Barrier Height and Film Thickness Dependence of the TMR

We calculate the tunnel magneto-resistance (TMR) ratio by employing the Keldysh formalism. The barrier height dependence of TMR has a minimum whose origin is different from that of Slonczewski's theory. The TMR decreases and saturates with increasing film thickness. When the barrier is low and thick, the barrier height dependence of modulated density of states in the electrodes is dominant to the behavior of TMR. In this case the TMR increases with increasing bias voltage. When the barrier is high and thin, the barrier height dependence of damping factor through the insulating film is dominant to the behavior of TMR. In this case the TMR decreases greatly with increasing bias voltage. The temperature dependence is small below the temperature where electrons can hop over the barrier. As the barrier becomes high, the TMR becomes weakly temperature dependent, and the bias dependence becomes large.