Commit 42497c41 authored by Anton Akhmerov's avatar Anton Akhmerov
Browse files

add 2d tight-binding

parent de2b3763
......@@ -25,7 +25,7 @@ build and upload the contents:
script:
# Compile lectures
- python code/band_structures.py
- python code/nfem.py
- python code/bands_2d.py
- gitbook install
- gitbook build
- "rsync -rv _book/* solidstate@tnw-tn1.tudelft.net:"
import numpy as np
import plotly.offline as py
import plotly.graph_objs as go
pi = np.pi
def E(k_x, k_y):
delta = np.array([-2*pi, 0, 2*pi])
H = np.diag(
((k_x + delta)[:, np.newaxis]**2
+ (k_y + delta)[np.newaxis]**2).flatten()
)
return tuple(np.linalg.eigvalsh(H + 5)[:3])
E = np.vectorize(E, otypes=(float, float, float))
figure_html = """
<head>
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
</head>
<body>{figure}
</body>
""".format
def plot_nfem():
momenta = np.linspace(-2*pi, 2*pi, 100)
kx, ky = momenta[:, np.newaxis], momenta[np.newaxis, :]
bands = E(kx, ky)
# Extended Brillouin zone scheme
pad = .3
first_BZ = ((abs(kx) < pi + pad) & (abs(ky) < pi + pad))
second_BZ = (
((abs(kx) > pi - pad) | (abs(ky) > pi - pad))
& ((abs(kx + ky) < 2*pi + pad) & (abs(kx - ky) < 2*pi + pad))
)
third_BZ = (
(abs(kx + ky) > 2*pi - pad) | (abs(kx - ky) > 2*pi - pad)
)
bands[0][~first_BZ] = np.nan
bands[1][~second_BZ] = np.nan
#bands[2][~third_BZ] = np.nan
# Actually plotting
fig = go.Figure(
data = [
go.Surface(
z=band / 5,
colorscale=color,
opacity=opacity,
showscale=False,
hoverinfo=False,
x=momenta,
y=momenta,
)
for band, color, opacity
in zip(bands[:2],
['#cf483d', '#3d88cf'],
(1, 0.9))
],
layout = go.Layout(
title='Nearly free electrons in 2D',
autosize=True,
width=500,
height=500,
hovermode=False,
scene=dict(
yaxis={"title": "k_y"},
xaxis={"title": "k_x"},
zaxis={"title": "E"},
)
)
)
with open('figures/nfem_2d.html', 'w') as f:
f.write(figure_html(figure=
py.plot(fig, show_link=False,
output_type='div', include_plotlyjs=False)
))
def plot_tb():
momenta = np.linspace(-pi, pi, 100)
kx, ky = momenta[:, np.newaxis], momenta[np.newaxis, :]
energies = -np.cos(kx) - np.cos(ky)
fig = go.Figure(
data = [
go.Surface(
z=energies,
colorscale='#3d88cf',
opacity=1,
showscale=False,
hoverinfo=False,
x=momenta,
y=momenta,
)
],
layout = go.Layout(
title='Tight-binding in 2D',
autosize=True,
width=500,
height=500,
hovermode=False,
scene=dict(
yaxis={"title": "k_y"},
xaxis={"title": "k_x"},
zaxis={"title": "E"},
)
)
)
with open('figures/tb_2d.html', 'w') as f:
f.write(figure_html(figure=
py.plot(fig, show_link=False,
output_type='div', include_plotlyjs=False)
))
if __name__ == '__main__':
plot_nfem()
plot_tb()
import numpy as np
import plotly.offline as py
import plotly.graph_objs as go
pi = np.pi
def E(k_x, k_y):
delta = np.array([-2*pi, 0, 2*pi])
H = np.diag(
((k_x + delta)[:, np.newaxis]**2
+ (k_y + delta)[np.newaxis]**2).flatten()
)
return tuple(np.linalg.eigvalsh(H + 5)[:3])
E = np.vectorize(E, otypes=(float, float, float))
momenta = np.linspace(-2*pi, 2*pi, 100)
kx, ky = momenta[:, np.newaxis], momenta[np.newaxis, :]
bands = E(kx, ky)
# Extended Brillouin zone scheme
pad = .3
first_BZ = ((abs(kx) < pi + pad) & (abs(ky) < pi + pad))
second_BZ = (
((abs(kx) > pi - pad) | (abs(ky) > pi - pad))
& ((abs(kx + ky) < 2*pi + pad) & (abs(kx - ky) < 2*pi + pad))
)
third_BZ = (
(abs(kx + ky) > 2*pi - pad) | (abs(kx - ky) > 2*pi - pad)
)
bands[0][~first_BZ] = np.nan
bands[1][~second_BZ] = np.nan
#bands[2][~third_BZ] = np.nan
# Actually plotting
fig = go.Figure(
data = [
go.Surface(
z=band / 5,
colorscale=color,
opacity=opacity,
showscale=False,
hoverinfo=False,
x=momenta,
y=momenta,
)
for band, color, opacity
in zip(bands[:2],
['#cf483d', '#3d88cf'],
(1, 0.9))
],
layout = go.Layout(
title='Nearly free electrons in 2D',
autosize=True,
width=500,
height=500,
hovermode=False,
scene=dict(
yaxis={"title": "k_y"},
xaxis={"title": "k_x"},
zaxis={"title": "E"},
)
)
)
out = rf"""
<head>
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
</head>
<body>{py.plot(fig, show_link=False, output_type='div', include_plotlyjs=False)}
</body>
"""
with open('figures/nfem_2d.html', 'w') as f:
f.write(out)
\ No newline at end of file
......@@ -82,7 +82,7 @@ Here the coupling strength $W = \langle \psi_+ | V(x) | \psi_- \rangle$ is the m
We need to diagonalize a 2x2 matrix Hamiltonian. The answer is
$$ E(\delta k) = E_0 \pm \sqrt{v^2\hbar^2\delta k^2 + |W|^2}$$
In one of the exercises you will encounter the details of this calculation.
Check out section 15.1.1 of the book for the details of this calculation.
#### Physical meaning of $W$
......@@ -176,7 +176,12 @@ The resulting band structure looks like this (in the extended Brillouin zone sch
Observe that the top of the first band is above the bottom of the lowest band. Therefore if $V$ is sufficiently weak, the material can be conducting even with 2 electrons per unit cell!
A larger $V$ makes the Fermi surface more square-like and eventually makes the material insulating.
A larger $V$ makes the Fermi surface more distorted and eventually makes the material insulating.
Let's compare the almost parabolic dispersion of the nearly free electron model with a tight-binding model in 2D.
We now have a dispersion relation $E = E_0 + 2t(\cos k_x a + \cos k_y a)$, which looks like this:
<iframe width="600", height="600" src="figures/tb_2d.html" frameBorder="0" align="center" style="border:0;">Your browser still doesn't support iframes??</iframe>
### Light adsorption
......
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