Inclusion of Airy's asymptotic functions for TMR voltage dependence phenomena in comparison with Kubo's formalism in nanostructures

Using the Airy functions for the nanostructure of the form F/I/F, in which F is a ferromagnetic metal and I is an insulator, one can investigate the rate of changes of TMR in both low as well as a high voltage cases. For some range of voltage, TMR decreases rapidly and even changes sign and when bias voltage is increased, the TMR oscillates. In this article, we have compared the result of the forgoing phenomenon with Kubo’s formalism in the low voltage regime.