# Simulations: more Chern insulators¶

As usual, start by grabbing the notebooks of this week (`w6_3dti`

). They are once again over here.

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).

** MoocSelfAssessment description**

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

**Discussion** entitled '3DTI' is available in the online version of the course.

### Electrostatic Coupling between Two Surfaces of a Topological Insulator
Nanodevice

http://arxiv.org/abs/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

http://arxiv.org/abs/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

http://arxiv.org/abs/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

http://arxiv.org/abs/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

http://arxiv.org/abs/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!

** Read one of the above papers and see how it is
related to the current topic.**

**In the live version of the course, you would need to write a summary which is then assessed by your peers.**

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

**Discussion** entitled '3DTI' is available in the online version of the course.