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Abstract The conventional, Rashba and Dreessselhaus spin orbit interactions have been investigated and discussed in detail. The Rashba and Dresselhaus interactions have been studied in an infinite cylindrical quantum wire in the case when the wavevector along the axis of the cylinder vanishes. The geometry is equivalent to a two dimensional circular disk. The study has led to the splitting of the energy levels due to the spin orbit interaction. The dispersion relation is determined from an equation which is derived to the first time in the present work. The conventional spin-orbit interaction has been explored in the presence of center or off-center impurity placed in a single spherical quantum dot (QD) or in the inner dot of a multilayered spherical quantum dot (MSQD). The calculations have been performed in the excited state (2p). The different masses of dots and barrriers have been taken into consideration. They arise due to the dependence of the barriers masses on the �� – concentration in a ���� − ��1−� ��� �� structure. In all previous treatments on a single quantum dot the conventional spin orbit interaction has been calculated for a central impurity assuming equal masses of dot and barrier. Also, to the best of our knowledge no studies have been reported in a MSQD for the conventional spin-orbit interaction. For both the QD and MSQD geometries, the variational method has been applied in the case of off-central and central impurities using two forms of the trial wavefunction, the conventional form and a new form that satisfies the required boundary conditions in the case of different masses. The new form has considerably improved the results of the spin-orbit interaction. In the case of central impurity the spin-orbit interaction has been also calculated using the exact solution of the radial Schrödinger equation in the presence of the impurity for the state (2p). The results are higher than those obtained using the variational method. However, the rsults obtained using the new form of the trial wavefunction are always nearer to the exact solution than those obtained using the old wavefunction. Also, the results obtained using the exact solution in the case of QD indicate that the consideration of different masses has a significant effect on the spin-orbit energy for � < 0.63�∗ where � is the radius of the dot. Moreover the comparison between the results of different and equal masses in the case of MSQD using the exact solution (center impurity) showed that spin orbit energy in the case of equal masses is higher than the results of different masses. On the other hand, in the case of off- center impurity (�� = �1 ⁄2) the results for different masses calculated using the new wavefunction are higher for most of the range considered. The variation of the confining potential �1 of the inner barrier in the case of MSQD has an appreciable effect for both center and off-center impurities. Since for the range of �� considered the electron is most probably localized in the outer dot the decrease of �1 leads to the increase of the spin orbit energy (��.𝑂. ). This effect is higher for the central impurity. Regarding the position of the off-center impurity at which the spin orbit energy takes its highest value it differs significantly in the case of a single QD from that of MSQD. In a single QD this position is related to the position at which the probability of finding the electron is maximum while in the case of MSQD, the spin orbit energy increases as the impurity moves away from the center. In the case of a single QD, the results have been compared with the results reported in earlier work. The results obtained using the exact solution for a central impurity are identical with the results of Yang et al [86] and of Özmen et al [60] to within 0.4%. Regarding, the other earlier treatments we have suggested some amendments to reconcile their results with the present results. In this connection, it is worthwhile pointing out that the comparison has been made only for central impurity since all earlier treatment were performed in this case. Moreover, some of the methods used in these treatments (Yang et al [86], Yakar et al [85] and Özmen et al [60]) cannot be applied for off-central impurities. The new analytical expressions obtained using the exact and variational methods for the binding and spin orbit energies in the case of single QD and in MSQD have facilitated the performance of the required computations in a fast and feasible way. Finally, we would like to confirm that in spite of the small order of magnitude of spin-orbit interaction with respect to the electron energy and binding energy it can be separately measured by spintronic devices where it plays the essential role. |