Schematics: Transparency tricks

Although latticegen is primarily focussed on generating numerical data to test algorithms (e.g. GPA) on, it is also possible to use it to generate more schematic renderings of lattices. To visualize bonds and combine different lattices, it is possible to play transparency tricks with matplotlibs imshow() and the diverging colormaps.

import numpy as np
import matplotlib.pyplot as plt

import latticegen

A hexagonal moiré lattice

By applying a hexagonal lattice of order=1 itself as transparency, we obtain something nicely respresenting the bonds in the lattice.

Note: In matplotlib, alpha values are clipped between 0 and 1. Here, the lattices are scaled to \(max = 1\) instead of using the normalize kwarg of anylattice_gen(), to obtain slightly moire transparent lattices overall.

r_k = 0.05
lattice1 = latticegen.hexlattice_gen(r_k, 0, order=1)
lattice1 = np.clip(lattice1 / lattice1.max(), 0, 1).compute()
lattice2 = latticegen.hexlattice_gen(r_k, 5, order=1)
lattice2 = np.clip(lattice2 / lattice2.max(), 0, 1).compute()

fig,axs = plt.subplots(ncols=3, figsize=[10,4])
for i in [0, 1]:
    axs[i].imshow(-lattice1.T, cmap='PiYG',
                      vmax=1,
                      vmin=-1,
                       alpha=lattice1.T
                     )
    axs[i + 1].imshow(lattice2.T, cmap='PiYG',
                   vmax=1,
                   vmin=-1,
                   alpha=lattice2.T)

A diatomic lattice

By combining two separate trigonal lattices, we can illustrate a diatomic hexagonal lattice such as the insulator hexagonal Boron Nitride (hBN). A different colormap is used to generate different colors.

(Any combination of colors can be obtained by creating a custom colormap)

r_hBN = r_k * (0.246 / 0.2504)
sublattice_a = latticegen.trilattice_gen(r_hBN, 0, order=1, normalize=True)
sublattice_a = sublattice_a.compute()
# Now add the second shifted sublattice lattice to get a hexagonal lattice
ks = latticegen.generate_ks(r_hBN, 0, sym=6)
x = np.array([ks[1], -ks[2]])
shift = (np.linalg.inv(x / r_hBN).T/(3*r_k)).sum(axis=0).T  # Don't ask, this works
sublattice_b = latticegen.trilattice_gen(r_hBN, 0, order=1,
                                         shift=shift, normalize=True)
sublattice_b = sublattice_b.compute()

fig, axs = plt.subplots(ncols=3, figsize=[10,4])
axs[0].set_title('Sublattice a')
axs[1].set_title('Both sublattices')
axs[2].set_title('Sublattice b')
for i in [0, 1]:
    axs[i].imshow(-sublattice_a.T, cmap='bwr',
                  vmax=1, vmin=-1,
                  alpha=sublattice_a.T)
    axs[i + 1].imshow(sublattice_b.T, cmap='bwr',
                      vmax=1, vmin=-1,
                      alpha=sublattice_b.T)

A moiré of a diatomic lattice and a hexagonal lattice

Putting both examples together to create an image of a moiré of a graphene lattice (green) on top of a hBN lattice (red/blue):

plt.figure(figsize=[10,10])
plt.imshow(-sublattice_a.T, cmap='bwr',
           vmax=1, vmin=-1,
           alpha=sublattice_a.T)
plt.imshow(sublattice_b.T, cmap='bwr',
           vmax=1, vmin=-1,
           alpha=sublattice_b.T)
plt.imshow(lattice2.T, cmap='PiYG',
           vmax=1, vmin=-1,
           alpha=lattice2.T*0.9)

<matplotlib.image.AxesImage at 0x7f40f579c2e0>