topological two-dimensional polymers supporting informationtransitive regular nets hcb, hxl, and...
TRANSCRIPT
Topological two-dimensional polymers
Supporting Information
Maximilian A. Springer1,2, Tsai-Jung Liu2, Agnieszka Kuc1, and
Thomas Heine1,2
1Helmholtz-Zentrum Dresden-Rossendorf, Institute of Resource Ecology, Research Site Leipzig,
Permoserstrasse 15, 04138 Leipzig, Germany2TU Dresden, Faculty for Chemistry and Food Chemistry, Bergstrasse 66c, 01069 Dresden,
Germany
Contents
1 Band structures and sketches for 2D nets 1
2 Technical tutorial for tight-binding calculations 30
1
Electronic Supplementary Material (ESI) for Chemical Society Reviews.This journal is © The Royal Society of Chemistry 2020
1 Band structures and sketches for 2D nets
On the following pages, Tables S1-S4 show the tight-binding band structures that were
calculated using scripts similar to the accompanying example script, based on the pythTB
package.[1] The Fermi level was estimated by integrating over a grid of equidistant points
in the first Brillouin zone. The investigated topologies are separated by the manifold of
faces and vertices. First, Table S1 shows the band structures for the face- and vertex-
transitive regular nets hcb, hxl, and sql. In Table S2 and S3, band structures for the
semiregular (one type of vertex but different types of faces) and demiregular nets (ex-
actly two types of vertices) can be found. Additionally, we show the tight-binding band
structures for selected topologies that do not fit into one of these groups in Table S4.
1
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14
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2 Technical tutorial for tight-binding calculations
In this review, we have used the pythTB package, which can be downloaded from
http://www.physics.rutgers.edu/pythtb/. Here, we explain the elementary steps to per-
form a tight-binding (TB) calculation with the example of the honeycomb (hcb) lattice
(this tutorial is written for python 3.6 and pythTB 1.7.2). For setting up the calcula-
tions, a python script with the following content is needed:
1. Load necessary modules:
from __future__ import print_function
from pythTB import *
import numpy as np
2. Define latticelat and fractional coordinates of sites orb:
lat = [[1, 0], [0.5, np.sqrt(3)/2]]
orb = [[1/3, 1/3], [2/3, 2/3]]
3. Initialise the pythTB class:
my_model = tb_model(dim_k, dim_r, lat, orb, nspin=2)
Here, dim_k and dim_r define the dimensionality in reciprocal and real space,
respectively. For the hcb lattice, dim_k = 2 and dim_r = 2 are correct settings
corresponding to both a two-dimensional real space and a two-dimensional re-
ciprocal space. nspin = 2 enables spin-polarised calculations, while the default
nspin = 1 is the setting for spin-restricted (diamagnetic) systems.
4. Setting on-site energies α for each site as a list with the same ordering as in orb:
my_model.set_onsite([alpha, alpha])
For different chemical potentials, different values for α are possible. Herein, we
use α = 0.
30
5. Setting hopping integrals β:
my_model.set_hop(beta, i, j, [A, B])
Hopping integrals need to be specified one by one. In this function, β is the
hopping integral for interaction between vertex i in the first cell and vertex j
in the cell replicated by the translation vector R[A, B]. Where A and B are in
units of the respective lattice vector, moving from the first cell to the replicated
cell. For first-neighbour hoppings within the first cell, A and B can be set to 0.
In pythTB, symmetric hoppings are set automatically. i -> j + R[A, B] and
i -> j - R[A, B] are set simultaneosly, and cannot be specified twice in the
script. For example, if we have
my_model.set_hop(beta, 1, 0, [0, 1])
then there should be no my_model.set_hop(beta, 0, 1, [0, -1]), since this
is the same translation vector in opposite direction. In addition, individual hop-
ping elements, e.g. β′, can be specified afterwards by setting the mode add:
my_model.set_hop(beta’, 1, 0, [0, 1], mode=’add’)
In this case, the final hopping from site 0 to site 1 in the cell with translation
vector ~R = (0, 1) would be β + β′. In the case of hcb, three hoppings have to be
set:
my_model.set_hop(beta, 0, 1, [0, 0])
my_model.set_hop(beta, 1, 0, [0, 1])
my_model.set_hop(beta, 1, 0, [1, 0])
6. Setting the path in k -space which should be used for the band structure calculation:
(k_vec, k_dist, k_node) = my_model.k_path(path, nk)
With path being a list with the fractional coordinates of high-symmetry points
in the Brillouin zone and nk the total number of calculated eigenvalues for this
path. Returns k_vec, a list of the k -points, for which the eigenvalue problem
should be solved. k_dist is the accumulated distance of each k -point to the first
k -point. k_node is the distance of high-symmetry points to the first k -point.
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These are, therefore, selected values from k_dist. For hexagonal cells, the setting
path = [[0.0, 0.0], [0.5, 0.5], [2/3, 1/3], [0.0, 0.0]]
will return a band structure along Γ -M -K -Γ . Additionally, the labels for these
high-symmetry points can be given:
label = [r’$\Gamma$’, ’M’, ’K’, r’$\Gamma$’]
7. Start the calculation and return eigenvalues for each k -point in k_vec:
evals = my_model.solve_all(k_vec)
8. Plotting can be done by, e.g., using matplotlib, which requires to load the package
via import matplotlib.pyplot as plt. Band structure plots can be created by
the following:
[plt.plot(k_vec, eigenvalue, color=’black’) for eigenvalue in evals]
plt.xticks(k_nodes, labels)
plt.ylabel(’Energy’)
plt.show()
9. Additional notes:
a) In order to use the formalism given by Kane and Mele[2] to take spin-orbit
coupling (SOC) into account, one needs to use the spin-polarised Hamiltonian
(thus, nspin = 2) and to manually define the Pauli matrix σz:
sigma_z = np.array([0, 0, 0, 1])
tsoc = -1.j*Lambda*sigma_z
With a SOC strength of λ. Note that for the Haldane model[3] is possible
for a spin-restricted calculation:
tsoc = Lambda*np.exp((1.j)*np.pi/2)
Note that if there are two equivalent paths electrons could use to travel from
a site to the second neighbour, the effect of SOC cancels out, e.g., two ver-
32
tices laying diagonally in a square show no spin-orbit contribution. Since the
second neighbour hopping includes both the normal hopping β2 and the SOC
contribution, two lines per second neighbour and the mode add can be used
to assign the hopping and SOC:
my_model.set_hop(beta_2, 0, 0, [1, 0])
my_model.set_hop(tsoc, 0, 0, [1, 0], mode=’add’)
b) Calculation details can be printed by my_model.display().
c) In order to estimate the Fermi level, it is necessary to integrate over the
whole Brillouin zone. In a k -grid of NG points, each grid point contributes
1/NG electrons (2/NG for spin-restricted calculations). The bands are occu-
pied from the lowest to higher bands, until the number of electrons matches
the number of electrons in the cell (see Section 3.1 and the example input file).
d) Obtaining the Chern number and Z2 invariant:
The pythtb.wf_array class calculates the Berry phase and related proper-
ties. After setting up the Hamiltonian, this class can be initialised with
my_array = wf_array(my_model, [nk, nk])
with nk being the number of grid points per axis.
my_array.solve_on_grid([x, y])
solves the model in a regular grid with the origin being at the point [x, y].
The Chern number equals the Berry phase
berry = [my_array.berry_flux([i]) for i in range(len(my_array))]
divided by 2π (with i being the band index). It is worth noting that for
calculating Chern number/Berry phase, a spin-restricted calculation is re-
quired. In order to get the Z2 invariant, the Wannier Charge Centres (WCC)
have to be calculated. An example for this is given on the pythTB website.[4]
33
References
1. http://www.physics.rutgers.edu/pythtb/, accessed 11 February 2020.
2. C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2005, 95, 226801.
3. F. D. M. Haldane, Phys. Rev. Lett., 1988, 61, 2015.
4. http://www.physics.rutgers.edu/pythtb/examples.html#kane-mele-model-using-spinor-
features, accessed 11 February 2020.
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