Differential Equation for the Transfer Matrix

An electron propagating through a crystal toward an interface can either reflect or transmit. The determination of its transmission and reflection probabilities represents an actual task in such fields as nanoelectronics, magnetoelectronics, or spin electronics. Within the framework of the effective mass approximation the problem can be reduced to the tunneling of the quantum particle through one-dimensional potential barrier. The tunneling process can be described by means of the transfer matrix, which contains all the information about the energetic dependence of the transmission and reflection coefficients. In the present work the differential equation for the transfer matrix of the arbitrary potential barrier is derived. The method proposed represents an alternative way of the calculation of the transfer matrix.