## The Simulation of Quantum Transport in Topological Insulators (TIs) and TI Based Nanoscale Field Effect Transistors (TI FETs)

Principal Supervisor (Director of Studies): **Dr G. Edwards, Department of Electronic & Electrical Engineering**

Second Supervisor: **Dr T. Papadopoulos, Department of Natural Sciences**

**Suitable Background for Prospective Student: **

Upper Second Class in Undergraduate Degree (BSc Physics, BEng Electrical & Electronic Engineering, BSc Applied Mathematics)

### Project Description:

Topological insulator (TI) materials have burst onto the scene to become a very important topic in condensed matter physics in the last decade (Qi & Zhang, 2011). Topological insulators are insulators in the bulk but have topologically protected edge current carrying states at the surface. Quantum Transport properties of topologically protected states are also of strong contemporary electrical engineering interest as they act as perfect conducting channels, in potential nanoelectronic interconnects and quantum functional devices. A 2D topological insulator system such as a CdTe/HgTe/CdTe quantum well, above the critical well width shows the Quantum Spin Hall (QSH) effect (Qi & Zhang, 2011). The conducting edge states are protected by time reversal symmetry and their carriers have their spin locked to the momentum (see Figure 1). At the upper edge the forward/backward moving states are tied to up/down spin states and at the lower edge this situation is reversed with the correspondence [forward/backward: down/up].

**Figure 1** The edge states for the 2D Quantum Spin Hall (QSH) Effect. The upper edge supports a forward moving state with spin up and a backward moving state with spin down, while for the lower edge the situation is the converse.

In this PhD project the Kwant Quantum Transport Simulator (Groth, et al., 2014), which is written in Python, to begin with, will be initially to explore the effect of non-magnetic impurities, defects and vacancies, initially in 2D for QSH systems, described by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian (Qi & Zhang, 2011). Non-magnetic impurities have spin conserving scattering which don’t destroy the topologically protected states. However, the presence of these impurities will alter the bulk conduction and will introduce quantum interference effects. Lattice vacancies can cause the formation of current vortices (Dang, et al., 2015). Kwant calculates the Scattering matrix for the system and gives the conductance through the Landauer - Buttiker formula. Kwant can also be used to examine the local density of states (DOS) and local current. Looking at the local DOS and currents will provide detailed information on the effects of impurities and vacancies on Quantum Transport. This work will then be extended by addressing the Quantum Transport in impure 3D topological insulators, from the Bi2Se3 family.

Then the extension of looking at the effect of magnetic gating, by a ferromagnetic adatom, in a TA, will be tackled (Dang, et al., 2016). Once getting a handle on quantum transport and its control by a single adatom gate, the next stage is to address the TI Field Effect Transistors (FETs) engineering device application, where the gate is now film of atoms. A ferromagnetic insulator layer is required to give the TI Dirac Cone a bandgap i.e. splitting the zero gap, essential for TI FET switching behaviour, with a high ratio of ‘on’ to ‘off’ current. The potential applied at the gate and source to drain bias gives a linear potential drop along the TI channel which can be treated in Kwant by discretising as a staircase potential. Kwant can then calculate the Scattering matrix for the system and then integrating the transmission coefficient, gives the channel current and consequently the transistor transfer characteristic. The next phase of the research would be to investigate a TI FET incorporating a quantum well resonant tunnelling region and to study the emerging 2D materials silicene and germanene, for the channel

The results below are for the subband dispersion (see Figure 2) and the probability density versus (*x*, *y*) over the quantum wire, for energy *E* = 0 (see Figure 3), both within the BHZ Hamiltonian model. The BHZ Hamiltonian parameters are (*A* = 3.65 eV Å, *B* = -68.6 eV Å2, *C* = 0 eV*, D* = -51.1 eV Å2, *M* =- 0.01 eV). The subband structure shows the edge states in the gap (see Figure 2) while the probability density plot (see Figure 3) shows that the edge states decay away from the edges along y moving into the material.

**Figure 2** The subband dispersion along *kx* for a 2D quantum wire with length *L* = 2000 *a* and width *W* = 1000 *a* for the BHZ Hamiltonian with confinement along y. The tight binding lattice constant is given by *a* = 2 nm.

**Figure 3** A plot of the probability density *P* as a function of (*x*,y) i.e. *P* = *P*(*x*,*y*) for a 2D quantum wire region with length *L* = 2000 *a* and width *W* = 1000 *a* for the BHZ Hamiltonian with confinement along y. The tight binding lattice constant is given by *a* = 2 nm. Note the topological surface states at the top and bottom surfaces decay rapidly, going into the wire, along *y*.

### References

Dang, X., Burton, J. D. & Tsymbal, E. Y., 2015. Local currents in a 2D topological insulator. *J. Phys.: Condens. Matter, *Volume 27, p. 505301.

Dang, X., Burton, J. D. & Tsymbal, E. Y., 2016. Magnetic gating of a 2D topological. *J. Phys.: Condens. Matter, *Volume 28, p. 38LT01.

Groth, C. W., Wimmer, M., Akhmerov, A. R. & Waintal, X., 2014. Kwant: a software package for quantum transport. *New Journal of Physics, *Volume 16, p. 063065.

Qi, X.-L. & Zhang, S.-C., 2011. Topological insulators and superconductors. *Reviews of Modern Physics, *Volume 83, p. 1057.

**Contact**

For **self-funding** PhD candidates interesting in the above project and the research area of nano-electronics in general, contact Dr Gerard Edwards, Tel. (01244) 512314 Email: gerard.edwards@chester.ac.uk