Moiré patterns in a pair of hexagonal lattices


A Moiré pattern is an interference pattern that occurs when two grids of repeating lines or shapes are rotated by a small amount relative to one another (oblig. xkcd).

A recent application concerned the experimental verification of Hofstadter's butterfly: Douglas Hofstadter (he of Gödel, Escher, Bach fame) predicted in a 1976 paper that the energy structure of the electrons in a two-dimensional lattice of atoms in a magnetic field has a fractal structure. The problem with the experimental verification of this prediction is that the lattice dimensions of a typical solid material are so small that it would be unfeasible to generate the necessary magnetic field to make the energy structure measurable.

In 2013, three research groups were able to verify the claim by placing a layer of graphene on a substrate layer of boron nitride (which has a very similar hexagonal atomic structure), at such an angle that the resulting Moiré pattern had a much larger repeating length scale.

The code below generates an SVG image of two hexagonal lattices, rotated by a specified amount, which shows the repeating Moiré pattern that results.

Hexagonal Moiré pattern

import numpy as np

# Image dimensions and grid lattice size, a.
width, height = 600, 400
a = 5

# a.sin(60) and a.cos(60)
gx, gy = a * np.sqrt(3)/2, a/2
# The number of unit cells, horizontally and vertically.
nx, ny = int(width / 2 / gx)+1, int(height / (a + gy))+1
# We'll need the coordinates of the centre of the image when we come to rotate.
cx, cy = width / 2, height / 2

def svg_line(x0, y0, x1, y1, th=0, cls=None):
    """Return the SVG for a single, line possibly rotated by th radians."""

    def rotate(x, y, th):
        """Rotate the coordinates (x,y) about the centre (cx,cy)."""
        c, s = np.cos(th), np.sin(th)
        xp, yp = x-cx, y-cy
        x, y = c*xp - s*yp, s*xp + c*yp
        return x+cx, y+cy

    if th != 0:
        x0, y0 = rotate(x0, y0, th)
        x1, y1 = rotate(x1, y1, th)

    # If an SVG class has been provided, add it to the line element.
    s_cls = 'class="{}"'.format(cls) if cls else ''
    return '<line x1="{}" y1="{}" x2="{}" y2="{}" {}/>'.format(
                                                    x0, y0, x1, y1, s_cls)

def add_unit_cell(s, x0, y0, th=0, cls=None):
    """Add a unit cell from the lattice to the SVG output.

    The "unit cell" consists of the arrangement of lines: \ /
    centred at the vertex where they meet. th is the angle of rotation, in
    radians and cls is an optional SVG class to add to the <line> element.


    s.append(svg_line(x0, y0, x0, y0+a, th, cls))
    s.append(svg_line(x0, y0, x0-gx, y0-gy, th, cls))
    s.append(svg_line(x0, y0, x0+gx, y0-gy, th, cls))

def svg_preamble(s):
    """The usual SVG preamble and style definitions."""

    s.append('<?xml version="1.0" encoding="utf-8"?>')
    s.append('<svg xmlns=""\n' + ' '*5 +
         'xmlns:xlink="" width="{}" height="{}" >'
            .format(width, height))
    <style type="text/css"><![CDATA[
    line {
        stroke-width: 2px;
        stroke: #000;
    .lattice1 {
        stroke: #824e4e;
    .lattice2 {
        stroke: #4a5f70;

s = []

# Angle of rotation of the lattices (degrees)
th = 10
# An internal variable to offset every other row of the unit cells when drawing
_ph = 0
for iy in range(ny):
    _ph = 0 if _ph else gx
    for ix in range(nx):
        add_unit_cell(s, ix*2*gx + _ph, iy*(a+gy), cls='lattice1')
        add_unit_cell(s, ix*2*gx + _ph, iy*(a+gy), np.radians(th), 'lattice2')


with open('lattice.svg', 'w') as fo:
    print('\n'.join(s), file=fo)
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Alfonsi Jessica 3 months, 2 weeks ago

Wonderful blog! It makes Python even more appealing to me, I just need a good problem to start learning by coding.
Have a look at my Mathematica Demonstration I published 5 years ago

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christian 3 months, 2 weeks ago

Thank you! I think we may have been reading the same papers about rotated graphene layers. I was going to write something about Hofstadter's Butterfly as well but never got round to it...

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Alfonsi Jessica 3 months, 1 week ago

I have been thinking often about plotting Hofstadter's butterfly for different lattices, square and hexagonal, but now I'm looking into other systems. There's a recent IOP ebook

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