used. Also, we have amended the relations that have been reported in Kuan et al [21] between the coefficients involved in the expansions of the eigenfunctions in the presence of the Rashba interactions. We believe that the modified relations are more accurate than these reported in Kuan et al [21].
In the second problem, the effect of the Rashba spin orbit interaction has been investigated in an infinite cylindrical quantum well wire (QWW) with finite confining potential. The effect has been considered in the absence and presence of an axial magnetic field. The solution of the radial Schrödinger equation has been obtained in the wire and in the barrier, in the absence of the Rashba spin-orbit interactions. The general solution in the presence of these interactions has then been expanded in terms of the obtained two solutions and in terms of the step function. The study should, however, be performed in two dimensions. For this reason, the wave vector along the axis of the wire has been chosen to be zero. The orthogonality of the elements of the resulting basis has been proved. The dependence of the Rashba coupling coefficient on the electron effective mass has also been taken into consideration. The results have been applied to the case of GaAs-Ga_(1-x) Al_x As cylindrical quantum wire.
Finally, the role of the Rashba effect has been explored in a spherical quantum dot confined by a radial parabolic potential. Also, external parallel magnetic and electric fields have been applied. The solution of the Schrödinger equation in the presence of the Rashba interactions has been derived by applying an approach that differs from the one used in an earlier treatment (Refs. Vaseghi et al [53], [55]). The wave function in the presence of these interactions has been expanded in terms of the eigenfunctions of the Hamiltonian in their absence. In our opinion the new form introduced for the wave function presents the exact solution in a more accurate manner. The coefficients of expansion have been chosen either to depend on the three quantum numbers involved or on the principal quantum number only. The results have shown that the Rashba interactions have a considerable effect on the electron energy levels and on their splitting. The variation of this effect with the applied fields and the Rashba coupling strength has been investigated.