RANGE OF VALIDITY OF WKB TUNNEL PROBABILTY, AND COMPARISON OF EXPERIMENTAL DATA AND THEORY

It is common to use in calculations on tunneling through thin insulating films the WKB tunnel probability, even though at the turning points of the barrier the potential changes extremely rapidly and would thus invalidate its use in this case. Furthermore, the lower limit of barrier size to which the WKB tunnel probability can be applied is not well defined. In this paper we derive the tunnel probability through a trapezoidal barrier (where the use of the WKB tunnel probability is expected to be invalid) using both an exact method and the WKB tunnel probability. Provided that Δs φ 1 2 > 4, where Δs is the distance between the turning points in angstrom units and φ is the mean barrier height in electron volts, the two results have the same exponential forms but differ by a pre-exponential factor, which is a very slowly varying function only of the barrier parameters; that is the two expressions have essentially the same functional forms. The pre-exponential factor is related to the rate of change of potential at the turning points. It is practically unity for a slowly-changing potential (for which case the WKB approximation is virtually exact), and takes larger values the greater the change of potential, amongst other things, and has a maximum value of about ten for a vertically rising potential at the turning points. These two extreme conditions embody conditions at the turning points of any practical (arbitrary) barrier; thus the functional form of the exact tunnel probability through an arbitrary practical barrier is preserved in the WKB tunnel probability for that barrier, subject to the conditions that s φ 1 2 > 4, which in itself corresponds to extremely small (arbitrary) barriers. The fact that the WKB tunnel probability is exact only to within the pre-exponential factor and since it is usually used in developing the theoretical current-voltage tunnel characteristics, means that functional, and not direct, correlation is to be expected between experimental data and theory of practical tunnel junctions in which the potential changes rapidly at the turning points. © 1969.