Effect of image force on tunneling

In this section, we will derive the correction factor for the tunneling current due to the image force. For a free metal surface, the image potential pertinent to this surface, Eq. (2.2), is always prc.sent. The the simple image potential is always an essential part of the free sample, and is always implied in any first- 

We choose the bisection plane in the barrier as the separation surface. The correction for tunneling current can be obtained from the correction for the wavefunction on the bisection plane. Using the Green s function method, following Eq. (2.42), the correction factor for the wavefunction at z = W/2 is 

Substituting Eq. (2.45) for Eq. (2.46), performing the elementary integrals, noticing that 

The wavefunction of another electrode gets the same factor. The tunneling current is proportional to the square of the matrix element. Thus, 

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