Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal‐insulator‐metal thin film sandwiches, given approximately by , where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.
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A. G. Chynoweth, Progress in Semiconductors, edited by A. F. Gibson (John Wiley & Sons, Inc., New York, 1960), Vol. 4, p. 97.
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D. Bohm, Quantum Theory (Prentice‐Hall, Inc., Englewood Cliffs, New Jersey, 1951), Chaps. 3 and 11.
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Jánossy13 has found the latter result to hold for the case of a δ‐function potential barrier also.
18.
19.
Here tunneling is defined as penetration of the barrier proper.
20.
The present analysis differs from that of MacColl where only the portion of the argument is taken into account. The remainder was neglected on the grounds that it is slowly varying over the range of interest. All of is included in the present analysis since the total magnitude of the time delay is of interest as well as the change in the delay.
21.
C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc., New York, 1956), 2nd ed., Chap. 10.
22.
Preferred values from Handbook of Chemistry and Physics (Chemical Rubber Publishing Company, Cleveland, Ohio, 1954), 36th ed., p. 2342.
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© 1962 The American Institute of Physics.
1962
The American Institute of Physics
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