Building the crystal

Model creation

The crystalline structure of the system is automatically built from the configuration file when we initialize the tightbinder.models.SlaterKoster model. By default the models are built with periodic boundary conditions (PBC), specially if the configuration file has Bravais vectors defined.

from tightbinder.fileparse import parse_config_file
from tightbinder.models import SlaterKoster

file = open("./examples/hBN.txt", "r")
config = parse_config_file(file)
model = SlaterKoster(config)

Then, the information about crystal can be accessed from the model via its attributes Crystal.bravais_lattice and Crystal.motif:

model.bravais_lattice
model.motif

Supercell

Once the model has been initialized for the first time, we can make changes to the crystal before initializing the Hamiltonian to obtain the energy spectrum. For instance, we can change the unit cell to describe instead a supercell made of 2x2 original unit cells:

from tightbinder.fileparse import parse_config_file
from tightbinder.models import SlaterKoster

file = open("./examples/hBN.txt", "r")
config = parse_config_file(file)
model = SlaterKoster(config).supercell(n1=2, n2=2)

Note that supercell() is a method from System, which is the base class for all model classes, meaning that building supercells is not restricted to SlaterKoster but is a general feature of any model.

Reducing dimensionality

In some cases we might be interested in removing the periodic boundary conditions after building the supercell, for instance if we want to look at the spectrum of a semi-infinite system such as a ribbon or a slab, or directly to study a finite system.

from tightbinder.fileparse import parse_config_file
from tightbinder.models import SlaterKoster

file = open("./examples/hBN.txt", "r")
config = parse_config_file(file)
model = SlaterKoster(config).supercell(n1=2).reduce(n2=2)

The reduce() method first builds a supercell in the directions specified, and then removes the Bravais vectors in those directions effectively switching to OBC along those boundaries. In the example shown here, we first double the unit cell along one direction (still periodic), and then the unit cell is doubled in the other direction but the periodicity is removed. Note that one can go from a fully periodic system to a finite one (PBC to OBC) either using the reduce() method, or simply changing the boundary attribute of model:

model.boundary = "OBC"

Amorphous supercell

The description of amorphous solids can be regarded as an extension of the supercell procedure. To capture accurately the physics of amorphous solids, we usually want to have a big supercell so that there is enough variation inside of it. Therefore, the procedure is to first produce a supercell starting from the crystalline system, and then move the positions of the atoms sampling the displacements from statistical distributions (either a uniform or gaussian distribution) with the amorphize() method.

from tightbinder.fileparse import parse_config_file
from tightbinder.models import SlaterKoster
from tightbinder.disorder import amorphize

file = open("./examples/hBN.txt", "r")
config = parse_config_file(file)
model = SlaterKoster(config).supercell(n1=2, n2=2)

disorder = 0.1
model = amorphize(model, disorder)

Adding vacancies

Instead of amorphizing the crystalline supercell, one could want instead to introduce vacancies in the system, i.e. to remove atoms. This can be done in an analogous way to the amorphize() method with introduce_vacancies():

from tightbinder.fileparse import parse_config_file
from tightbinder.models import SlaterKoster
from tightbinder.disorder import introduce_vacancies

file = open("./examples/hBN.txt", "r")
config = parse_config_file(file)
model = SlaterKoster(config).supercell(n1=2, n2=2)

probability = 0.1
model = introduce_vacancies(model, probability)

Crystal visualization

After initializing the model, it is useful to visualize the model to ensure that the subyacent crystal lattice was also appropiately built, specially if we have performed any manipulation of the lattice (see e.g supercell or disorder). The library currently provides two different ways to present the crystal lattice: either plotting the positions of the atoms via Matplotlib, or with a 3D render using VPython. Both cases are illustrated below:

model.visualize() # Uses VPython to render the crystal

Note that the visualize() method uses VPython, which opens a browser window if run the code is ran from terminal. For a simpler way of plotting the crystal, one can use the method plot_crystal():

model.plot_crystal() # Plots the crystal projection onto the xy plane.

Note that both methods are actually implemented by the base class Crystal and so are not specific to any model class.

Bonds between atoms

The bonds between atoms are usually determined when the Hamiltonian of the system is built with the method initialize_hamiltonian(). However, in some cases we might be interested in knowing them beforehand, either to check whether they are correct or to modify the list of bonds. This can be done calling the method find_neighbours(), and they can be visualized with plot_wireframe().

from tightbinder.fileparse import parse_config_file
from tightbinder.models import SlaterKoster
import matplotlib.pyplot as plt

file = open("./examples/hBN.txt", "r")
config = parse_config_file(file)
model = SlaterKoster(config)

model.find_neighbours()
model.plot_wireframe()

plt.show()

Note

In general, the use of explicit use of find_neighbours() is not advised since it is already used in initialize_hamiltonian(), which also takes care of building the Hamiltonian from those bonds. Therefore, one can also use plot_wireframe() after initialize_hamiltonian() instead.