Abstract
We know from Bloch’s theorem that the wave function ψ of an electron in a semiconducting or insulating crystal can be written
where k is the wave vector of the state and u(x) is a periodic function with the lattice period of the crystal. Over reasonable distances the u(x) part of the function averages out, and we can view the propagating part e ikx as a quasi-particle behaving in much the same way as a particle in free space. The dynamics of this particle are given by the dispersion relation between the wave vector k and the energy E. For a semiconductor or insulator with a forbidden gap the dispersion relation is something like the one shown in Fig. 1. Note that we have plotted E versus k 2 rather than the more conventional E versus k. The reason is as follows: It can be shown rather generally that near one of the band edges (E c or E υ ) the energy is quadratic in k,and is usually written E=ћ 2 k 2/2m* by analogy with the free electron. Here m* is the effective mass of the quasi-particle. This form is shown by the straight line in Fig. 1. For E>E c, and E<E υ , k 2 is positive; therefore k is real, and propagating solutions result. For energies in the forbidden gap E c <E<E υ , k 2 is negative, k is imaginary, and exponentially damped solutions result. It is these damped solutions with which we are concerned in tunneling problems.
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References
K. K. Thornber, T. C. McGill, and C. A. Mead, J. Appl. Phys. 38: 2384 (1967).
R. Stratton, G. Lewicki, and C. A. Mead, J. Phys. Chem. Solids 27: 1599 (1966).
G. Lewicki, “Electron Tunneling Through Thin Films of Aluminum Nitride,” available as University Microfilms 66–10, 586.
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S. Pollack, Trans. Met. Soc. AIME 233: 497 (1965).
C. A. Mead, Solid-State Electron. 9: 1023 (1966).
J. van Laar, and J. J. Scheer, Surface Science 3: 189 (1965).
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© 1969 Plenum Press
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Mead, C.A. (1969). Some Properties of Exponentially Damped Wave Functions. In: Burstein, E., Lundqvist, S. (eds) Tunneling Phenomena in Solids. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-1752-4_9
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DOI: https://doi.org/10.1007/978-1-4684-1752-4_9
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