5. Resonance and Transmission Coefficients
Calculating the transmission coefficient, the probability that an electron wave of a certain energy will tunnel through a potential barrier, is a Partial Differential Equation Boundary Value Problem. First we declare that we are interested in the time independent solution, i.e. the steady state solution. This reduces our problem to Ordinary Differential Equations. Second we find the solution for the electron's wave function where it is under the influence of a constant potential, known as a free particle solution.
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So graphically our system looks like:
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| Figure 5: One dimensional, single barrier scattering |
Finally we apply boundary conditions to reduce the solution to an algebraic problem. The algebraic problem is attacked with a method known as the transfer matrix.
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Resonance is the phenomenon of a jump in transmission over a small range of incoming energies. Note in Figure 6 that the single barrier exhibits a resonance at E/V ~ 1.6. We will see that for a double barrier we will have a resonance at electron energies E < V!
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| Figure 6: Transmission coefficient for a single barrier as a function of electron energy divided by barrier energy. |