Topologically Non-trivial RTD: Final Report

AbstractTopological insulators are materials that exhibit a unique quantum state of matter. This state is often referred to as the quantum spin hall state and has not been observed in any other type of material. This state of matter has very unique transport properties and in the last 10 years there has been much speculation about potential applications to take advantage of these properties. In this paper, the potential and applicability of a resonance tunneling diode constructed of topological insulator material is investigated. Conventional resonance tunneling diode properties are briefly reviewed and then used as comparison to a theoretical topological insulator resonance tunneling diode.

First proposed in 1973 by Chang and Esaki, resonant tunneling has since become a very well understood and well utilized property in devices(Chang 1974). Due to the fabrication capabilities at the time, the devices that could be created were too thick and defective to properly utilize this effect. Early experiments on this effect were promising, but only showed weak features at unrealistically low temperatures. However, since then fabrication techniques have greatly improved and tunneling effects can now be well observed and utilized. One of the primary uses of this property is in the *resonant tunneling diode*(RTD), in which the double potential barrier system allows for strong diode control(Mizuta 1995).

In this report, the analysis of a theoretical RTD composed of topological insulator material will be carried out and used to compare the properties of this topologically non-trivial RTD to those of a conventional RTD. Topological insulators will be discussed in the following sections, but these materials exhibit unique edge state transport properties and could be useful in a RTD.

As previously stated, an RTD is a tunneling device that makes use of resonances in the tunneling probability for diode control. This resonance arises from the interaction of the potential barriers, which can be thought of as a double potential barrier with adjustable positions relative to each other. When an electron tunnels through one barrier, it will be temporarily bound between the two barriers until it can tunnel out(Mizuta 1995). The relative positional adjustment of the barriers, which in actuality is the result of applying a bias to the device, results in adjustability of the amount of time the electron is bound. At a certain bias, the tunneling probability spikes and the electron sees virtually no barriers; this is the resonance condition, which can be seen in figures 1a and 1b. Using tunneling probability, current can be calculated and has been shown to exhibit unique properties that include a negative differential resistance region, which can be seen in figure 2. These properties have made RTD’s very useful for switching devices as well as a platform for studying the wave nature of electrons.

Using the WKB approximation(Bohm 2012), one can determine an analytical expression for the transmission probability. For double barrier systems, the WKB method is known to lack accuracy, but the relative simplicity of this method makes it useful for coarse analysis. Using the WKB approximation, a general expression for the transmission coefficient of a double barrier system can be found(Ferry 1994)^{,}(Chowdhury 2015). \[T_{tot}(E) = \frac{T_1T_2}{(1-\sqrt{R_1R_2})^2+4\sqrt{R_1R_2}cos^2(\Phi)}\] \[\Phi = k_1b+\frac{\theta_{L12}+\theta_{R21}-\theta_{L11}-\theta_{R11}}{2}\] For this analysis, the assumption is made that the band diagram slopes are relatively flat. Under this assumption, the above expressions can be used and the system can be considered two offset square barriers. Solving for the transmission probability, resonant peaks can be seen in figures 1a and 1b.

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