# Simulations¶

As usual, start by grabbing the notebooks of this week (w5_qshe). They are once again over here.

### Kane-Mele model¶

The first known implementation of quantum spin Hall effect is the Kane-Mele model, introduced in this paper. It is a doubled copy of the Haldane model (get that one from the previous week's notebooks), with spin up and spin down having next-nearest neighbor hoppings complex conjugate of each other due to spin-orbit coupling.

Implement the Kane-Mele model and add a staggered onsite potential to also be able to create a trivial gap. Calculate the scattering matrix topological invariant of that model.

How would you add disorder and calculate the topological invariant? (Hint: you need to add disorder to the scattering region, and make leads on both sides conducting)

### Quantum Hall regime¶

The helical edge states of quantum spin Hall effect survive for some time when a magnetic field is added. Make a Hall bar out of the BHZ model. Can you reproduce the experimental results? What do you see? Are the inversion symmetry breaking terms important?

What about conductance in a two terminal geometry: can you see the crossover from quantum spin Hall regime to quantum Hall regime?

MoocSelfAssessment description

In the live version of the course, you would need to share your solution and grade yourself.

Discussion Quantum spin Hall effect is available in the EdX version of the course.

# Review assignment¶

### Observation of Quantum Spin Hall States in InAs/GaSb Bilayers under Broken Time-Reversal Symmetry (arXiv:1306.1925)¶

Lingjie Du, Ivan Knez, Gerard Sullivan, Rui-Rui Du

Topological insulators (TIs) are a novel class of materials with nontrivial surface or edge states. Time-reversal symmetry (TRS) protected TIs are characterized by the Z2 topological invariant and their helical property becomes lost in an applied magnetic field. Currently there exist extensive efforts searching for TIs that are protected by symmetries other than TRS. Here we show, a topological phase characterized by a spin Chern topological invariant is realized in an inverted electron-hole bilayer engineered from indium arsenide-gallium antimonide (InAs/GaSb) semiconductors which retains robust helical edges under a strong magnetic field. Wide conductance plateaus of 2e2/h value are observed; they persist to 12T applied in-plane magnetic field without evidence for transition to a trivial insulator. In a perpendicular magnetic field up to 8T, there exists no signature to the bulk gap closing. While the Fermi energy remains inside the bulk gap, the longitudinal conductance increases from 2e2/h in strong magnetic fields suggesting a trend towards chiral edge transport. Our findings are first evidences for a quantum spin Hall (QSH) insulator protected by a spin Chern invariant. These results demonstrate that InAs/GaSb bilayers are a novel system for engineering the robust helical edge channels much needed for spintronics and for creating and manipulating Majorana particles in solid state.

Hint: A better material?

### Corner Junction as a Probe of Helical Edge States (arXiv:0808.1723)¶

Chang-Yu Hou, Eun-Ah Kim, Claudio Chamon

We propose and analyze inter-edge tunneling in a quantum spin Hall corner junction as a means to probe the helical nature of the edge states. We show that electron-electron interactions in the one-dimensional helical edge states result in Luttinger parameters for spin and charge that are intertwined, and thus rather different than those for a quantum wire with spin rotation invariance. Consequently, we find that the four-terminal conductance in a corner junction has a distinctive form that could be used as evidence for the helical nature of the edge states.

Hint: What happens when edge states meet.

### Engineering a robust quantum spin Hall state in graphene via adatom deposition (arXiv:1104.3282)¶

Conan Weeks, Jun Hu, Jason Alicea, Marcel Franz, Ruqian Wu

The 2007 discovery of quantized conductance in HgTe quantum wells delivered the field of topological insulators (TIs) its first experimental confirmation. While many three-dimensional TIs have since been identified, HgTe remains the only known two-dimensional system in this class. Difficulty fabricating HgTe quantum wells has, moreover, hampered their widespread use. With the goal of breaking this logjam we provide a blueprint for stabilizing a robust TI state in a more readily available two-dimensional material---graphene. Using symmetry arguments, density functional theory, and tight-binding simulations, we predict that graphene endowed with certain heavy adatoms realizes a TI with substantial band gap. For indium and thallium, our most promising adatom candidates, a modest 6% coverage produces an estimated gap near 80K and 240K, respectively, which should be detectable in transport or spectroscopic measurements. Engineering such a robust topological phase in graphene could pave the way for a new generation of devices for spintronics, ultra-low-dissipation electronics and quantum information processing.

Hint: A completely different approach.

### Induced Superconductivity in the Quantum Spin Hall Edge (arXiv:1312.2559)¶

Sean Hart, Hechen Ren, Timo Wagner, Philipp Leubner, Mathias Mühlbauer, Christoph Brüne, Hartmut Buhmann, Laurens W. Molenkamp, Amir Yacoby

Topological insulators are a newly discovered phase of matter characterized by a gapped bulk surrounded by novel conducting boundary states. Since their theoretical discovery, these materials have encouraged intense efforts to study their properties and capabilities. Among the most striking results of this activity are proposals to engineer a new variety of superconductor at the surfaces of topological insulators. These topological superconductors would be capable of supporting localized Majorana fermions, particles whose braiding properties have been proposed as the basis of a fault-tolerant quantum computer. Despite the clear theoretical motivation, a conclusive realization of topological superconductivity remains an outstanding experimental goal. Here we present measurements of superconductivity induced in two-dimensional HgTe/HgCdTe quantum wells, a material which becomes a quantum spin Hall insulator when the well width exceeds d_{C}=6.3 nm. In wells that are 7.5 nm wide, we find that supercurrents are confined to the one-dimensional sample edges as the bulk density is depleted. However, when the well width is decreased to 4.5 nm the edge supercurrents cannot be distinguished from those in the bulk. These results provide evidence for superconductivity induced in the helical edges of the quantum spin Hall effect, a promising step toward the demonstration of one-dimensional topological superconductivity. Our results also provide a direct measurement of the widths of these edge channels, which range from 180 nm to 408 nm.

### Helical edge resistance introduced by charge puddles (arXiv:1303.1766)¶

Jukka I. Väyrynen, Moshe Goldstein, Leonid I. Glazman

We study the influence of electron puddles created by doping of a 2D topological insulator on its helical edge conductance. A single puddle is modeled by a quantum dot tunnel-coupled to the helical edge. It may lead to significant inelastic backscattering within the edge because of the long electron dwelling time in the dot. We find the resulting correction to the perfect edge conductance. Generalizing to multiple puddles, we assess the dependence of the helical edge resistance on temperature and doping level, and compare it with recent experimental data.

Hint: Sources of back-scattering in QSHE edge.

### Bonus: Find your own paper to review!¶

Do you know of another paper that fits into the topics of this week, and you think is good? Then you can get bonus points by reviewing that paper instead!

MoocSelfAssessment description

In the live version of the course, you would need to share your solution and grade yourself.

Do you have questions about what you read? Would you like to suggest other papers? Tell us:

Discussion Quantum spin Hall effect is available in the EdX version of the course.