Obtaining the band structure

Hamiltonian initialization

Once the model is built and prepared as desired (e.g. modifying the unit cell), the first thing to do is to initialize the Hamiltonian, which is done with tightbinder.system.System.initialize_hamiltonian().

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)

model.initialize_hamiltonian()

Generate kpoints

Usually, we want to look at the spectrum of the Bloch Hamiltonian \(H(k)\) at k points along high symmetry paths in the irreducible Brillouin zone (IBZ). We can define this reciprocal path by specifying the high symmetry points that compose it, with the method tightbinder.crystal.Crystal.high_symmetry_path(). For instance, since hBN has a triangular lattice, we define the path in the IBZ to be \(\Gamma - K - M - \Gamma\):

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)

nk = 100
points = ["G", "K", "M", "G"]
kpoints = model.high_symmetry_path(nk, points)

Note

The method to generate the high symmetry path currently works only with 1D and 2D systems, since the high symmetry points have been specified only for the possible Bravais lattice in those dimensions. For 3D the library still works, but one has to specify manually this high symmetry path.

Obtaining the spectrum

Once the model and its Hamiltonian have been initialized, and we know over which k points we are going to evaluate the Bloch Hamiltonian, we can obtain the spectrum calling the tightbinder.system.System.solve() method, which returns a tightbinder.result.Result object.

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)

nk = 100
points = ["G", "K", "M", "G"]
kpoints = model.high_symmetry_path(nk, points)

model.initialize_hamiltonian()

result = model.solve(kpoints)

Plotting the bands

Once we have the Result object with the eigenvalues and eigenvectors of the system, we can plot the eigenvalues to analyze the bands of the system. This is done with the tightbinder.result.Result.plot_along_path() method of the Result class.

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)

nk = 100
points = ["G", "K", "M", "G"]
kpoints = model.high_symmetry_path(nk, points)

model.initialize_hamiltonian()

result = model.solve(kpoints)
result.plot_along_path(points)

Spectrum of a finite system

So far we have shown how to obtain the band structure of a crystalline system. In case that we want to get the spectrum of a finite system, the procedure is simpler since we do not need to generate the k points first. When the spectrum is not a function of \(k\), it can be plotted instead using the method tightbinder.result.Result.plot_spectrum().

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).reduce(n1=5, n2=5)

model.initialize_hamiltonian()

result = model.solve()
result.plot_spectrum()

Note

The same code works for a periodic system, but in that case the call to solve() diagonalizes the Bloch Hamiltonian at \(k=0\) only.