RTDs have been investigated in great detail by various groups since the phenomena was first observed experimentally. Bohm\cite{Bohm2012} and Iogansen\cite{Iogansen1964} solved the double barrier setup using the well accepted WKB approximation approach.
Here, the WKB approximation is used to calculate the transmission coefficient, which has a functional dependence on the energy of the incident electron. This derives the resonant energies of the electron in the barrier, the energies at which the barriers become transparent to the incident electron wave-packet.
The transmission coefficient as derived by Bohm\cite{Bohm2012}, for energies near the resonant energies \(E_N\), is given by:
\[T(E) \approx \frac{1}{1+ \frac{\tau^2_0}{\hbar^2} (E - E_N)^2 \Theta^4 }\]
where, \(\Theta = exp(kx)\) after the electron crosses the barriers and \(\tau_0\) is the classical time for the electron to cross the well and return.
The \(T(E)\) goes to unity when the energies of the electron are at resonant energies (Fig \ref{fig:WKB}). This expression can then be used to calculate the current for various setups and biases by incorporating the Fermi distribution over all energies.
Although this does finally lead to negative resistance, but does not give a clear picture of it’s origin. It basically gives the energies the electron needs to travel through without impedance, but not what the electrons “do”.