Spin-orbit coupling and tunneling current in a parabolic quantum dot

We propose an approach to explore the signature of the spin-orbit interaction in a quantum dot subjected to a tilted magnetic field. The spin-orbit coupling within the dot manifests itself as an anticrossing of the energy levels as the tilt angle is varied. The anticrossing gap has a nonmonotonic dependence on the magnitude of the magnetic field and exhibits a peak at some finite values of the magnetic field. From the dependence of the tunneling current through the quantum dot on the bias voltage and the tilt angle, the anticrossing gap, and most importantly the spin-orbit coupling strength, can be determined. In this paper, we propose an approach to determine the strength of the SO interaction in the QDs. It is based on an analysis of the behavior of the electronic QD energy levels in a tilted magnetic field. The tilted magnetic field has an ad- vantage over parallel and perpendicular fields because it in- troduces the Zeeman splitting of the energy levels and modi- fies the orbital motion of the electron within the QD as well. The relative strength of these two contributions in the elec- tron dynamics can be varied by changing the tilt angle. With- out the SO interaction, the energy spectrum of the QD has a strong dependence on the direction of the magnetic field, thus exhibiting regions of level crossings at different tilt angles. The levels that cross have the opposite spin direc- tions, and without the SO interaction there is no mixing be- tween them. The SO interaction couples the different spin states. In this case we should expect an anticrossing of the energy levels as a function of the tilt angle. Interestingly, the strength of the anticrossing characterizes the strength of the SO coupling. The most accurate way to study experimentally the structure of the energy spectra around the anticrossing region is to measure the tunneling current through the quan- tum dot. Transport spectroscopy is a powerful tool to study a variety of phenomena related to the correlation and interac- tion effects in a QD. 8 The main idea of the tunneling spec- trosopy at a finite bias voltage is that the tunneling current depends on the number of available for tunneling channels in the QD. In the following, we study the tunneling transport through a QD in a tilted magnetic field and show that the tunneling current is dependent on the tilt angle and the bias voltage within the anticrossing region. The energy range of the anticrossing region is usually smaller than the energy of the interelectron interaction, which can be estimated to be about 7 meV. 3 The tunneling process can then be described by a single-electron picture. The Hamiltonian of an electron in a parabolic QD in a tilted magnetic field is