# Topics for self-study¶

## Simulations: more Chern insulators¶

Simulations of the three-dimensional systems are hard, mostly because they take a lot of computational power. That’s why we’ll do something relatively simple this time.

### Breaking time-reversal¶

One mechanism of opening the gap on the surface of a topological insulator is to bring it into contact with a ferromagnet, which creates an effective Zeeman field.

By calculating dispersion of a slab of 3D TI, observe the effect of Zeeman field pointing in different directions on the surface state dispersion. Find out which direction of the Zeeman field opens the gap in the surface state.

Make a domain wall between different orientations of Zeeman field. Are there any modes in this domain wall?

### Many invariants¶

The BHZ model is rather rich and allows to produce every possible topological invariant. Can you find the parameter values that produce all the desired values of the invariants? (Hint: you need to make the model anisotropic).

**Now share your results:**

## Review assignment¶

### Electrostatic Coupling between Two Surfaces of a Topological Insulator Nanodevice (arXiv:1410.0655)¶

Valla Fatemi, Benjamin Hunt, Hadar Steinberg, Stephen L. Eltinge, Fahad Mahmood, Nicholas P. Butch, Kenji Watanabe, Takashi Taniguchi, Nuh Gedik, Ray Ashoori, Pablo Jarillo-Herrero

We report on electronic transport measurements of dual-gated nano-devices of the low-carrier density topological insulator Bi1.5Sb0.5Te1.7Se1.3. In all devices the upper and lower surface states are independently tunable to the Dirac point by the top and bottom gate electrodes. In thin devices, electric fields are found to penetrate through the bulk, indicating finite capacitive coupling between the surface states. A charging model allows us to use the penetrating electric field as a measurement of the inter-surface capacitance \(C_{TI}\) and the surface state energy-density relationship \(\mu\)(n), which is found to be consistent with independent ARPES measurements. At high magnetic fields, increased field penetration through the surface states is observed, strongly suggestive of the opening of a surface state band gap due to broken time-reversal symmetry.

**Hint:** What enters the measurement of a Dirac point conductance.

### Seeing the magnetic monopole through the mirror of topological surface states (arXiv:0811.1303)¶

Xiao-Liang Qi, Rundong Li, Jiadong Zang, Shou-Cheng Zhang

Existence of the magnetic monopole is compatible with the fundamental laws of nature, however, this illusive particle has yet to be detected experimentally. In this work, we show that an electric charge near the topological surface state induces an image magnetic monopole charge due to the topological magneto-electric effect. The magnetic field generated by the image magnetic monopole can be experimentally measured, and the inverse square law of the field dependence can be determined quantitatively. We propose that this effect can be used to experimentally realize a gas of quantum particles carrying fractional statistics, consisting of the bound states of the electric charge and the image magnetic monopole charge.

**Hint:** Consequences of magneto-electric effect.

### Z2 phase diagram of three-dimensional disordered topological insulator via scattering matrix approach (arXiv:1401.7461)¶

Björn Sbierski, Piet W. Brouwer

The role of disorder in the field of three-dimensional time reversal invariant topological insulators has become an active field of research recently. However, the computation of Z2 invariants for large, disordered systems still poses a considerable challenge. In this paper we apply and extend a recently proposed method based on the scattering matrix approach, which allows the study of large systems at reasonable computational effort with few-channel leads. By computing the Z2 invariant directly for the disordered topological Anderson insulator, we unambiguously identify the topological nature of this phase without resorting to its connection with the clean case. We are able to efficiently compute the Z2 phase diagram in the mass-disorder plane. The topological phase boundaries are found to be well described by the self consistent Born approximation, both for vanishing and finite chemical potential.

**Hint:** Weak and strong topological insulators with disorder.

### Imaging the Nanoscale Band Structure of Topological Sb (arXiv:1311.1758)¶

Anjan Soumyanarayanan, Michael M. Yee, Yang He, Hsin Lin, Dillon R. Gardner, Arun Bansil, Young S. Lee, Jennifer E. Hoffman

Many promising building blocks of future electronic technology - including non-stoichiometric compounds, strongly correlated oxides, and strained or patterned films - are inhomogeneous on the nanometer length scale. Exploiting the inhomogeneity of such materials to design next-generation nanodevices requires a band structure probe with nanoscale spatial resolution. To address this demand, we report the first simultaneous observation and quantitative reconciliation of two candidate probes - Landau level spectroscopy and quasiparticle interference imaging - which we employ here to reconstruct the multi-component surface state band structure of the topological semimetal antimony(Sb). We thus establish the technique of band structure tunneling microscopy (BSTM), whose unique advantages include nanoscale access to non-rigid band structure deformation, empty state dispersion, and magnetic field dependent states. We use BSTM to elucidate the relationship between bulk conductivity and surface state robustness in topological materials, and to quantify essential metrics for spintronics applications.

**Hint:** Topological, but not insulator.

### Aharonov-Bohm oscillations in disordered topological insulator nanowires (arXiv:1005.3762)¶

J. H. Bardarson, P. W. Brouwer, J. E. Moore

A direct signature of electron transport at the metallic surface of a topological insulator is the Aharonov-Bohm oscillation observed in a recent study of Bi_2Se_3 nanowires [Peng et al., Nature Mater. 9, 225 (2010)] where conductance was found to oscillate as a function of magnetic flux \(\phi\) through the wire, with a period of one flux quantum \(\phi_0=h/e\) and maximum conductance at zero flux. This seemingly agrees neither with diffusive theory, which would predict a period of half a flux quantum, nor with ballistic theory, which in the simplest form predicts a period of \(\phi_0\) but a minimum at zero flux due to a nontrivial Berry phase in topological insulators. We show how h/e and h/2e flux oscillations of the conductance depend on doping and disorder strength, provide a possible explanation for the experiments, and discuss further experiments that could verify the theory.

**Hint:** Threading flux through a topological insulator.

### 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!