From 2ef5f643a23b138e1806bd69ca00cc1d63813226 Mon Sep 17 00:00:00 2001
From: Anton Akhmerov <anton.akhmerov@gmail.com>
Date: Thu, 26 Jul 2018 22:39:08 +0200
Subject: [PATCH] use inline code execution
---
.gitlab-ci.yml | 5 +-
code/band_structures.py | 227 ------------------
code/common.py | 35 ++-
code/heat_capacity.py | 106 --------
code/misc.py | 75 ------
code/semiconductors.py | 70 ------
docs/mathjaxhelper.js | 5 -
execute.py | 43 ++++
mkdocs.yml | 5 +-
{docs => src}/figures/DOS.svg | 0
{docs => src}/figures/DOS_periodic.svg | 0
.../figures/E_F_and_carrier_density.svg | 0
{docs => src}/figures/RKKY.svg | 0
{docs => src}/figures/adsorption.svg | 0
.../figures/band_diagram_question.svg | 0
.../figures/band_diagram_solution.svg | 0
.../figures/band_structure_sketch.svg | 0
{docs => src}/figures/bonding.svg | 0
.../figures/bonding_with_repulsion.svg | 0
{docs => src}/figures/bragg.svg | 0
{docs => src}/figures/bravais1.svg | 0
{docs => src}/figures/bravais2.svg | 0
{docs => src}/figures/brillouin.svg | 0
{docs => src}/figures/brillouin_mod.svg | 0
{docs => src}/figures/crystalfield.svg | 0
{docs => src}/figures/cubic.svg | 0
{docs => src}/figures/cubic_mod.svg | 0
{docs => src}/figures/direct_indirect.svg | 0
{docs => src}/figures/dos_1d_tb.svg | 0
{docs => src}/figures/einstein.svg | 0
{docs => src}/figures/electric_field.svg | 0
{docs => src}/figures/electrons.svg | 0
{docs => src}/figures/ewald.svg | 0
.../extended_nearly_free_electron_bands.svg | 0
{docs => src}/figures/fcc.svg | 0
{docs => src}/figures/fcc_mod.svg | 0
{docs => src}/figures/fermi_circle.svg | 0
.../figures/fermi_circle_periodic.svg | 0
{docs => src}/figures/fermi_level.svg | 0
{docs => src}/figures/free_electron.svg | 0
{docs => src}/figures/graphite.svg | 0
{docs => src}/figures/graphite_mod.svg | 0
{docs => src}/figures/hall_effect.svg | 0
{docs => src}/figures/harmonic.svg | 0
{docs => src}/figures/holes.svg | 0
{docs => src}/figures/hund.svg | 0
.../figures/interatomic_interaction.svg | 0
{docs => src}/figures/lattice_potential.svg | 0
{docs => src}/figures/laue.svg | 0
{docs => src}/figures/laue_mod.svg | 0
{docs => src}/figures/matthiessen.svg | 0
{docs => src}/figures/metal_semiconductor.svg | 0
{docs => src}/figures/miller.svg | 0
{docs => src}/figures/models.svg | 0
.../figures/nearly_free_electron_bands.svg | 0
{docs => src}/figures/packing.svg | 0
{docs => src}/figures/phonons1.svg | 0
{docs => src}/figures/phonons2.svg | 0
{docs => src}/figures/phonons5.svg | 0
{docs => src}/figures/pn_charge_density.svg | 0
{docs => src}/figures/pn_junction_bias.svg | 0
{docs => src}/figures/realistic.svg | 0
{docs => src}/figures/reciprocal.svg | 0
{docs => src}/figures/reciprocal_mod.svg | 0
.../reduced_nearly_free_electron_bands.svg | 0
{docs => src}/figures/semiconductor.svg | 0
{docs => src}/figures/single_atom.svg | 0
{docs => src}/figures/soft_xray.svg | 0
{docs => src}/figures/solar_cell.svg | 0
{docs => src}/figures/stacking.svg | 0
{docs => src}/figures/superexchange.svg | 0
{docs => src}/figures/transport.svg | 0
{docs => src}/figures/two_atoms.svg | 0
{docs => src}/figures/xray.svg | 0
{docs => src}/index.md | 0
{docs => src}/lecture_1.md | 90 ++++++-
{docs => src}/lecture_2.md | 74 +++++-
{docs => src}/lecture_3.md | 46 +++-
{docs => src}/lecture_4.md | 168 ++++++++++++-
{docs => src}/lecture_5.md | 0
{docs => src}/lecture_6.md | 0
{docs => src}/lecture_7.md | 67 +++++-
{docs => src}/lecture_8.md | 27 ++-
83 files changed, 509 insertions(+), 534 deletions(-)
delete mode 100644 code/band_structures.py
delete mode 100644 code/heat_capacity.py
delete mode 100644 code/misc.py
delete mode 100644 code/semiconductors.py
delete mode 100644 docs/mathjaxhelper.js
create mode 100644 execute.py
rename {docs => src}/figures/DOS.svg (100%)
rename {docs => src}/figures/DOS_periodic.svg (100%)
rename {docs => src}/figures/E_F_and_carrier_density.svg (100%)
rename {docs => src}/figures/RKKY.svg (100%)
rename {docs => src}/figures/adsorption.svg (100%)
rename {docs => src}/figures/band_diagram_question.svg (100%)
rename {docs => src}/figures/band_diagram_solution.svg (100%)
rename {docs => src}/figures/band_structure_sketch.svg (100%)
rename {docs => src}/figures/bonding.svg (100%)
rename {docs => src}/figures/bonding_with_repulsion.svg (100%)
rename {docs => src}/figures/bragg.svg (100%)
rename {docs => src}/figures/bravais1.svg (100%)
rename {docs => src}/figures/bravais2.svg (100%)
rename {docs => src}/figures/brillouin.svg (100%)
rename {docs => src}/figures/brillouin_mod.svg (100%)
rename {docs => src}/figures/crystalfield.svg (100%)
rename {docs => src}/figures/cubic.svg (100%)
rename {docs => src}/figures/cubic_mod.svg (100%)
rename {docs => src}/figures/direct_indirect.svg (100%)
rename {docs => src}/figures/dos_1d_tb.svg (100%)
rename {docs => src}/figures/einstein.svg (100%)
rename {docs => src}/figures/electric_field.svg (100%)
rename {docs => src}/figures/electrons.svg (100%)
rename {docs => src}/figures/ewald.svg (100%)
rename {docs => src}/figures/extended_nearly_free_electron_bands.svg (100%)
rename {docs => src}/figures/fcc.svg (100%)
rename {docs => src}/figures/fcc_mod.svg (100%)
rename {docs => src}/figures/fermi_circle.svg (100%)
rename {docs => src}/figures/fermi_circle_periodic.svg (100%)
rename {docs => src}/figures/fermi_level.svg (100%)
rename {docs => src}/figures/free_electron.svg (100%)
rename {docs => src}/figures/graphite.svg (100%)
rename {docs => src}/figures/graphite_mod.svg (100%)
rename {docs => src}/figures/hall_effect.svg (100%)
rename {docs => src}/figures/harmonic.svg (100%)
rename {docs => src}/figures/holes.svg (100%)
rename {docs => src}/figures/hund.svg (100%)
rename {docs => src}/figures/interatomic_interaction.svg (100%)
rename {docs => src}/figures/lattice_potential.svg (100%)
rename {docs => src}/figures/laue.svg (100%)
rename {docs => src}/figures/laue_mod.svg (100%)
rename {docs => src}/figures/matthiessen.svg (100%)
rename {docs => src}/figures/metal_semiconductor.svg (100%)
rename {docs => src}/figures/miller.svg (100%)
rename {docs => src}/figures/models.svg (100%)
rename {docs => src}/figures/nearly_free_electron_bands.svg (100%)
rename {docs => src}/figures/packing.svg (100%)
rename {docs => src}/figures/phonons1.svg (100%)
rename {docs => src}/figures/phonons2.svg (100%)
rename {docs => src}/figures/phonons5.svg (100%)
rename {docs => src}/figures/pn_charge_density.svg (100%)
rename {docs => src}/figures/pn_junction_bias.svg (100%)
rename {docs => src}/figures/realistic.svg (100%)
rename {docs => src}/figures/reciprocal.svg (100%)
rename {docs => src}/figures/reciprocal_mod.svg (100%)
rename {docs => src}/figures/reduced_nearly_free_electron_bands.svg (100%)
rename {docs => src}/figures/semiconductor.svg (100%)
rename {docs => src}/figures/single_atom.svg (100%)
rename {docs => src}/figures/soft_xray.svg (100%)
rename {docs => src}/figures/solar_cell.svg (100%)
rename {docs => src}/figures/stacking.svg (100%)
rename {docs => src}/figures/superexchange.svg (100%)
rename {docs => src}/figures/transport.svg (100%)
rename {docs => src}/figures/two_atoms.svg (100%)
rename {docs => src}/figures/xray.svg (100%)
rename {docs => src}/index.md (100%)
rename {docs => src}/lecture_1.md (74%)
rename {docs => src}/lecture_2.md (84%)
rename {docs => src}/lecture_3.md (92%)
rename {docs => src}/lecture_4.md (68%)
rename {docs => src}/lecture_5.md (100%)
rename {docs => src}/lecture_6.md (100%)
rename {docs => src}/lecture_7.md (91%)
rename {docs => src}/lecture_8.md (95%)
diff --git a/.gitlab-ci.yml b/.gitlab-ci.yml
index b2b491e2..13db2d37 100644
--- a/.gitlab-ci.yml
+++ b/.gitlab-ci.yml
@@ -24,11 +24,8 @@ build and upload the contents:
script:
# Compile lectures
- - python code/band_structures.py
- python code/bands_2d.py
- - python code/misc.py
- - python code/semiconductors.py
- - python code/heat_capacity.py
+ - python execute.py
- mkdocs build
- "rsync -rv site/* solidstate@tnw-tn1.tudelft.net:"
only:
diff --git a/code/band_structures.py b/code/band_structures.py
deleted file mode 100644
index 7c4e380b..00000000
--- a/code/band_structures.py
+++ /dev/null
@@ -1,227 +0,0 @@
-import os
-
-import numpy as np
-import matplotlib
-import mpl_toolkits
-matplotlib.use('agg')
-from matplotlib import pyplot
-
-import common
-from common import draw_classic_axes
-
-pi = np.pi
-
-
-def fermi_dirac():
- fig = pyplot.figure()
- ax = fig.add_subplot(1,1,1)
- xvals = np.linspace(0, 2, 200)
- mu = .75
- beta = 20
- ax.plot(xvals, xvals < mu, ls='dashed', label='$T=0$')
- ax.plot(xvals, 1/(np.exp(beta * (xvals-mu)) + 1),
- ls='solid', label='$T>0$')
- ax.set_xlabel(r'$\varepsilon$')
- ax.set_ylabel(r'$f(\varepsilon, T)$')
- ax.set_yticks([0, 1])
- ax.set_yticklabels(['$0$', '$1$'])
- ax.set_xticks([mu])
- ax.set_xticklabels([r'$\mu$'])
- ax.set_ylim(-.1, 1.1)
- ax.legend()
- draw_classic_axes(ax)
- pyplot.tight_layout()
- fig.savefig('fermi_dirac.svg')
-
-
-def phonons_1d():
- k = np.linspace(-2*pi, 6*pi, 500)
- fig, ax = pyplot.subplots()
-
- pyplot.plot(k, np.abs(np.sin(k/2)))
-
- ax.set_ylim(bottom=0, top=1.2)
- ax.set_xlabel('$ka$')
- ax.set_ylabel(r'$\omega$')
- pyplot.xticks(list(2*pi*np.arange(-1, 4)) + [-pi, pi],
- [r'$-2\pi$', '$0$', r'$2\pi$', r'$4\pi$', r'$6\pi$',
- r'$-\pi$', r'$\pi$'])
- pyplot.yticks([1], [r'$2\sqrt{\frac{\kappa}{m}}$'])
- pyplot.vlines([-pi, pi], 0, 1.1, linestyles='dashed')
- pyplot.hlines([1], .1*pi, 1.3*pi, linestyles='dashed')
- draw_classic_axes(ax)
- ax.annotate(s='', xy=(-pi, -.15), xytext=(pi, -.15),
- arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
- ax.text(0, -.25, '1st Brillouin zone', ha='center')
- ax.set_ylim(bottom=-.7)
- pyplot.savefig('phonons3.svg')
-
-
-def aliasing():
- x = np.linspace(-.2, 2.8, 500)
- fig, ax = pyplot.subplots()
- ax.plot(x, np.cos(pi*(x - .3)), label=r'$k=\pi/a$')
- ax.plot(x, np.cos(3*pi*(x - .3)), label=r'$k=3\pi/a$')
- ax.plot(x, np.cos(5*pi*(x - .3)), label=r'$k=5\pi/a$')
- sites = np.arange(3) + .3
- ax.scatter(sites, np.cos(pi*(sites - .3)), c='k', s=64, zorder=5)
- ax.set_xlabel('$x$')
- ax.set_ylabel('$u_n$')
- ax.set_xlim((-.1, 3.2))
- ax.set_ylim((-1.3, 1.3))
- ax.legend(loc='lower right')
- draw_classic_axes(ax)
- ax.annotate(s='', xy=(.3, -1.1), xytext=(1.3, -1.1),
- arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
- ax.text(.3 + .5, -1.25, '$a$', ha='center')
- pyplot.savefig('phonons4.svg')
-
-
-def tight_binding_1d():
- pyplot.figure()
- k = np.linspace(-pi, pi, 300)
- pyplot.plot(k, -np.cos(k))
- pyplot.xlabel('$ka$'); pyplot.ylabel('$E$')
- pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$'])
- pyplot.yticks([-1, 0, 1], ['$E_0+2t$', '$E_0$', '$E_0-2t$'])
- pyplot.savefig('tight_binding_1d.svg')
-
-
-def meff_1d_tb():
- pyplot.figure(figsize=(8, 5))
- k = np.linspace(-pi, pi, 300)
- meff = 1/np.cos(k)
- color = list(matplotlib.rcParams['axes.prop_cycle'])[0]['color']
- pyplot.plot(k[meff > 0], meff[meff > 0], c=color)
- pyplot.plot(k[meff < 0], meff[meff < 0], c=color)
- pyplot.ylim(-5, 5)
- pyplot.xlabel('$ka$'); pyplot.ylabel('$m_{eff}$')
- pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$']);
- pyplot.savefig('meff_1d_tb.svg')
-
-
-def dispersion_2m(k, kappa=1, M=1.4, m=1, acoustic=True):
- Mm = M*m
- m_harm = (M + m) / Mm
- root = kappa * np.sqrt(m_harm**2 - 4*np.sin(k/2)**2 / Mm)
- if acoustic:
- root *= -1
- return np.sqrt(kappa*m_harm + root)
-
-
-def phonons_1d_2masses():
- # TODO: Add panels with eigenvectors
- k = np.linspace(-2*pi, 6*pi, 300)
- fig, ax = pyplot.subplots()
- ax.plot(k, dispersion_2m(k, acoustic=False), label='optical')
- ax.plot(k, dispersion_2m(k), label='acoustic')
- ax.set_xlabel('$ka$')
- ax.set_ylabel(r'$\omega$')
- pyplot.xticks([-pi, 0, pi], [r'$-\pi$', '$0$', r'$\pi$'])
- pyplot.yticks([], [])
- pyplot.vlines([-pi, pi], 0, 2.2, linestyles='dashed')
- pyplot.legend()
- pyplot.xlim(-1.75*pi, 3.5*pi)
- pyplot.ylim(bottom=0)
- draw_classic_axes(ax, xlabeloffset=.2)
- pyplot.savefig('phonons6.svg')
-
-
-def DOS_2masses():
- matplotlib.rcParams['font.size'] = 24
- k = np.linspace(-.25*pi, 1.5*pi, 300)
- k_dos = np.linspace(0, pi, 20)
- fig, (ax, ax2) = pyplot.subplots(ncols=2, sharey=True, figsize=(10, 5))
- ax.plot(k, dispersion_2m(k, acoustic=False), label='optical')
- ax.plot(k, dispersion_2m(k), label='acoustic')
- ax.vlines(k_dos, 0, dispersion_2m(k_dos, acoustic=False),
- colors=(0.5, 0.5, 0.5, 0.5))
- ax.hlines(
- np.hstack((dispersion_2m(k_dos, acoustic=False), dispersion_2m(k_dos))),
- np.hstack((k_dos, k_dos)),
- 1.8*pi,
- colors=(0.5, 0.5, 0.5, 0.5)
- )
- ax.set_xlabel('$ka$')
- ax.set_ylabel(r'$\omega$')
- ax.set_xticks([0, pi])
- ax.set_xticklabels(['$0$', r'$\pi$'])
- ax.set_yticks([], [])
- ax.set_xlim(-pi/4, 2*pi)
- ax.set_ylim((0, dispersion_2m(0, acoustic=False) + .2))
- draw_classic_axes(ax, xlabeloffset=.2)
-
- k = np.linspace(0, pi, 1000)
- omegas = np.hstack((
- dispersion_2m(k, acoustic=False), dispersion_2m(k)
- ))
- ax2.hist(omegas, orientation='horizontal', bins=75)
- ax2.set_xlabel(r'$g(\omega)$')
- ax2.set_ylabel(r'$\omega$')
-
- # Truncate the singularity in the DOS
- max_x = ax2.get_xlim()[1]
- ax2.set_xlim((0, max_x/2))
- draw_classic_axes(ax2, xlabeloffset=.1)
- matplotlib.rcParams['font.size'] = 16
- pyplot.savefig('phonons8.svg')
-
-
-def consistency_1_2_masses():
- k = np.linspace(0, 2*pi, 300)
- k_dos = np.linspace(0, pi, 20)
- fig, ax = pyplot.subplots()
- ax.plot(k, dispersion_2m(k, acoustic=False), label='optical')
- ax.plot(k, dispersion_2m(k), label='acoustic')
- omega_max = dispersion_2m(0, acoustic=False)
- ax.plot(k, omega_max * np.sin(k/4), label='equal masses')
- ax.set_xlabel('$ka$')
- ax.set_ylabel(r'$\omega$')
- ax.set_xticks([0, pi, 2*pi])
- ax.set_xticklabels(['$0$', r'$\pi/2a$', r'$\pi/a$'])
- ax.set_yticks([], [])
- ax.set_xlim(-pi/8, 2*pi+.4)
- ax.set_ylim((0, dispersion_2m(0, acoustic=False) + .2))
- ax.legend(loc='lower right')
- pyplot.vlines([pi, 2*pi], 0, 2.2, linestyles='dashed')
- draw_classic_axes(ax, xlabeloffset=.2)
- pyplot.savefig('phonons7.svg')
-
-
-def DOS_finite_phonon_chain(n, output_name):
- rhs = 2 * np.eye(n) - np.eye(n, k=1) - np.eye(n, k=-1)
- rhs[0, 0] -= 1
- rhs[-1, -1] -= 1
- pyplot.figure()
- pyplot.hist(np.sqrt(np.abs(np.linalg.eigvalsh(rhs))), bins=30)
- pyplot.xlabel("$\omega$")
- pyplot.ylabel("Number of levels")
- pyplot.savefig(output_name)
-
-
-def DOS_finite_electron_chain(n, output_name):
- rhs = - np.eye(n, k=1) - np.eye(n, k=-1)
- pyplot.figure()
- pyplot.hist(np.linalg.eigvalsh(rhs), bins=30)
- pyplot.xlabel("$E$")
- pyplot.ylabel("Number of levels")
- pyplot.savefig(output_name)
-
-
-def main():
- os.chdir('docs/figures')
- fermi_dirac()
- phonons_1d()
- phonons_1d_2masses()
- consistency_1_2_masses()
- DOS_2masses()
- aliasing()
- meff_1d_tb()
- tight_binding_1d()
- DOS_finite_phonon_chain(3, '3_phonons.svg')
- DOS_finite_phonon_chain(300, '300_phonons.svg')
- DOS_finite_electron_chain(3, '3_electrons.svg')
- DOS_finite_electron_chain(300, '300_electrons.svg')
-
-if __name__ == "__main__":
- main()
diff --git a/code/common.py b/code/common.py
index e51a0987..df2d1fb3 100644
--- a/code/common.py
+++ b/code/common.py
@@ -1,22 +1,33 @@
-import os
-
-import numpy as np
+from IPython import get_ipython
import matplotlib
-import mpl_toolkits
-matplotlib.rcParams['text.usetex'] = True
-matplotlib.rcParams['figure.figsize'] = (8, 5)
-matplotlib.rcParams['font.size'] = 16
-matplotlib.rcParams['savefig.transparent'] = True
+
+def configure_plotting():
+ # Configure matplotlib
+ ip = get_ipython()
+ ip.config['InlineBackend']['figure_format'] = 'svg'
+
+ # Fix for https://stackoverflow.com/a/36021869/2217463
+ ip.config['InlineBackend']['rc'] = {}
+ ip.enable_matplotlib(gui='inline')
+
+ matplotlib.rcParams['text.usetex'] = False
+ matplotlib.rcParams['figure.figsize'] = (10, 7)
+ matplotlib.rcParams['font.size'] = 16
+
def draw_classic_axes(ax, x=0, y=0, xlabeloffset=.1, ylabeloffset=.07):
ax.set_axis_off()
x0, x1 = ax.get_xlim()
y0, y1 = ax.get_ylim()
- ax.annotate(ax.get_xlabel(), xytext=(x1, y), xy=(x0, y),
- arrowprops=dict(arrowstyle="<-"), va='center')
- ax.annotate(ax.get_ylabel(), xytext=(x, y1), xy=(x, y0),
- arrowprops=dict(arrowstyle="<-"), ha='center')
+ ax.annotate(
+ ax.get_xlabel(), xytext=(x1, y), xy=(x0, y),
+ arrowprops=dict(arrowstyle="<-"), va='center'
+ )
+ ax.annotate(
+ ax.get_ylabel(), xytext=(x, y1), xy=(x, y0),
+ arrowprops=dict(arrowstyle="<-"), ha='center'
+ )
for pos, label in zip(ax.get_xticks(), ax.get_xticklabels()):
ax.text(pos, y - xlabeloffset, label.get_text(),
ha='center', va='bottom')
diff --git a/code/heat_capacity.py b/code/heat_capacity.py
deleted file mode 100644
index 41282ed3..00000000
--- a/code/heat_capacity.py
+++ /dev/null
@@ -1,106 +0,0 @@
-import os
-
-import matplotlib
-matplotlib.use('agg')
-from matplotlib import pyplot
-
-import numpy as np
-from scipy.optimize import curve_fit
-from scipy.integrate import quad
-
-import common
-from common import draw_classic_axes
-
-
-# Data from Einstein's paper
-T = [222.4, 262.4, 283.7, 306.4, 331.3, 358.5, 413.0, 479.2, 520.0, 879.7, 1079.7, 1258.0],
-c = [0.384, 0.578, 0.683, 0.798, 0.928, 1.069, 1.343, 1.656, 1.833, 2.671, 2.720, 2.781]
-
-def plot_n_e():
- fig, (ax, ax2) = pyplot.subplots(ncols=2, figsize=(10, 5))
- omega = np.linspace(0.1, 2)
- ax.plot(omega, 1/(np.exp(omega) - 1))
- ax.set_ylim(0, top=3)
- ax.set_xlim(left=0)
- ax.set_xlabel(r'$\hbar \omega$')
- ax.set_xticks([0, 1])
- ax.set_xticklabels(['$0$', '$k_B T$'])
- ax.set_ylabel('$n$')
- ax.set_yticks([1, 2])
- ax.set_yticklabels(['$1$', '$2$'])
- draw_classic_axes(ax, xlabeloffset=.2)
- T = np.linspace(0.01, 2)
- ax2.plot(T, 1/2 + 1/(np.exp(1/T)-1))
- ax2.set_ylim(bottom=0)
- ax2.set_xlabel('$k_B T$')
- ax2.set_xticks([0, 1])
- ax2.set_xticklabels(['$0$', r'$\hbar \omega$'])
- ax2.set_ylabel(r"$\bar\varepsilon$")
- ax2.set_yticks([1/2])
- ax2.set_yticklabels([r'$\hbar\omega/2$'])
- draw_classic_axes(ax2, xlabeloffset=.15)
- fig.savefig('bose_einstein.svg')
-
-
-def c_einstein(T, T_E=1):
- x = T_E / T
- return 3 * x**2 * np.exp(x) / (np.exp(x) - 1)**2
-
-
-def integrand(y):
- return y**4 * np.exp(y) / (np.exp(y) - 1)**2
-
-
-@np.vectorize
-def c_debye(T, T_D=1):
- x = T / T_D
- return 9 * x**3 * quad(integrand, 0, 1/x)[0]
-
-
-def plot_einstein():
- T = np.linspace(0.01, 1.5, 500)
- fig, ax = pyplot.subplots()
-
- ax.plot(T, c_einstein(T)/3)
- ax.fill_between(T, c_einstein(T)/3, 1, alpha=0.5)
-
- ax.set_ylim(bottom=0, top=1.2)
- ax.set_xlabel('$T$')
- ax.set_ylabel(r'$\omega$')
- ax.set_xticks([1])
- ax.set_xticklabels([r'$\hbar \omega/k_B$'])
- ax.set_yticks([1])
- ax.set_yticklabels(['$k_B$'])
- pyplot.hlines([1], 0, 1.5, linestyles='dashed')
- draw_classic_axes(ax)
- pyplot.savefig('zeropoint.svg')
-
-
-def plot_debye():
- T = np.linspace(0.01, 1.5, 500)
- fig, ax = pyplot.subplots()
-
- ax.plot(T, c_einstein(T), label="Einstein model")
- ax.plot(T, c_debye(T), label="Debye model")
-
- ax.set_ylim(bottom=0, top=3.4)
- ax.set_xlabel('$T$')
- ax.set_ylabel(r'$\omega$')
- ax.set_xticks([1])
- ax.set_xticklabels([r'$\Theta_D$'])
- ax.set_yticks([3])
- ax.set_yticklabels(['$3Nk_B$'])
- ax.legend(loc='lower right')
- pyplot.hlines([3], 0, 1.5, linestyles='dashed')
- draw_classic_axes(ax, xlabeloffset=0.3)
- pyplot.savefig('debye.svg')
-
-
-def main():
- os.chdir('docs/figures')
- plot_n_e()
- plot_einstein()
- plot_debye()
-
-if __name__ == "__main__":
- main()
diff --git a/code/misc.py b/code/misc.py
deleted file mode 100644
index 28345843..00000000
--- a/code/misc.py
+++ /dev/null
@@ -1,75 +0,0 @@
-import os
-
-import matplotlib
-matplotlib.use('agg')
-from matplotlib import pyplot
-
-import numpy as np
-from scipy.optimize import curve_fit
-from scipy.integrate import quad
-
-import common
-from common import draw_classic_axes
-
-def plot_curie():
- B = np.linspace(-2, 2, 1000)
- fig, ax = pyplot.subplots()
-
- sqrt_plus = lambda x: np.sqrt(x * (x >= 0))
- ax.plot(B, np.tanh(B * 3), label="low $T$")
- ax.plot(B, np.tanh(B), label="high $T$")
-
- ax.legend()
- ax.set_ylabel("$M$")
- ax.set_xlabel("$B$")
- draw_classic_axes(ax, xlabeloffset=.2)
- fig.savefig('curie.svg')
-
-
-def plot_free_electron_DOS():
- E = np.linspace(0, 2, 500)
- fig, ax = pyplot.subplots()
-
- ax.plot(E, np.sqrt(E))
-
- ax.set_ylabel(r"$g(\varepsilon)$")
- ax.set_xlabel(r"$\varepsilon$")
- draw_classic_axes(ax, xlabeloffset=.2)
- fig.savefig('sqrt.svg')
-
-
-def plot_finite_T_FEM():
- E = np.linspace(0, 2, 500)
- fig, ax = pyplot.subplots()
- ax.plot(E, np.sqrt(E), linestyle='dashed')
- ax.text(1.7, 1.4, r'$g(\varepsilon)\propto \sqrt{\varepsilon}$', ha='center')
- ax.fill_between(E, np.sqrt(E) * (E < 1), alpha=.3)
-
- n = np.sqrt(E) / (1 + np.exp(20*(E-1)))
- ax.plot(E, n)
- ax.fill_between(E, n, alpha=.5)
- w = .17
- ax.annotate(s='', xy=(1, 1), xytext=(1-w, 1),
- arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
- ax.text(1-w/2, 1.1, r'$\sim k_BT$', ha='center')
- ax.plot([1-w, 1+w], [1, 0], c='k', linestyle='dashed')
- ax.annotate(s='', xy=(1, 0), xytext=(1, 1),
- arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
- ax.text(1.2, .7, r'$g(\varepsilon_F)$', ha='center')
- ax.set_xticks([1])
- ax.set_xticklabels([r'$\varepsilon_F$'])
-
- ax.set_ylabel(r"$g(\varepsilon)$")
- ax.set_xlabel(r"$\varepsilon$")
- draw_classic_axes(ax, xlabeloffset=.2)
-
- fig.savefig('FD_DOS.svg')
-
-def main():
- os.chdir('docs/figures')
- plot_curie()
- plot_free_electron_DOS()
- plot_finite_T_FEM()
-
-if __name__ == "__main__":
- main()
diff --git a/code/semiconductors.py b/code/semiconductors.py
deleted file mode 100644
index 3d578281..00000000
--- a/code/semiconductors.py
+++ /dev/null
@@ -1,70 +0,0 @@
-import os
-
-import matplotlib
-matplotlib.use('agg')
-from matplotlib import pyplot
-
-import numpy as np
-from scipy.optimize import curve_fit
-from scipy.integrate import quad
-
-import common
-from common import draw_classic_axes
-
-E_V, E_C, E_F = -1.2, 1.8, .4
-E_D, E_A = E_C - .7, E_V + .5
-m_h, m_e = 1, .5
-sqrt_plus = lambda x: np.sqrt(x * (x >= 0))
-
-
-def plot_dos():
- E = np.linspace(-3, 3, 1000)
- fig, ax = pyplot.subplots()
-
- n_F = 1/(np.exp(2*(E - E_F)) + 1)
- g_e = m_e * sqrt_plus(E - E_C)
- g_h = m_h * sqrt_plus(E_V - E)
- ax.plot(E, g_h, label="$g_e$")
- ax.plot(E, g_e, label="$g_h$")
- ax.plot(E, 10 * g_h * (1-n_F), label="$n_h$")
- ax.plot(E, 10 * g_e * n_F, label="$n_e$")
- ax.plot(E, n_F, label="$n_F$", linestyle='dashed')
- ax.set_ylim(top=1.5)
-
- ax.set_xlabel('$E$')
- ax.set_ylabel('$g$')
- ax.set_xticks([E_V, E_C, E_F])
- ax.set_xticklabels(['$E_V$', '$E_C$', '$E_F$'])
- ax.legend()
- draw_classic_axes(ax, xlabeloffset=.2)
- fig.savefig('intrinsic_DOS.svg')
-
-def plot_doping():
- E = np.linspace(-3, 3, 1000)
- fig, ax = pyplot.subplots()
-
- n_F = 1/(np.exp(2*(E - E_F)) + 1)
- g_e = m_e * sqrt_plus(E - E_C)
- g_h = m_h * sqrt_plus(E_V - E)
- ax.plot(E, g_h, label="$g_e$")
- ax.plot(E, g_e, label="$g_h$")
-
- sigma = 0.01
- g_D = np.exp(-(E_D - E)**2 / sigma**2)
- g_A = .7 * np.exp(-(E_A - E)**2 / sigma**2)
- ax.plot(E, g_D, label='$g_D$')
- ax.plot(E, g_A, label='$g_A$')
- ax.legend()
- ax.set_xticks([E_V, E_C, E_A, E_D])
- ax.set_xticklabels(['$E_V$', '$E_C$', '$E_A$', '$E_D$'])
- draw_classic_axes(ax, xlabeloffset=.2)
- fig.savefig('doping.svg')
-
-
-def main():
- os.chdir('docs/figures')
- plot_dos()
- plot_doping()
-
-if __name__ == "__main__":
- main()
diff --git a/docs/mathjaxhelper.js b/docs/mathjaxhelper.js
deleted file mode 100644
index bec7b177..00000000
--- a/docs/mathjaxhelper.js
+++ /dev/null
@@ -1,5 +0,0 @@
-MathJax.Hub.Config({
- config: ["MMLorHTML.js"],
- jax: ["input/TeX", "output/HTML-CSS", "output/NativeMML"],
- extensions: ["MathMenu.js", "MathZoom.js"]
-});
diff --git a/execute.py b/execute.py
new file mode 100644
index 00000000..71c7b23e
--- /dev/null
+++ b/execute.py
@@ -0,0 +1,43 @@
+from pathlib import Path
+import shutil
+
+import nbconvert
+import notedown
+from traitlets.config import Config
+
+reader = notedown.MarkdownReader()
+
+src = Path('src')
+target = Path('docs')
+shutil.rmtree(target, ignore_errors=True)
+target.mkdir(exist_ok=True)
+shutil.copytree(src / 'figures', target / 'figures')
+
+exporter = nbconvert.MarkdownExporter(
+ config=Config(dict(
+ MarkdownExporter=dict(
+ preprocessors=[
+ nbconvert.preprocessors.ExecutePreprocessor,
+ nbconvert.preprocessors.ExtractOutputPreprocessor,
+ ],
+ exclude_input=True
+ )
+ ))
+)
+
+writer = nbconvert.writers.FilesWriter(build_directory=str(target))
+
+for source_file in src.glob('*.md'):
+ fname = source_file.name[:-len('.md')]
+ notebook = reader.reads(source_file.read_text())
+
+ output, resources = exporter.from_notebook_node(
+ notebook,
+ resources={
+ 'unique_key': fname,
+ 'output_files_dir': 'figures',
+ 'metadata': {'path': str(Path('.') / 'code')}
+ }
+ )
+
+ writer.write(output, resources, fname)
diff --git a/mkdocs.yml b/mkdocs.yml
index e55b2c3a..a6b4cc1f 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -13,7 +13,8 @@ pages:
- Crystal structure and diffraction: 'lecture_5.md'
- Tight binding and nearly free electrons: 'lecture_6.md'
- Semiconductors: 'lecture_7.md'
- - Magnetism: 'lecture_8.md'
+ - Attic:
+ - Magnetism: 'lecture_8.md'
theme:
name: material
@@ -22,7 +23,6 @@ theme:
primary: 'white'
accent: 'indigo'
-
markdown_extensions:
- mdx_math:
enable_dollar_delimiter: True
@@ -33,6 +33,5 @@ markdown_extensions:
extra_javascript:
- 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.3/MathJax.js?config=TeX-AMS_HTML'
- - 'mathjaxhelper.js'
copyright: "Copyright © 2017-2018 Delft University of Technology, CC-BY-SA-NC 4.0."
diff --git a/docs/figures/DOS.svg b/src/figures/DOS.svg
similarity index 100%
rename from docs/figures/DOS.svg
rename to src/figures/DOS.svg
diff --git a/docs/figures/DOS_periodic.svg b/src/figures/DOS_periodic.svg
similarity index 100%
rename from docs/figures/DOS_periodic.svg
rename to src/figures/DOS_periodic.svg
diff --git a/docs/figures/E_F_and_carrier_density.svg b/src/figures/E_F_and_carrier_density.svg
similarity index 100%
rename from docs/figures/E_F_and_carrier_density.svg
rename to src/figures/E_F_and_carrier_density.svg
diff --git a/docs/figures/RKKY.svg b/src/figures/RKKY.svg
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rename from docs/figures/RKKY.svg
rename to src/figures/RKKY.svg
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rename from docs/figures/adsorption.svg
rename to src/figures/adsorption.svg
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rename from docs/figures/band_diagram_question.svg
rename to src/figures/band_diagram_question.svg
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similarity index 100%
rename from docs/figures/band_diagram_solution.svg
rename to src/figures/band_diagram_solution.svg
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similarity index 100%
rename from docs/figures/band_structure_sketch.svg
rename to src/figures/band_structure_sketch.svg
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rename from docs/figures/bonding.svg
rename to src/figures/bonding.svg
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rename to src/figures/bonding_with_repulsion.svg
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rename from docs/figures/bragg.svg
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rename from docs/figures/crystalfield.svg
rename to src/figures/crystalfield.svg
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rename from docs/figures/cubic.svg
rename to src/figures/cubic.svg
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rename from docs/figures/cubic_mod.svg
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rename from docs/figures/direct_indirect.svg
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rename from docs/figures/dos_1d_tb.svg
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diff --git a/docs/figures/transport.svg b/src/figures/transport.svg
similarity index 100%
rename from docs/figures/transport.svg
rename to src/figures/transport.svg
diff --git a/docs/figures/two_atoms.svg b/src/figures/two_atoms.svg
similarity index 100%
rename from docs/figures/two_atoms.svg
rename to src/figures/two_atoms.svg
diff --git a/docs/figures/xray.svg b/src/figures/xray.svg
similarity index 100%
rename from docs/figures/xray.svg
rename to src/figures/xray.svg
diff --git a/docs/index.md b/src/index.md
similarity index 100%
rename from docs/index.md
rename to src/index.md
diff --git a/docs/lecture_1.md b/src/lecture_1.md
similarity index 74%
rename from docs/lecture_1.md
rename to src/lecture_1.md
index 8a43009e..babc7992 100644
--- a/docs/lecture_1.md
+++ b/src/lecture_1.md
@@ -1,3 +1,14 @@
+```python
+from matplotlib import pyplot
+
+import numpy as np
+from scipy.optimize import curve_fit
+from scipy.integrate import quad
+
+from common import draw_classic_axes, configure_plotting
+
+configure_plotting()
+```
# Lecture 1 – Phonons and specific Heat
_Based on chapter 2 of the book_
@@ -25,7 +36,31 @@ $$
n(\omega,T)=\frac{1}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}\Rightarrow\bar{\varepsilon}=\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}
$$
-
+```python
+
+fig, (ax, ax2) = pyplot.subplots(ncols=2, figsize=(10, 5))
+omega = np.linspace(0.1, 2)
+ax.plot(omega, 1/(np.exp(omega) - 1))
+ax.set_ylim(0, top=3)
+ax.set_xlim(left=0)
+ax.set_xlabel(r'$\hbar \omega$')
+ax.set_xticks([0, 1])
+ax.set_xticklabels(['$0$', '$k_B T$'])
+ax.set_ylabel('$n$')
+ax.set_yticks([1, 2])
+ax.set_yticklabels(['$1$', '$2$'])
+draw_classic_axes(ax, xlabeloffset=.2)
+T = np.linspace(0.01, 2)
+ax2.plot(T, 1/2 + 1/(np.exp(1/T)-1))
+ax2.set_ylim(bottom=0)
+ax2.set_xlabel('$k_B T$')
+ax2.set_xticks([0, 1])
+ax2.set_xticklabels(['$0$', r'$\hbar \omega$'])
+ax2.set_ylabel(r"$\bar\varepsilon$")
+ax2.set_yticks([1/2])
+ax2.set_yticklabels([r'$\hbar\omega/2$'])
+draw_classic_axes(ax2, xlabeloffset=.15)
+```
The term $\frac{1}{2}\hbar\omega$ is the _zero point energy_, which follows from the uncertainty principle.
@@ -40,7 +75,31 @@ C = \frac{\partial\bar{\varepsilon}}{\partial T}
\end{multline}
$$
-
+```python
+# Data from Einstein's paper
+T = [222.4, 262.4, 283.7, 306.4, 331.3, 358.5, 413.0, 479.2, 520.0, 879.7, 1079.7, 1258.0],
+c = [0.384, 0.578, 0.683, 0.798, 0.928, 1.069, 1.343, 1.656, 1.833, 2.671, 2.720, 2.781]
+
+def c_einstein(T, T_E=1):
+ x = T_E / T
+ return 3 * x**2 * np.exp(x) / (np.exp(x) - 1)**2
+
+T = np.linspace(0.01, 1.5, 500)
+fig, ax = pyplot.subplots()
+
+ax.plot(T, c_einstein(T)/3)
+ax.fill_between(T, c_einstein(T)/3, 1, alpha=0.5)
+
+ax.set_ylim(bottom=0, top=1.2)
+ax.set_xlabel('$T$')
+ax.set_ylabel(r'$\omega$')
+ax.set_xticks([1])
+ax.set_xticklabels([r'$\hbar \omega/k_B$'])
+ax.set_yticks([1])
+ax.set_yticklabels(['$k_B$'])
+pyplot.hlines([1], 0, 1.5, linestyles='dashed')
+draw_classic_axes(ax)
+```
The dashed line signifies the classical value, $k_{\rm B}$. Shaded area $=\frac{1}{2}\hbar\omega$, the zero point energy that cannot be removed through cooling.
@@ -126,4 +185,29 @@ $$
where $x=\frac{\hbar\omega}{k_{\rm B}T}$ and $\Theta_{\rm D}\equiv\frac{\hbar\omega_{\rm D}}{k_{\rm B}}$, the _Debye temperature_.
-
+```python
+def integrand(y):
+ return y**4 * np.exp(y) / (np.exp(y) - 1)**2
+
+@np.vectorize
+def c_debye(T, T_D=1):
+ x = T / T_D
+ return 9 * x**3 * quad(integrand, 0, 1/x)[0]
+
+T = np.linspace(0.01, 1.5, 500)
+fig, ax = pyplot.subplots()
+
+ax.plot(T, c_einstein(T), label="Einstein model")
+ax.plot(T, c_debye(T), label="Debye model")
+
+ax.set_ylim(bottom=0, top=3.4)
+ax.set_xlabel('$T$')
+ax.set_ylabel(r'$\omega$')
+ax.set_xticks([1])
+ax.set_xticklabels([r'$\Theta_D$'])
+ax.set_yticks([3])
+ax.set_yticklabels(['$3Nk_B$'])
+ax.legend(loc='lower right')
+pyplot.hlines([3], 0, 1.5, linestyles='dashed')
+draw_classic_axes(ax, xlabeloffset=0.3)
+```
diff --git a/docs/lecture_2.md b/src/lecture_2.md
similarity index 84%
rename from docs/lecture_2.md
rename to src/lecture_2.md
index 8af060a5..e4d0bc16 100644
--- a/docs/lecture_2.md
+++ b/src/lecture_2.md
@@ -1,3 +1,12 @@
+```python
+from matplotlib import pyplot
+
+import numpy as np
+
+from common import draw_classic_axes, configure_plotting
+
+configure_plotting()
+```
# Lecture 2 – Free electron model
_Based on chapters 3 and 4 of the book_
@@ -51,7 +60,7 @@ How fast do electrons travel through a copper wire? Let's take $E$ = 1 volt/m, $
$\rightarrow v=\mu E=\frac{e\tau}{m_{\rm e}}E=\frac{10^{-19}\times 2.5\times 10^{-14}}{10^{-30}}=2.5\times10^{-3}=2.5$ mm/s ! (= 50 $\mu$m @ 50 Hz AC)
### Hall effect
-Consider a conductive wire in a magnetic field ${\bf B} \rightarrow$ electrons are deflected in a direction perpendicular to ${\bf B}$ and ${\bf j}$.
+Consider a conductive wire in a magnetic field ${\bf B} \rightarrow$ electrons are deflected in a direction perpendicular to ${\bf B}$ and ${\bf j}$.
@@ -94,7 +103,7 @@ Comparable to phonons, but: electrons are _fermions_.
- Only 2 (due to spin) allowed per $k$-value
- Fill up from the lowest energy until you run out of electrons
-
+
$\rightarrow$ Calculate when you are out of electrons $\rightarrow$ _Fermi energy_.
@@ -117,7 +126,16 @@ $$
g(\varepsilon)=\frac{ {\rm d}N}{ {\rm d}\varepsilon}=\frac{Vm^{3/2}\sqrt{2\varepsilon}}{\pi^2\hbar^3}\propto\sqrt{\varepsilon}
$$
-
+```python
+E = np.linspace(0, 2, 500)
+fig, ax = pyplot.subplots()
+
+ax.plot(E, np.sqrt(E))
+
+ax.set_ylabel(r"$g(\varepsilon)$")
+ax.set_xlabel(r"$\varepsilon$")
+draw_classic_axes(ax, xlabeloffset=.2)
+```
Similarly,
@@ -133,7 +151,7 @@ $$
with $\varepsilon_{\rm F}$ the _Fermi energy_ = highest filled energy at $T=0$.
$$
-\varepsilon_{\rm F}=\frac{\hbar^2}{2m}\left( 3\pi^2\frac{N}{V} \right)^{2/3}\equiv \frac{\hbar^2 k_{\rm F}^2}{2m},\
+\varepsilon_{\rm F}=\frac{\hbar^2}{2m}\left( 3\pi^2\frac{N}{V} \right)^{2/3}\equiv \frac{\hbar^2 k_{\rm F}^2}{2m},\
k_{\rm F}=\left( 3\pi^2\frac{N}{V} \right)^{1/3}
$$
@@ -155,7 +173,27 @@ $$
f(\varepsilon,T)=\frac{1}{ {\rm e}^{(\varepsilon-\mu)/k_{\rm B}T}+1}
$$
-
+```python
+fig = pyplot.figure()
+ax = fig.add_subplot(1,1,1)
+xvals = np.linspace(0, 2, 200)
+mu = .75
+beta = 20
+ax.plot(xvals, xvals < mu, ls='dashed', label='$T=0$')
+ax.plot(xvals, 1/(np.exp(beta * (xvals-mu)) + 1),
+ ls='solid', label='$T>0$')
+ax.set_xlabel(r'$\varepsilon$')
+ax.set_ylabel(r'$f(\varepsilon, T)$')
+ax.set_yticks([0, 1])
+ax.set_yticklabels(['$0$', '$1$'])
+ax.set_xticks([mu])
+ax.set_xticklabels([r'$\mu$'])
+ax.set_ylim(-.1, 1.1)
+ax.legend()
+draw_classic_axes(ax)
+pyplot.tight_layout()
+```
+
Chemical potential $\mu=\varepsilon_{\rm F}$ if $T=0$. Typically $\varepsilon_{\rm F}/k_{\rm B}$~70 000 K (~7 eV), whereas room temperature is only 300 K (~30 meV) $\rightarrow$ thermal smearing occurs only very close to Fermi surface.
@@ -167,7 +205,31 @@ $$
We can use this to calculate the electronic contribution to the heat capacity.
-
+```python
+E = np.linspace(0, 2, 500)
+fig, ax = pyplot.subplots()
+ax.plot(E, np.sqrt(E), linestyle='dashed')
+ax.text(1.7, 1.4, r'$g(\varepsilon)\propto \sqrt{\varepsilon}$', ha='center')
+ax.fill_between(E, np.sqrt(E) * (E < 1), alpha=.3)
+
+n = np.sqrt(E) / (1 + np.exp(20*(E-1)))
+ax.plot(E, n)
+ax.fill_between(E, n, alpha=.5)
+w = .17
+ax.annotate(s='', xy=(1, 1), xytext=(1-w, 1),
+ arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+ax.text(1-w/2, 1.1, r'$\sim k_BT$', ha='center')
+ax.plot([1-w, 1+w], [1, 0], c='k', linestyle='dashed')
+ax.annotate(s='', xy=(1, 0), xytext=(1, 1),
+ arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+ax.text(1.2, .7, r'$g(\varepsilon_F)$', ha='center')
+ax.set_xticks([1])
+ax.set_xticklabels([r'$\varepsilon_F$'])
+
+ax.set_ylabel(r"$g(\varepsilon)$")
+ax.set_xlabel(r"$\varepsilon$")
+draw_classic_axes(ax, xlabeloffset=.2)
+```
Electrons in the top triangle are being excited to the bottom triangle due to temperature increase. Number of excited electrons $\approx\frac{1}{2}g(\varepsilon_{\rm F})k_{\rm B}T=n_{\rm exc}$. Total extra energy $E(T)-E(0)=n_{\rm exc}k_{\rm B}T=\frac{1}{2}g(\varepsilon_{\rm F})k_{\rm B}^2T^2$.
diff --git a/docs/lecture_3.md b/src/lecture_3.md
similarity index 92%
rename from docs/lecture_3.md
rename to src/lecture_3.md
index f0c7645d..c8cbb70c 100644
--- a/docs/lecture_3.md
+++ b/src/lecture_3.md
@@ -1,3 +1,14 @@
+```python
+from matplotlib import pyplot
+
+import numpy as np
+
+from common import draw_classic_axes, configure_plotting
+
+configure_plotting()
+
+pi = np.pi
+```
# Atoms and bonds
@@ -246,7 +257,7 @@ Same as a single covalent bond, only more atoms in a line. Considering 3 atoms:
$$
E \begin{pmatrix}
c_1 \\ c_2 \\ c_3
-\end{pmatrix} =
+\end{pmatrix} =
\begin{pmatrix}
E_0 & t & 0 \\
t & E_0 & t \\
@@ -263,22 +274,45 @@ Diagonalizing large matrices is unwieldy, but let's try and check it numerically
Eigenfrequencies of 3 atoms: `[0.0 1.0 1.732050]`
-
+```python
+def DOS_finite_phonon_chain(n):
+ rhs = 2 * np.eye(n) - np.eye(n, k=1) - np.eye(n, k=-1)
+ rhs[0, 0] -= 1
+ rhs[-1, -1] -= 1
+ pyplot.figure()
+ pyplot.hist(np.sqrt(np.abs(np.linalg.eigvalsh(rhs))), bins=30)
+ pyplot.xlabel("$\omega$")
+ pyplot.ylabel("Number of levels")
+
+DOS_finite_phonon_chain(3)
+```
Energies of 3 orbitals: `[-1.41421356 0.0 1.41421356]`
-
+```python
+def DOS_finite_electron_chain(n):
+ rhs = - np.eye(n, k=1) - np.eye(n, k=-1)
+ pyplot.figure()
+ pyplot.hist(np.linalg.eigvalsh(rhs), bins=30)
+ pyplot.xlabel("$E$")
+ pyplot.ylabel("Number of levels")
+
+DOS_finite_electron_chain(3)
+```
### From 3 atoms to 300
Frequencies of many phonons:
-
+```python
+DOS_finite_phonon_chain(300)
+```
Energies of many electrons:
-
-
+```python
+DOS_finite_electron_chain(300)
+```
## Summary
* Electrons in atoms occupy "shells" in the energetically favorable order
diff --git a/docs/lecture_4.md b/src/lecture_4.md
similarity index 68%
rename from docs/lecture_4.md
rename to src/lecture_4.md
index a93448b2..55be75d7 100644
--- a/docs/lecture_4.md
+++ b/src/lecture_4.md
@@ -1,3 +1,16 @@
+```python
+import matplotlib
+from matplotlib import pyplot
+
+import numpy as np
+
+from common import draw_classic_axes, configure_plotting
+
+configure_plotting()
+
+pi = np.pi
+```
+
# Electrons and phonons in 1D
Last lecture:
@@ -10,7 +23,7 @@ This lecture:
* Infinitely many atoms
* Main idea: use *periodicity* in space, similar to periodicity in time
-## Equations of motion
+## Equations of motion
### Phonons
@@ -53,7 +66,24 @@ In that case $c_n/c_0 = \exp(2 \pi n p a/L) = \exp(2 \pi n p/N)$, and changing $
We can see that the solutions with different $k$ (or different $p$) are identical by plotting them:
-
+```python
+x = np.linspace(-.2, 2.8, 500)
+fig, ax = pyplot.subplots()
+ax.plot(x, np.cos(pi*(x - .3)), label=r'$k=\pi/a$')
+ax.plot(x, np.cos(3*pi*(x - .3)), label=r'$k=3\pi/a$')
+ax.plot(x, np.cos(5*pi*(x - .3)), label=r'$k=5\pi/a$')
+sites = np.arange(3) + .3
+ax.scatter(sites, np.cos(pi*(sites - .3)), c='k', s=64, zorder=5)
+ax.set_xlabel('$x$')
+ax.set_ylabel('$u_n$')
+ax.set_xlim((-.1, 3.2))
+ax.set_ylim((-1.3, 1.3))
+ax.legend(loc='lower right')
+draw_classic_axes(ax)
+ax.annotate(s='', xy=(.3, -1.1), xytext=(1.3, -1.1),
+ arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+ax.text(.3 + .5, -1.25, '$a$', ha='center');
+```
Let's count how many different solutions we expect to find. We have a system with $N$ degrees of freedom (either $u_n$ or $c_n$), and therefore we need $N$ normal modes (or eigenstates).
@@ -73,7 +103,27 @@ $$\omega = \sqrt{\frac{2\kappa}{m}}\sqrt{1-\cos(ka)} = 2\sqrt{\frac{\kappa}{m}}|
So we arrive to the dispersion relation
-
+```python
+k = np.linspace(-2*pi, 6*pi, 500)
+fig, ax = pyplot.subplots()
+
+pyplot.plot(k, np.abs(np.sin(k/2)))
+
+ax.set_ylim(bottom=0, top=1.2)
+ax.set_xlabel('$ka$')
+ax.set_ylabel(r'$\omega$')
+pyplot.xticks(list(2*pi*np.arange(-1, 4)) + [-pi, pi],
+ [r'$-2\pi$', '$0$', r'$2\pi$', r'$4\pi$', r'$6\pi$',
+ r'$-\pi$', r'$\pi$'])
+pyplot.yticks([1], [r'$2\sqrt{\frac{\kappa}{m}}$'])
+pyplot.vlines([-pi, pi], 0, 1.1, linestyles='dashed')
+pyplot.hlines([1], .1*pi, 1.3*pi, linestyles='dashed')
+draw_classic_axes(ax)
+ax.annotate(s='', xy=(-pi, -.15), xytext=(pi, -.15),
+ arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+ax.text(0, -.25, '1st Brillouin zone', ha='center')
+ax.set_ylim(bottom=-.7);
+```
Here $k_p = 2\pi p / L$ for $0 ≤ p < N$ are *all the normal modes* of the crystal, and therefore we can rederive a better Debye model!
@@ -96,7 +146,14 @@ E = E_0 + 2t\cos(ka),
$$
so we come to the dispersion relation:
-
+```python
+pyplot.figure()
+k = np.linspace(-pi, pi, 300)
+pyplot.plot(k, -np.cos(k))
+pyplot.xlabel('$ka$'); pyplot.ylabel('$E$')
+pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$'])
+pyplot.yticks([-1, 0, 1], ['$E_0+2t$', '$E_0$', '$E_0-2t$']);
+```
Usually electron dispersion has multiple options for $E(k)$, each called a *band*. The complete dispersion relation is also called a *band structure*.
@@ -147,8 +204,17 @@ Substituting, we get
$$m_{eff} = \hbar^2\left(\frac{d^2 E(k)}{dk^2}\right)^{-1}$$
-
-
+```python
+pyplot.figure(figsize=(8, 5))
+k = np.linspace(-pi, pi, 300)
+meff = 1/np.cos(k)
+color = list(matplotlib.rcParams['axes.prop_cycle'])[0]['color']
+pyplot.plot(k[meff > 0], meff[meff > 0], c=color)
+pyplot.plot(k[meff < 0], meff[meff < 0], c=color)
+pyplot.ylim(-5, 5)
+pyplot.xlabel('$ka$'); pyplot.ylabel('$m_{eff}$')
+pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$']);
+```
### density of states
@@ -251,7 +317,30 @@ $$\omega^2=\frac{\kappa(M+m)}{Mm}\pm \kappa\left\{\left(\frac{M+m}{Mm}\right)^2-
Looking at the eigenvectors we see that for every $k$ there are now two values of $\omega$: one corresponding to in-phase motion (–) and anti-phase (+).
-
+```python
+def dispersion_2m(k, kappa=1, M=1.4, m=1, acoustic=True):
+ Mm = M*m
+ m_harm = (M + m) / Mm
+ root = kappa * np.sqrt(m_harm**2 - 4*np.sin(k/2)**2 / Mm)
+ if acoustic:
+ root *= -1
+ return np.sqrt(kappa*m_harm + root)
+
+# TODO: Add panels with eigenvectors
+k = np.linspace(-2*pi, 6*pi, 300)
+fig, ax = pyplot.subplots()
+ax.plot(k, dispersion_2m(k, acoustic=False), label='optical')
+ax.plot(k, dispersion_2m(k), label='acoustic')
+ax.set_xlabel('$ka$')
+ax.set_ylabel(r'$\omega$')
+pyplot.xticks([-pi, 0, pi], [r'$-\pi$', '$0$', r'$\pi$'])
+pyplot.yticks([], [])
+pyplot.vlines([-pi, pi], 0, 2.2, linestyles='dashed')
+pyplot.legend()
+pyplot.xlim(-1.75*pi, 3.5*pi)
+pyplot.ylim(bottom=0)
+draw_classic_axes(ax, xlabeloffset=.2)
+```
'A' is called the _acoustic branch_, 'B' is called the _optical branch_.
@@ -261,9 +350,46 @@ The quantity $\frac{ {\rm d}k}{ {\rm d}\omega}$ can be calculated from the dispe
When plotted, the DOS may look something like this:
-
-
-Note that normally $g(\omega)$ is plotted along the vertical axis and $\omega$ along the horizontal axis – the plot above is just to demonstrate the relation between the dispersion and the DOS. The singularities in $g(\omega)$ at the bottom and top of each branch are called _van Hove singularities_.
+```python
+matplotlib.rcParams['font.size'] = 24
+k = np.linspace(-.25*pi, 1.5*pi, 300)
+k_dos = np.linspace(0, pi, 20)
+fig, (ax, ax2) = pyplot.subplots(ncols=2, sharey=True, figsize=(10, 5))
+ax.plot(k, dispersion_2m(k, acoustic=False), label='optical')
+ax.plot(k, dispersion_2m(k), label='acoustic')
+ax.vlines(k_dos, 0, dispersion_2m(k_dos, acoustic=False),
+ colors=(0.5, 0.5, 0.5, 0.5))
+ax.hlines(
+ np.hstack((dispersion_2m(k_dos, acoustic=False), dispersion_2m(k_dos))),
+ np.hstack((k_dos, k_dos)),
+ 1.8*pi,
+ colors=(0.5, 0.5, 0.5, 0.5)
+)
+ax.set_xlabel('$ka$')
+ax.set_ylabel(r'$\omega$')
+ax.set_xticks([0, pi])
+ax.set_xticklabels(['$0$', r'$\pi$'])
+ax.set_yticks([], [])
+ax.set_xlim(-pi/4, 2*pi)
+ax.set_ylim((0, dispersion_2m(0, acoustic=False) + .2))
+draw_classic_axes(ax, xlabeloffset=.2)
+
+k = np.linspace(0, pi, 1000)
+omegas = np.hstack((
+ dispersion_2m(k, acoustic=False), dispersion_2m(k)
+))
+ax2.hist(omegas, orientation='horizontal', bins=75)
+ax2.set_xlabel(r'$g(\omega)$')
+ax2.set_ylabel(r'$\omega$')
+
+# Truncate the singularity in the DOS
+max_x = ax2.get_xlim()[1]
+ax2.set_xlim((0, max_x/2))
+draw_classic_axes(ax2, xlabeloffset=.1)
+matplotlib.rcParams['font.size'] = 16
+```
+
+Note that normally $g(\omega)$ is plotted along the vertical axis and $\omega$ along the horizontal axis – the plot above is just to demonstrate the relation between the dispersion and the DOS. The singularities in $g(\omega)$ at the bottom and top of each branch are called _van Hove singularities_.
### Consistency check with 1 atom per cell
@@ -275,7 +401,25 @@ For $M\rightarrow m$, the dispersion diagram should come back to the diagram obt
To reconsile the two pictures let's plot two unit cells in the reciprocal space. (Definition: each of these unit cells is called a **Brillouin zone**, a concept that will come up later.)
-
+```python
+k = np.linspace(0, 2*pi, 300)
+k_dos = np.linspace(0, pi, 20)
+fig, ax = pyplot.subplots()
+ax.plot(k, dispersion_2m(k, acoustic=False), label='optical')
+ax.plot(k, dispersion_2m(k), label='acoustic')
+omega_max = dispersion_2m(0, acoustic=False)
+ax.plot(k, omega_max * np.sin(k/4), label='equal masses')
+ax.set_xlabel('$ka$')
+ax.set_ylabel(r'$\omega$')
+ax.set_xticks([0, pi, 2*pi])
+ax.set_xticklabels(['$0$', r'$\pi/2a$', r'$\pi/a$'])
+ax.set_yticks([], [])
+ax.set_xlim(-pi/8, 2*pi+.4)
+ax.set_ylim((0, dispersion_2m(0, acoustic=False) + .2))
+ax.legend(loc='lower right')
+pyplot.vlines([pi, 2*pi], 0, 2.2, linestyles='dashed')
+draw_classic_axes(ax, xlabeloffset=.2)
+```
Doubling the lattice constant "folds" the band structure on itself, doubling all the bands.
@@ -289,7 +433,7 @@ $\rightarrow \rho_{\rm S}(k)=$ (no. of $k$-values / unit of $k$) $=\frac{L}{\pi}
For open boundary conditions $\rightarrow$ running waves $\rightarrow k=$ $...\frac{-4\pi}{L},\frac{-2\pi}{L},0,\frac{2\pi}{L},\frac{4\pi}{L}...$
-$\rightarrow \rho_{\rm R}(k)=\frac{L}{2\pi}$, which is lower than for the case of closed boundary conditions, however, in this case the entire $k$-space is filled.
+$\rightarrow \rho_{\rm R}(k)=\frac{L}{2\pi}$, which is lower than for the case of closed boundary conditions, however, in this case the entire $k$-space is filled.
## Summary
diff --git a/docs/lecture_5.md b/src/lecture_5.md
similarity index 100%
rename from docs/lecture_5.md
rename to src/lecture_5.md
diff --git a/docs/lecture_6.md b/src/lecture_6.md
similarity index 100%
rename from docs/lecture_6.md
rename to src/lecture_6.md
diff --git a/docs/lecture_7.md b/src/lecture_7.md
similarity index 91%
rename from docs/lecture_7.md
rename to src/lecture_7.md
index f4ab67aa..d7949cfd 100644
--- a/docs/lecture_7.md
+++ b/src/lecture_7.md
@@ -1,3 +1,23 @@
+```python
+from matplotlib import pyplot
+
+import numpy as np
+from scipy.optimize import curve_fit
+from scipy.integrate import quad
+
+from common import draw_classic_axes, configure_plotting
+
+configure_plotting()
+
+def sqrt_plus(x):
+ return np.sqrt(x * (x >= 0))
+
+# Band structure parameters.
+E_V, E_C, E_F = -1.2, 1.8, .4
+E_D, E_A = E_C - .7, E_V + .5
+m_h, m_e = 1, .5
+```
+
# Semiconductor physics
## Review of band structure properties
@@ -102,7 +122,27 @@ Observe that because we are describing particles in the valence band as holes, $
### Part 1: pristine semiconductor
-
+```python
+E = np.linspace(-3, 3, 1000)
+fig, ax = pyplot.subplots()
+
+n_F = 1/(np.exp(2*(E - E_F)) + 1)
+g_e = m_e * sqrt_plus(E - E_C)
+g_h = m_h * sqrt_plus(E_V - E)
+ax.plot(E, g_h, label="$g_e$")
+ax.plot(E, g_e, label="$g_h$")
+ax.plot(E, 10 * g_h * (1-n_F), label="$n_h$")
+ax.plot(E, 10 * g_e * n_F, label="$n_e$")
+ax.plot(E, n_F, label="$n_F$", linestyle='dashed')
+ax.set_ylim(top=1.5)
+
+ax.set_xlabel('$E$')
+ax.set_ylabel('$g$')
+ax.set_xticks([E_V, E_C, E_F])
+ax.set_xticklabels(['$E_V$', '$E_C$', '$E_F$'])
+ax.legend()
+draw_classic_axes(ax, xlabeloffset=.2)
+```
Electrons
$$ E = E_G + {p^2}/{2m_e}$$
@@ -198,7 +238,26 @@ Likewise an acceptor adds an extra state at $E_A$ (close to the top of the valen
### Density of states with donors and acceptors
-
+```python
+E = np.linspace(-3, 3, 1000)
+fig, ax = pyplot.subplots()
+
+n_F = 1/(np.exp(2*(E - E_F)) + 1)
+g_e = m_e * sqrt_plus(E - E_C)
+g_h = m_h * sqrt_plus(E_V - E)
+ax.plot(E, g_h, label="$g_e$")
+ax.plot(E, g_e, label="$g_h$")
+
+sigma = 0.01
+g_D = np.exp(-(E_D - E)**2 / sigma**2)
+g_A = .7 * np.exp(-(E_A - E)**2 / sigma**2)
+ax.plot(E, g_D, label='$g_D$')
+ax.plot(E, g_A, label='$g_A$')
+ax.legend()
+ax.set_xticks([E_V, E_C, E_A, E_D])
+ax.set_xticklabels(['$E_V$', '$E_C$', '$E_A$', '$E_D$'])
+draw_classic_axes(ax, xlabeloffset=.2)
+```
All donor/acceptor states at the same energy:
$$g_A(E) = N_A \delta(E-E_A),\quad g_D(E) = N_D \delta(E- (E_G - E_D))$$
@@ -336,7 +395,9 @@ See the book for details.
Density of states in a doped semiconductor:
-
+```python
+fig
+```
Charge balance determins the number of electrons and holes as well as the position of the Fermi level.
diff --git a/docs/lecture_8.md b/src/lecture_8.md
similarity index 95%
rename from docs/lecture_8.md
rename to src/lecture_8.md
index 894df24b..9fda813c 100644
--- a/docs/lecture_8.md
+++ b/src/lecture_8.md
@@ -1,3 +1,15 @@
+```python
+from matplotlib import pyplot
+
+import numpy as np
+from scipy.optimize import curve_fit
+from scipy.integrate import quad
+
+from common import draw_classic_axes, configure_plotting
+
+configure_plotting()
+```
+
# Lecture 8 – Magnetism
_Based on chapters 19 and 20 of the book_
@@ -81,7 +93,18 @@ $$
with $C$ a constant. This is known as the _Curie law_, and it shows that at high temperature a paramagnetic material becomes less susceptible to magnetisation.
-
+```python
+B = np.linspace(-2, 2, 1000)
+fig, ax = pyplot.subplots()
+
+ax.plot(B, np.tanh(B * 3), label="low $T$")
+ax.plot(B, np.tanh(B), label="high $T$")
+
+ax.legend()
+ax.set_ylabel("$M$")
+ax.set_xlabel("$B$")
+draw_classic_axes(ax, xlabeloffset=.2)
+```
### Paramagnetism: free spin $J$
For an atom with magnetic moments $L$ and $S$ the Hamiltonian will become:
@@ -193,4 +216,4 @@ $$
{\mathcal H}=\sum_i \left( -J \sigma_i \sigma_{i+1}+ g\mu_{\rm B}B\sigma_i\right)
$$
-where $\sigma_i=\pm S$.
\ No newline at end of file
+where $\sigma_i=\pm S$.
--
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