The free layer thickness $\left(t_{\text{free}}\right)$ in double barrier magnetic tunnel junctions (DMTJs) based on crystalline MgO barriers and CoFeB ferromagnetic layers has been varied from 0.5 to 3.0 nm in order to investigate its effect on the magnetic and electrical properties. One obvious feature of DMTJs with $t_{\text{free}}\le 1\text{nm}$ is the absence of sharp free layer switching in the TMR curves, which can be explained by the superparamagnetic nature of discontinuous CoFeB layer, which breaks into nanodots when it is very thin. Normal free layer switch is observed when $t_{\text{free}}=2.0$ and 3.0 nm. Another difference is a rapid increase in junction resistance and tunnel magnetoresistance at low temperature for DMTJs with thin $t_{\text{free}}$ , which is attributed to the Coulomb blockade effect. We also observed a small conductance peak in the $dI/dV$ curve at low bias only in the parallel configuration and at temperatures below 100 K. This is related to the Kondo scattering process on the nanodots, which constitutes the discontinuous free layer. We found no Coulomb staircase existing in the $I\text{-}V$ curves; this may be due to the microsize of the junctions.

We put forward a new ray tracing method for electromagnetic wave propagation based on piecewise conformal transformations. Our approach is based on our previous research, where conformal transformation was focused on giving an accurate formula for the effective Earth radius. In this article, according to the height interval of atmospheric refractive index data, we approximate the electromagnetic wave propagation path by the combination of circular arcs with different curvature radius and find the effective Earth radius of each arc using conformal transformations. Then, upon using the effective Earth radius, we transform the circular arc path into a straight path, thus simplifying the calculations needed to describe the propagation of electromagnetic waves. We compare our approach to the traditional ray tracing method for several typical atmospheric refractive index profiles and prove that our new method may achieve ray tracing with higher accuracy using fewer path points. We also consider nonuniform atmosphere and irregular terrain and compare the performance of our ray tracing scheme with that of the open-source software PETOOL, which is based on the parabolic equation method. Result proves that the new method may be effectively used for ray tracing in complex environments.