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Topological materials form a broad family including insulators, semimetals, and superconductors, of which perhaps the best known are topological insulators. For concreteness, I will focus on topological insulators as these are the ones specifically mentioned in the question. Topological insulator. A topological insulator is an insulator whose Hamiltonian ...


11

Not a very thorough answer, but it should get the ball rolling. Spontaneous magnetization or extrinsically magnetic TIs have been achieved through defect engineering in non-magnetic Topological Insulators. This is typically done via doping of 3d magentic atoms (e.g. $\ce{Fe}$, $\ce{Mn}$). A recent review (2019) published in Nature Reviews Physics on the ...


10

A Weyl point is a crossing of two bands. A two-band crossing can be described using the general Hamiltonian: $$ \hat{H}(\mathbf{k})=d_0(\mathbf{k})+d_1(\mathbf{k})\sigma_1+d_2(\mathbf{k})\sigma_2+d_3(\mathbf{k})\sigma_3 $$ where $\sigma_i$ are the Pauli matrices. The eigenvalues are given by $E_{\pm}=d_0\pm\sqrt{d_1^2+d_2^2+d_3^2}$, so that a band crossing ...


10

Magnetic order and topological order can exist simultaneously. In fact, what one may call the very first proposal of an instrinsic topological material was Haldane's model from 1988, which is an example of this. In this tight binding model, based on a hexagonal 2-dimensional lattice (think graphene), we have a next-nearest-neighbour complex hopping term ...


9

The bulk band structure of a topological insulator would look just like any other insulator, with the Fermi level in the gap between the valence and conduction bands. If your band structure includes the surface states, then you would see some states in the gap that cross the Fermi level. There's an example on Wikipedia: An idealized band structure for a ...


9

There is a whole zoo of topological phases, and hopefully someone will provide a more complete answer, but here are some thoughts. Symmetry and dimension. The topological classification of a material with a gap (topological insulator or topological superconductor) depends on (i) symmetry and (ii) dimension. These relations are summarized by the so-called ...


9

There are two different questions in your post: The band structure that you are reproducing is not that of a semiconductor. There are bands crossing the Fermi level, so the material is metallic. Looking at the link that you include in the post, there is a plot of the density of states (DOS) right next to the plot you reproduce here of the band structure, ...


9

Pseudopotentials (PPs) describe the effective interaction between the valence electrons and a nuclei screened by frozen core electrons. This approximation makes DFT calculations less computationally expensive as only valence electrons are treated explicitly and the resulting valence wavefunctions no longer oscillate rapidly near the cores to ensure ...


7

Topological magnon band structures Another way of combining topology and magnetism is to consider a magnetic insulator with non-trivial magnon band structure. This setting is somewhat different from the usual picture of topological electron band structure in that i) the band structure represents only quasiparticle excitations, ii) the quasiparticles are ...


7

Question 1. eq. 3.102 defines a matrix in "band-space", e.g. it's NxN where N = number of bands being considered. The right hand side is a regular vector inner product for fixed m,n. The vectors on the right hand side are in any basis you want, so long as they span the same Kohn-Sham/band subspace. Question 2. I'm not sure where you're lost. You ...


7

The surface Green's function method is described in detail here, but here is a summary. Principal Layers: Block Tridiagonal Form We consider a semi-infinite system (a surface with an infinite bulk below it) and we split it into so-called principal layers. A principal layer is a collection of atomic layers such that each principal layer only interacts with ...


6

You can see the open-source package WannierTools: https://github.com/quanshengwu/wannier_tools A brief description: We present an open-source software package WannierTools, a software for the investigation of novel topological materials. This code works in the tight-binding framework, which can be generated by another software package Wannier90. It can ...


6

Finding topological invariant number(called Z$_2$ number) can give information about topological invariance. Different codes are available for such calculations such as Z2pack or wanniertools and others.


5

As a first step, you can use the WannierTools. There are five typical examples in there. Bi2Se3 (3D strong TI) MoS2 (2D QSHE) WTe2 (Type II Weyl semimetal) IrF4 (Nodal Chain metals) FeSi (Weyl point in Phonon system)


5

Arkadiy Simonov and Andrew Goodwin have a nice review out on the arXiv regarding designing disorder into materials, that can elicit unique topological states [1], and there is are some older reviews discussing specifically crystallography [2,3]. For a more robust mathematical treatment, this paper focuses on abstraction with a set of good physical examples [...


5

Consider the energy eigenstate $|\psi_{n\mathbf{k}}\rangle$, where $n$ is the band you are interested in and $\mathbf{k}$ is wave vector of the TRIM point. Then, if the system has inversion symmetry, these energy eigenstates are also eigenstates of the parity operator $\hat{\pi}$. This means that to determine the parity of that state, you can calculate the ...


4

Spin-orbit coupling is an effect that is dependent on the relativistic effect. So you should use fully relativistic PP(pseudopotentials) whatever you use. Another thing is that PP often varies from your material structure. So, the efficiency of PP is highly dependent on what your simulation output parameter is. There are a lot of things related to the ...


4

Topological semimetals do have interesting edge states. Here I give two examples: Weyl semimetals. The edge states are called "Fermi arcs", which are constant-energy states that connect the projections of the Weyl points onto the surface. The Wikipedia article has a very nice depiction of Fermi arcs. Nodal line semimetals. The edge states are ...


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