Path Integral Approach to Transport in Resonant Tunneling Diodes


Resonant Tunneling Diodes (RTDs) are two terminal electronic devices in which the transport is achieved by the electrons tunneling through a resonant state in a double potential barrier setup. This is one of the first devices to take advantage of the quantum mechanical electron-wave property (Chang 1974). Since the improvement in crystal growth and nano-fabrication methods, RTDs has been of great interest both as a theoretical study of quantum transport physics (Kluksdahl 1989) and functional quantum devices in the laboratories. The specific property of interest is the region of negative resistance (or conductance) it shows under certain conditions, which is not an intuitive classical result. This can be seen in the figure \ref{fig:graph}, where there are regions where the conductance goes negative at certain applied voltages for a GaAs/GaAlAs double barrier RTD.

Here the electron transport through resonant tunneling diodes are analyzed using path integrals. The negative differential resistance (NDR) is studied by comparing the results of path integrals with traditional WKB approximation method, and is quantitatively shown to lend a better understanding to the origin of the NDR.

\label{fig:graph} Current and conductance characteristics of a double-barrier structure of GaAs between two Ga\(_{0.3}\)Al\(_{0.7}\)As, as shown in the energy diagram. Both the thicknesses and the calculated quasistationary states of the structure are indicated in the diagram. Arrows in the curves indicate the observed voltages of singularities corresponding to these resonant states (Chang 1974).

Past Work

RTDs have been investigated in great detail by various groups since the phenomena was first observed experimentally. Bohm(Bohm 2012) and Iogansen(Iogansen 1964) 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(Bohm 2012), 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”.