Case of Parabolic Potential Barrier

In the case of a parabolic potential barrier, the Schrodinger equation to be solved is given by 

It should be noted that the coefficients of the wave running to the right are taken to be -C and -C as in Equation (2.2). By using the asymptotic expressions of Weber functions and the above WKB solutions, the connection between the coefficients 

The coefficients and represent the incoming waves to the potential barrier and the coefficients C f, and C represent the outgoing waves from the barrier. The matrix 5 that connects these sets of coefficients has physical meaning of scattering matrix defined as 

This tunneling scattering matrix can be obtained easily from the above expression of M-matrix as 

The diagonal (off-diagonal) elements of this matrix represent tunneling (reflection) amplitudes. At T Vmax the parameter e is negative [e = - e ] [see Equation (2.17)] and the tunneling probability is given by 

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For the case of parabolic potential barriers, the conductivity as a function of temperature is described by the simple expression /52/ ... 

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