Although RTD’s have been extensively tested and studied for more than twenty years, the field of topological insulators has only come into focus in the past five years. Without going into excessive detail, a topological insulator is a material that can exhibit a unique quantum state of matter. This is known as the quantum spin hall(QSH) state and leads to numerous unique material and transport properties. Matter in this state is insulating in the bulk region, but has conducting edge states\cite{Hasan_2010},\cite{Qi_2011}. Moreover, these conduction edge states are spin polarized and propagate in opposite directions. This state of matter has been experimentally observed in HgTe/CdTe quantum wells\cite{Konig_2007},\cite{Roth_2009}. Unlike the quantum hall state, a magnetic field is not required to change to the QSH phase. Also, these states are topologically protected against scattering. These properties give topological insulators the potential to be extremely useful in devices that would require spin control and low energy dissipation\cite{Konig_2007}.
For topological insulators like HgTe/CdTe quantum wells, effective energy dispersions have been found that represent the bands of interest using a two band model\cite{Qi_2011}. \[E\pm=\epsilon(k)\pm \sqrt{A^2(kx^2+ky^2)+M^2(k)}\] \[\epsilon(k)=C-D(kx^2+ky^2)\] \[M(k)=M-B(kx^2+ky^2)\] In this model, A, B, C, and D are parameters that depend on the material geometry. Refer to table 1 for parameter values at different quantum well thicknesses. Using this scheme, the band structure for HgTe/CdTe topological insulators can be calculated. As shown in figure 3a-c, the structure depends on the quantum well thickness greatly. At a critical thickness (\(\approx 0.63\)Å), the bands become inverted, indicating a toplogical phase change.
l*6cr d (Å) & A (eV Å) & B (eV Å2) & D (eV Å2) & M (eV)
55 & 3.87 & -48.0 & -30.6 & 0.009
61 & 3.78 & -55.3 & -37.8 & -0.00015
70 & 3.65 & -68.6 & -51.2 & -0.010