m = bandcalc.generate_reciprocal_lattice_basis(rec_m)
# Real space moire lattice vectors
moire_lattice = bandcalc.generate_lattice_by_shell(m, shells)
if potential == "MoS2":
# Moire potential coefficients
V = 6.6 * 1e-3 * np.exp(-1j * 94 * np.pi / 180) # in eV
Vj = np.array([V if i % 2 else np.conjugate(V) for i in range(1, 7)])
potential_matrix = bandcalc.calc_potential_matrix_from_coeffs(
rec_moire_lattice, Vj)
elif potential == "off":
potential_matrix = bandcalc.calc_potential_matrix(rec_moire_lattice)
## Calculate eigenstates for every k point in the MBZ
k_points = bandcalc.generate_monkhorst_pack_set(rec_m, 30).astype(np.float32)
hamiltonian = bandcalc.calc_hamiltonian(rec_moire_lattice, potential_matrix,
mass)
## Calculate MBZ and choose some K-points for the k-path
vor_m = Voronoi(rec_moire_lattice)
sorted_vertices = np.array(
sorted(vor_m.vertices, key=lambda x: np.abs(x.view(complex))))[:6]
sorted_vertices = np.array(
sorted(sorted_vertices, key=lambda x: np.angle(x.view(complex))))
points = np.array([[0, 0], sorted_vertices[0], sorted_vertices[1],
sorted_vertices[3]])
k_names = [r"$\gamma$", r"$\kappa'$", r"$\kappa''$", r"$\kappa$"]
path = bandcalc.generate_k_path(points, N)
## Calculate bandstructure
rec_m = b1-bc.rotate_lattice(b2, angle)
# Complete reciprocal moire lattice
rec_moire_lattice = bc.generate_lattice_by_shell(rec_m, shells)
# Real space moire basis vectors
m = bc.generate_reciprocal_lattice_basis(rec_m)
# Real space moire lattice vectors
moire_lattice = bc.generate_lattice_by_shell(m, 2)
# Moire potential coefficients
Vj = np.array([V if i%2 else np.conjugate(V) for i in range(1, 7)])
potential_matrix = bc.calc_potential_matrix_from_coeffs(rec_moire_lattice, Vj)
k_points = bc.generate_monkhorst_pack_set(rec_m, 40)
## Calculate the band structure
# The bandstructure is slightly different from the bandstructure in the
# original paper, but that is most likely just a small difference in
# some parameters, like the lattice constants
hamiltonian = bc.calc_hamiltonian(rec_moire_lattice, potential_matrix, mass)
bandstructure = np.sort(
np.real(
bc.calc_bandstructure(k_points, hamiltonian)
)
)*1e3 # in meV
cell_area = bc.get_volume_element_regular_grid(moire_lattice)
n = filling/cell_area
m_len = np.abs(m[0].view(complex)[0])
# Calculate first shell reciprocal moire vectors
G = bc.generate_twisted_lattice_by_shell(b1, b2, angle, 1)
GT = G[0]
GB = G[1]
GM = GT-GB
GM = np.array(sorted(GM, key=lambda x: np.abs(x.view(complex))))[1:]
GM = np.array(sorted(GM, key=lambda x: np.angle(x.view(complex))))
# Moire potential coefficients
Vj = np.array([V if i%2 else np.conjugate(V) for i in range(1, 7)])
potential_matrix = bc.calc_potential_matrix_from_coeffs(rec_moire_lattice, Vj)
## Calculate eigenstates for every k point in the MBZ
k_points = bc.generate_monkhorst_pack_set(rec_m, 30).astype(np.float32)
hamiltonian = bc.calc_hamiltonian(rec_moire_lattice, potential_matrix, mass)
# Wannier functions should be centered around potential minima
potential_shift = bc.calc_moire_potential_minimum_position(GM, np.abs(V), np.angle(V))
print(f"Potential shift: {potential_shift}")
R0 = potential_shift
R1 = m[0]+potential_shift
R2 = 2*m[0]+potential_shift
## Calculate the wannier function
x_min, x_max, n_x = -2*m_len, 2*m_len, 100
y_min, y_max, n_y = -2*m_len, 2*m_len, 100
x, y = make_xy(x_min, x_max, n_x, y_min, y_max, n_y)
GM = np.array(sorted(GM, key=lambda x: np.abs(x.view(complex))))[1:]
GM = np.array(sorted(GM, key=lambda x: np.angle(x.view(complex))))
# Real space grid
size = np.linspace(-10, 10, 100)
grid = np.meshgrid(size, size)
# Reciprocal space grid
size_r = np.linspace(-3, 3, 100)
grid_r = np.meshgrid(size_r, size_r)
# Real space moire potential
moire_potential = bandcalc.calc_moire_potential_on_grid(grid, GM, Vj)
# Reciprocal space moire potential
mp_moire = bandcalc.generate_monkhorst_pack_set(m, 100)
moire_potential_pointwise = bandcalc.calc_moire_potential(mp_moire, GM, Vj)
rec_moire_potential = bandcalc.calc_moire_potential_reciprocal_on_grid(
mp_moire, grid_r, moire_potential_pointwise)
# Results
fig, axs = plt.subplots(nrows=2, ncols=2)
contour = bandcalc.plot_moire_potential(axs[0,0], grid, np.real(moire_potential), alpha=0.4)
moire_lattice = bandcalc.generate_lattice_by_shell(m, 2)
bandcalc.plot_lattice(axs[0,0], moire_lattice, "ko")
vor_m = Voronoi(moire_lattice)
voronoi_plot_2d(vor_m, axs[0,0], show_points=False, show_vertices=False)
bandcalc.plot_lattice(axs[0,0], mp_moire, "b,", alpha=0.4)
rec_moire_lattice = bandcalc.generate_lattice_by_shell(rec_m, 3)
import matplotlib.pyplot as plt from scipy.spatial import Voronoi, voronoi_plot_2d #pylint: disable=E0611 import bandcalc # Constants a = 1 N = 1000 # Reciprocal lattice vectors b1 = np.array([2 * np.pi / (np.sqrt(3) * a), 2 * np.pi / a]) b2 = np.array([2 * np.pi / (np.sqrt(3) * a), -2 * np.pi / a]) b = np.vstack([b1, b2]) lattice = bandcalc.generate_lattice_by_shell(b, 1) # Monkhorst-Pack lattice mp_lattice = bandcalc.generate_monkhorst_pack_set(b, 10) # Results fig, ax = plt.subplots() vor = Voronoi(lattice) voronoi_plot_2d(vor, ax, show_points=False, show_vertices=False) bandcalc.plot_lattice(ax, lattice, "ko", alpha=0.3) bandcalc.plot_lattice(ax, mp_lattice, "k.") ax.set_xlim([lattice[:, 0].min() - 2, lattice[:, 0].max() + 2]) ax.set_ylim([lattice[:, 1].min() - 2, lattice[:, 1].max() + 2]) plt.show()