diff --git a/execute.py b/execute.py
index ba5559f437c3f2c7d8c710e528d3f804f8e0dfa9..ad07dfce31da864144d3821e6827538fb71b4a3c 100644
--- a/execute.py
+++ b/execute.py
@@ -9,7 +9,7 @@ from traitlets.config import Config
from nbconvert_fix import ExtractOutputPreprocessor
-reader = notedown.MarkdownReader()
+reader = notedown.MarkdownReader(code_regex='fenced')
mimetypes.add_type('application/vnd.plotly.v1+json', '.json')
diff --git a/mkdocs.yml b/mkdocs.yml
index bdf45e6f46c064914789fbf93d250287915fb35c..86ca71eb648ecba5f158976d17d22e740b3b79d0 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -31,6 +31,7 @@ markdown_extensions:
permalink: True
- admonition
- pymdownx.details
+ - abbr
extra_javascript:
- 'https://cdn.plot.ly/plotly-latest.min.js'
diff --git a/src/lecture_1.md b/src/lecture_1.md
index 7395af369de673596b48d2122fa47427ba2a3001..9c49766fcd2d5760e20aeaee37e9ef4525195ef6 100644
--- a/src/lecture_1.md
+++ b/src/lecture_1.md
@@ -14,14 +14,13 @@ configure_plotting()
_(based on chapter 2 of the book)_
Exercises: 2.3, 2.4, 2.5, 2.6, 2.8
-In this lecture we will:
+!!! summary "Learning goals"
-- discuss specific heat of a solid based on atomic vibrations (_phonons_)
-- disregard periodic lattice $\Rightarrow$ consider homogeneous medium
- - _(chapter 9: discuss phonons in terms of atomic masses and springs)_
-- discuss the Einstein model
-- discuss the Debye model
-- introduce reciprocal space, periodic boundary conditions and _density of states_
+ After this lecture you will be able to:
+
+ - Explain quantum mechanical effects on the heat capacity of solids (Einstein model)
+ - Compute the expected particle number, energy, and heat capacity of a quantum harmonic oscillator (a single boson)
+ - Write down the total thermal energy of a material
### Einstein model
Before solid state physics: heat capacity per atom $C=3k_{\rm B}$ (Dulong-Petit). Each atom is (classical) harmonic oscillator in three directions. Experiments showed that this law breaks down at low temperatures, where $C$ reduces to zero ($C\propto T^3$).
@@ -128,6 +127,13 @@ $$E=\int\limits_0^\infty\left(\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}
$g(\omega)$ is the _density of states_: the number of normal modes found at each position along the $\omega$-axis. How do we calculate $g(\omega)$?
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - Describe the concept of reciprocal space and allowed momenta
+ - Write down the total energy of phonons given the temperature and the dispersion relation
+ - Estimate heat capacity due to phonons in high temperature and low temperature regimes of the Debye model
#### Reciprocal space, periodic boundary conditions
Each normal mode can be described by a _wave vector_ ${\bf k}$. A wave vector represents a point in _reciprocal space_ or _k-space_. We can find $g(\omega)$ by counting the number of normal modes in k-space and then converting those to $\omega$.
diff --git a/src/lecture_2.md b/src/lecture_2.md
index 19fcad125cb3cf54bd2a29f5e0ff924c93c15b14..056e6163aec7d3846ded733978d71ad15430a2ae 100644
--- a/src/lecture_2.md
+++ b/src/lecture_2.md
@@ -8,17 +8,17 @@ from common import draw_classic_axes, configure_plotting
configure_plotting()
```
-# Lecture 2 – Free electron model
+# Lectures 2A & 2B – Drude theory and the free electron model
_(based on chapters 3–4 of the book)_
Exercises 3.1, 3.3, 4.2, 4.5, 4.6, 4.7
-In this lecture we will:
+!!! summary "Learning goals"
-- consider electrons as charged point particles travelling through a solid
-- discuss Drude theory
-- discuss the Hall experiment
-- discuss specific heat of a solid based due to electrons
-- introduce mobility, Hall resistance and the Fermi energy
+ After this lecture you will be able to:
+
+ - discuss the basics of 'Drude theory', which describes electron motion in metals.
+ - use Drude theory to analyze transport of electrons through conductors in electric and magnetic fields.
+ - describe central terms such as the mobility and the Hall resistance.
### Drude theory
Ohm's law states that $V=IR=I\rho\frac{l}{A}$. In this lecture we will investigate where this law comes from. We will use the theory developed by Paul Drude in 1900, which is based on three assumptions:
@@ -86,6 +86,18 @@ where $R_{\rm H}=-\frac{1}{ne}$ is the _Hall resistance_. So by measuring the Ha
While most materials have $R_{\rm H}>0$, interestingly some materials are found to have $R_{\rm H}<0$. This would imply that the charge carriers either have a positive charge, or a negative mass. We will see later (chapter 17) how to interpret this.
### Sommerfeld theory (free electron model)
+
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - calculate the electron density of states in 1D, 2D, and 3D using the Sommerfeld free-electron model.
+ - express the number and energy of particles in a system in terms of integrals over k-space.
+ - use the Fermi distribution to extend the previous learning goal to finite T.
+ - calculate the electron contribution to the specific heat of a solid.
+ - describe central terms such as the Fermi energy, Fermi temperature, and Fermi wavevector.
+
+
Atoms in a metal provide conduction electrons from their outer shells (often s-shells). These can be described as waves in the crystal, analogous to phonons. Hamiltonian of a free electron:
$$
diff --git a/src/lecture_3.md b/src/lecture_3.md
index 41863a489598c86118dd7205521cbfca85da4088..b5a913c5dd0c9a7652a821fc7b42a04b1824a0cb 100644
--- a/src/lecture_3.md
+++ b/src/lecture_3.md
@@ -14,6 +14,14 @@ pi = np.pi
_(based on chapters 5–8 of the book)_
Exercises 5.2, 6.2, 6.3, 6.4, 6.5
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - Describe the shell-filling model of atoms
+ - Derive the LCAO model
+ - Obtain the spectrum of the LCAO model of several orbitals
+
## Looking back
So far we have:
@@ -133,6 +141,16 @@ An occupied $\psi_-$ increases the molecule energy as the atoms move closer ⇒
Therefore if each atom has a single electron in the outermost shell, these atoms attract, if there are 0 or 2 electrons, the net electron force cancels (but electrostatic repulsion remains).
+*[LCAO]: Linear Combination of Atomic Orbitals
+
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - Explain the origins of interatomic forces
+ - Compute vibrational spectra of small molecules in 1D
+ - Formulate Hamiltonians and equations of motion of bulk materials (but not yet solve them)
+
## Adding repulsion
If bringing two atoms closer would keep increasing energy, any covalent bond would collapse; eventually the two atoms must start repelling (at least when the nuclei get close, but really already much earlier).
diff --git a/src/lecture_4.md b/src/lecture_4.md
index 0f6550c65a5d1941dacc94c23640cec1223a0b92..9728744cee72857b66283ba753205e097619cec5 100644
--- a/src/lecture_4.md
+++ b/src/lecture_4.md
@@ -15,6 +15,14 @@ pi = np.pi
_(based on chapters 9–11 of the book)_
Exercises 9.2, 9.4, 10.1, 10.2, 11.2, 11.5
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - formulate equations of motion for electrons and phonons in 1D.
+ - solve these equations to arrive at the corresponding dispersion relations.
+ - derive the group velocity, effective mass, and density of states from the dispersion relation.
+
Last lecture:
* Movement of a few atoms (Newton's equations)
@@ -245,6 +253,15 @@ Also a sanity check: when the energy is close to the bottom of the band, $E = E_
## More complex systems
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - formulate equations of motion for electrons or phonons in 1D for systems with more than one degree of freedom per unit cell.
+ - solve these equations to arrive at the dispersion relation.
+ - derive the group velocity, effective mass, and density of states.
+ - explain what happens with the band structure when the periodicity of the lattice is increased or reduced.
+
### More hoppings in LCAO
Consider electrons in a 1D atomic chain again
diff --git a/src/lecture_5.md b/src/lecture_5.md
index 79aa22f1d83943a109bf3b5c5925b8139bf04394..b1c6920bfbdcc241d98a304071836c8d784fad4e 100644
--- a/src/lecture_5.md
+++ b/src/lecture_5.md
@@ -3,12 +3,14 @@
_based on chapters 12–14, (up to and including 14.2) of the book_
Exercises 12.3, 12.4, 13.3, 13.4, 14.2
-In this lecture we will:
+!!! summary "Learning goals"
-- discuss how to classify crystal structures
-- review some common crystal structures
-- expand on the topics of reciprocal space and Brillouin zone
-- consider diffraction experiments on crystals
+ After this lecture you will be able to:
+
+ - Describe any crystal using crystallographic terminology, and interpret this terminology
+ - Compute the volume filling fraction given a crystal structure
+ - Determine the primitive, conventional, and Wigner-Seitz unit cells of a given lattice
+ - Determine the Miller planes of a given lattice
### Crystal classification
@@ -100,6 +102,14 @@ Miller index 0 means that the plane is parallel to that axis (intersection at "$
If a crystal is symmetric under $90^\circ$ rotations, then $(100)$, $(010)$ and $(001)$ are physically indistinguishable. This is indicated with $\{100\}$. $[100]$ is a vector. In a cubic crystal, $[100]$ is perpendicular to $(100)$ $\rightarrow$ proof in problem set.
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - Define the reciprocal space, and explain its relevance
+ - Construct a reciprocal lattice from a given real space lattice
+ - Compute the intensity of X-ray diffraction of a given crystal
+
### Reciprocal lattice
For every real-space lattice
diff --git a/src/lecture_6.md b/src/lecture_6.md
index fa031cb38f6185c207cf6706a7871592f2e20ad2..430a3439a7e2649b16f30c5537d81f4e3e8c6907 100644
--- a/src/lecture_6.md
+++ b/src/lecture_6.md
@@ -10,20 +10,23 @@ pi = np.pi
_(based on chapters 15–16 of the book)_
Exercises 15.1, 15.3, 15.4, 16.1, 16.2
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - formulate a general way of computing the electron band structure - the **Bloch theorem**.
+ - recall that in a periodic potential all electron states are Bloch waves.
+ - derive the electron band structure when the interaction with the lattice is weak using the **Nearly free electron model**.
+
Let's summarize what we learned about electrons so far:
* Free electrons form a Fermi sea ([lecture 2](lecture_1.md))
* Isolated atoms have discrete orbitals ([lecture 3](lecture_3.md))
* When orbitals hybridize we get *LCAO* or *tight-binding* band structures ([lecture 4](lecture_4.md))
-In this lecture we:
+The nearly free electron model (the topic of this lecture) helps to understand the relation between tight-binding and free electron models. It describes the properties of metals.
-* Formulate a general way of computing the electron band structure, the **Bloch theorem**.
-* Derive the electron band structure when the interaction with the lattice is weak using the **Nearly free electron model**. This model:
- - Helps to understand the relation between tight-binding and free electron models
- - Describes the properties of metals.
-
-All the different limits can be put onto a single scale as a function of the strength of the lattice potential $V(x)$:
+These different models can be organized as a function of the strength of the lattice potential $V(x)$:
@@ -142,6 +145,13 @@ Extended BZ (n-th band within n-th BZ):
* Easy to relate to free electron model
* Contains discontinuities
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - examine 1D and 2D band structures and argue if you expect the corresponding material to be an insulator/semiconductor or a conductor.
+ - describe how the light absorption spectrum of a material relates to its band structure.
+
## Band structure
How are material properties related to the band structure?
@@ -309,7 +319,7 @@ Here $E'_G\approx 0.9eV$ and $E_G\approx 0.8 eV$. The two steps visible steps ar
-The band structure has two band gaps: *direct*, the band gap at $k=0$, $E'_G$ and *indirect* gap $E_G$ at any $k$. In Ge $E_G > E'_G$, and therefore it is an *indirect band gap semiconductor*. Silicon also has an indirect band gap. Direct band gap materials are for example GaAs and InAs.
+The band structure has two band gaps: *direct*, the band gap at $k=0$, $E'_G$ and *indirect* gap $E_G$ at any $k$. In Ge $E_G < E'_G$, and therefore it is an *indirect band gap semiconductor*. Silicon also has an indirect band gap. Direct band gap materials are for example GaAs and InAs.
Photons carry very little momentum and a very high energy since $E = c \hbar k$ and $c$ is large. Therefore to excite electrons at $E_G$, despite a lower photon energy is sufficient, there is not enough momentum. Then an extra phonon is required. Phonons may have a very large momentum at room temperature, and a very low energy since atomic mass is much higher than electron mass.
diff --git a/src/lecture_7.md b/src/lecture_7.md
index 5574d0d5d885a2292fb7be043625ec0889a7a501..335b8a39012a71c1924d3c0ad57b79e804a70bc6 100644
--- a/src/lecture_7.md
+++ b/src/lecture_7.md
@@ -22,6 +22,14 @@ m_h, m_e = 1, .5
_(based on chapters 17–18 of the book)_
Exercises 17.1–17.4, 18.1, 18.2
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - Describe the concept of holes, and apply this concept to describe the properties of semiconductors
+ - Compute the density of states of electrons in semiconductors
+ - Compute the density of charge carriers and chemical potential as a function of temperature
+
## Review of band structure properties
* Group velocity $v=\hbar^{-1}\partial E(k)/\partial k$
@@ -213,6 +221,13 @@ $n_i$ is the **intrinsic carrier concentration**, and for a pristine semiconduct
> $$n_e n_h = n_i^2$$
> is the **law of mass action**. The name is borrowed from chemistry, and describes the equilibrium concentration of two reagents in a reaction $A+B \leftrightarrow AB$. Here electrons and hole constantly split and recombine.
+!!! summary "Learning goals"
+
+ After this lecture you will be able to:
+
+ - Compute carrier density and Fermi level position of doped semiconductors
+ - Describe the functioning principles of semiconducting devices
+
## Adding an impurity to semiconductor
* Typical semiconductors are group IV (Si, Ge, GaAs).