Abstract

Tunneling of a particle through a potential barrier that is absolutely opaque classically is one of the most striking features of quantum mechanics. Whereas stationary tunneling rates have been calculated and experimentally verified for a number of different systems, thus far only very few experiments provided some (indirect) information of the temporal dynamics of particle tunneling. In solid states systems the major difficulty stems from the fact that the barrier traversal times of electrons are of the order of 10^-14 - 10^{- 15} seconds. Clearly and unambiguously interpretable experimental studies of the temporal aspects of tunneling would be important not only for semiconductor device physics but also from a fundamental physical point of view because of a few surprising and yet unverified theoretical findings from the past. Following the motion of the electron wave packet, Hartman¹ calculated the time taken by an electron to tunnel through a potential barrier (group delay), and found that the tune ling time becomes independent of the barrier thickness for opaque bariers. This paper presents what is to our knowledge the first direct experimental verification of the essential qualitative conclusions from Hartman's work¹, based on following the propagation of wave-packets through barriers having different thicknesses. These experiments have been made feasible by the availability of a highly stable ~10fs laser source that allows subfemtosecond time resolution². Relating the results of such an optical experiment to electron tunneling is warranted by the formal analogy between the time-independent Schrödinger equation and the Helmholtz wave equation describing the propagation of monochromatic em waves³ . The major problems in measuring the group delay originate generally from i) a distortion of the shape of the transmitted wave packet, and ii) a shifted energy spectrum, making the propagation velocity on the two sides of the barrier different⁴. Both these effects can severely impair the accuracy of time-of--flight measurements. Using photonic bandgap materials as the optical barrier⁵ provides ideal conditions for tunneling experiments because i) the group velocities of the incident and transmitted wave packets are equal and known (owing to free propagation in vacuum or air), and ii) the dispersion and transmission curves are slowly varying over a broad frequency range around the center of the photonic bandgap. This is revealed by Fig. 1 showing the group delay and intensity transmittivity for multilayer dielectric mirrors, which exhibit a ID photonic band gap.

© 1994 Optical Society of America