fix remaining list formatting

docs/index.md

@@ -32,11 +32,6 @@
 * Week 8: Magnetism  
   (based on chapters 19-20)
 
-# Practice exams
-
-## Exams from years 16-17
-[Exams archive](https://solidstate.quantumtinkerer.tudelft.nl/exams_2016_2017.zip), [solutions minitests 2017 archive](https://solidstate.quantumtinkerer.tudelft.nl/solutions_2017_minitests.zip) *(caution, those solutions are not always correct)*, [solutions final](https://solidstate.quantumtinkerer.tudelft.nl/final_2017/).
-
 ## Test results
 
 ### Minitest 1
@@ -47,6 +42,12 @@
 
 [Results report](https://solidstate.quantumtinkerer.tudelft.nl/summary_report_midterm.html), [solution](https://solidstate.quantumtinkerer.tudelft.nl/solutions_midterm.pdf)
 
+# Practice exams
+
+## Exams from years 16-17
+
+[Exams archive](https://solidstate.quantumtinkerer.tudelft.nl/exams_2016_2017.zip), [solutions minitests 2017 archive](https://solidstate.quantumtinkerer.tudelft.nl/solutions_2017_minitests.zip) *(caution, those solutions are not always correct)*, [solutions final](https://solidstate.quantumtinkerer.tudelft.nl/final_2017/).
+
 ## Course book
 
 The Oxford Solid State Basics by Steven H. Simon

docs/lecture_1.md

@@ -3,9 +3,12 @@
 _Based on chapter 2 of the book_
 
 In this lecture we will:
+
 - 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_

docs/lecture_2.md

@@ -32,6 +32,7 @@ $$
 $\sigma$ is the conductivity, which is the inverse of resistivity: $\rho=\frac{1}{\sigma}$. If we now take $j=\frac{I}{A}$ and $E=\frac{V}{l}$, we retrieve Ohm's Law: $\frac{I}{A}=\frac{V}{\rho l}$.
 
 Scattering is caused by collisions with:
+
 - Phonons: $\tau_{\rm ph}(T)$ ($\tau_{\rm ph}\rightarrow\infty$ as $T\rightarrow 0$)
 - Impurities/vacancies: $\tau_0$
 
@@ -90,6 +91,7 @@ $$
 ![](figures/free_electron.svg)
 
 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
    
@@ -118,6 +120,7 @@ $$
 ![](figures/sqrt.svg)
 
 Similarly,
+
 - For 1D: $g(\varepsilon) = \frac{2 V}{\pi} \frac{ {\rm d}k}{ {\rm d}\varepsilon} \propto 1/\sqrt{\varepsilon}$
 - For 2D: $g(\varepsilon) = \frac{k V}{\pi} \frac{ {\rm d}k}{ {\rm d}\varepsilon} \propto \text{constant}$
 
@@ -175,6 +178,7 @@ $$
 $T_{\rm F}=\frac{\varepsilon_{\rm F}}{k_{\rm B}}$ is the _Fermi temperature_.
 
 How does $C_{V,e}$ relate to the phonon contribution $C_{V,p}$?
+
 - At room temperature, $C_{V,p}=3Nk_{\rm B}\gg C_{V,e}$
 - Near $T=0$, $C_{V,p}\propto T^3$ and $C_{V,e}\propto T$ $\rightarrow$ competition.
 

docs/lecture_3.md

@@ -53,6 +53,7 @@ Single electron states have 4 quantum numbers: $|n, l, l_z, \sigma\rangle$
 ![](figures/single_atom.svg)
 
 Quantum numbers:
+
 * $n=1,2,\ldots$ is the azimuthal (principal) quantum number
 * $l=0, 1, \ldots, n-1$ is the angular momentum (also known as $s, p, d, f$ orbitals)
 * $l_z=-l, -l+1\ldots, l$ is the $z$-component of angular momentum
@@ -61,7 +62,6 @@ Quantum numbers:
 It turns out that electrons in all atoms occupy these orbitals, only the energies are very different.
 
 * Aufbau principle: *first fill a complete shell (all electrons with the same $n, l$) before going to the next one*
-
 * Madelung's rule: *first occupy the shells with the lowest $l+n$, and of those with equal $l+n$ those with smaller $n$*
 
 Therefore shell-filling order is 1s, 2s, 2p, 3s, 3p, 4s, 4d, ...

docs/lecture_5.md

@@ -11,7 +11,9 @@ In this lecture we will:
 - consider diffraction experiments on crystals
 
 ### Crystal classification
+
 - **_Lattice_**
+
   - periodic pattern of *lattice points*, which all have an identical view
   - lattice points are not necessarily the same as atom positions
   - there can be multiple atoms per lattice point
@@ -19,34 +21,43 @@ In this lecture we will:
   - multiple lattices with different point densities possible
 
 - **_Lattice vectors_**
+
   - from lattice point to lattice point
   - $N$ vectors for $N$ dimensions
   - multiple combinations possible
   - not all combinations provide full coverage
 
 - **_Unit cell_**
+
   - spanned by lattice vectors
   - has 4 corners in 2D, 8 corners in 3D
   - check if copying unit cell along lattice vectors gives full lattice
 
 - **_Primitive unit cell_**
+
   - smallest possible $\rightarrow$ no identical points skipped
   - not always most practical choice
 
 - **_Basis_**
+
   - only now we care about the contents (i.e. atoms)
   - gives element and position of atoms
   - properly count partial atoms $\rightarrow$ choose which belongs to unit cell
   - positions in terms of lattice vectors, *not* Cartesian coordinates!
 
 ### Example: graphite
+
 ![](figures/graphite_mod.svg)
+
 1. Choose origin (can be atom, not necessary)
 2. Find other lattice points that are identical
 3. Choose lattice vectors, either primitive (red) or not primitive (blue)
+
   - lengths of lattice vectors and angle(s) between them fully define the crystal lattice
   - for graphite: $|{\bf a}_1|=|{\bf a}_2|$ = 0.246 nm = 2.46 Å, $\gamma$ = 60$^{\circ}$
+
 4. Specify basis
+
   - using ${\bf a}_1$ and ${\bf a}_2$: C$(0,0)$, C$\left(\frac{2}{3},\frac{2}{3}\right)$
   - using ${\bf a}_1$ and ${\bf a}_{2}'$: C$(0,0)$, C$\left(0,\frac{1}{3}\right)$, C$\left(\frac{1}{2},\frac{1}{2}\right)$, C$\left(\frac{1}{2},\frac{5}{6}\right)$
 
@@ -80,6 +91,7 @@ Compare this to _body centered cubic_ (bcc), which consists of a cube with atoms
 Question: is 74% the largest possible filling factor? $\rightarrow$ Kepler conjecture (1571 – 1630). Positive proof by Hales _et al._ in 2015!
 
 Crystal structures that are related to fcc:
+
 1. ionic crystals (Fig. 1.13), e.g. NaCl
 2. zincblende (Fig. 1.15), e.g. diamond
 
@@ -237,7 +249,6 @@ $$
 The structure factor can cause destructive interference due to the contents of the unit cell, even though the Laue condition is met. Examples:
 
 - Simplest case: 1 atom at each lattice point, ${\bf r}_1=(0,0,0)\rightarrow S=f_1$. In this case each reciprocal lattice point gives one interference peak, none of which is cancelled.
-
 - Example 2: conventional cell of the fcc lattice.
 
 ![](figures/fcc_mod.svg)

docs/lecture_6.md

@@ -10,6 +10,7 @@ In this lecture we:
 
 * 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.
 
@@ -20,9 +21,7 @@ All the different limits can be put onto a single scale as a function of the str
 ## Bloch theorem
 
 > All Hamiltonian eigenstates in a crystal have the form
-
 > $$ \psi_n(\mathbf{r}) = u_n(\mathbf{r})e^{i\mathbf{kr}} $$
-
 > with $u_n(\mathbf{r})$ having the same periodicity as the lattice potential $V(\mathbf{r})$, and index $n$ labeling electron bands with energies $E_n(\mathbf{k})$.
 
 In other words: any electron wave function in a crystal is a product of a periodic part that describes electron motion within a unit cell and a plane wave.
@@ -66,20 +65,21 @@ $$H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} =
 $$
 
 Here we used $\delta p = \hbar \delta k$, and we expanded the quadratic function into a linear plus a small correction.  
-**Question: can you calculate** $E_0$ **and the velocity** $v$**?**
+**Question: can you calculate $E_0$ and the velocity $v$?**
 
 Without $V(x)$ the two wave functions $\psi_+$ and $\psi_-$ are independent since they have a different momentum. When $V(x)$ is present, it may couple these two states.
 
 So in presence of $V(x)$ the Hamiltonian becomes
 
-$$H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} = 
+$$
+H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} = 
 \begin{pmatrix} E_0 + v \hbar \delta k & W \\ W^* & E_0 - v \hbar \delta k\end{pmatrix}
 \begin{pmatrix}\alpha \\ \beta \end{pmatrix},
 $$
 
 Here the coupling strength $W = \langle \psi_+ | V(x) | \psi_- \rangle$ is the matrix element of the potential between two states. *(This where we need to apply the perturbation theory, and this is very similar to the LCAO Hamiltonian)*.
 
-**Question: how does our solution satisfy the Bloch theorem? What is** $u(x)$ **in this case?**
+**Question: how does our solution satisfy the Bloch theorem? What is $u(x)$ in this case?**
 
 #### Dispersion relation near the avoided level crossing
 

docs/lecture_7.md

 # Semiconductor physics
 
 ## Review of band structure properties
+
 * Group velocity $v=\hbar^{-1}\partial E(k)/\partial k$
 * Effective mass $m_{eff} = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1}$
 * Density of states $g(E) = \sum_{\textrm{FS}}dk \times (dn/dk) \times (dk/dE)$ the amount of states per infinitesimal interval of energy at each energy.
@@ -113,6 +114,7 @@ $$ E = {p^2}/{2m_h}$$
 $$ g(E) = (2m_h)^{3/2}\sqrt{E_h}/2\pi^2\hbar^3$$
 
 **The key algorithm of describing the state of a semiconductor:**
+
 1. Write down the density of states, assuming a certain position of the Fermi level
 2. Calculate the total amount of electrons and holes, equate the difference to the total amount of electrons $-$ holes available.
 3. Use physics intuition to simplify the equations (this is important!)
@@ -243,4 +245,4 @@ Charge balance determins the number of electrons and holes as well as the positi
 
 If dopant concentrations are low, then $n_e = n_h = n_i \equiv \sqrt{N_C N_V}e^{-E_G/2kT}$.
 
-If dopant concentrations are high, then in $n$-doped semiconductor $n_e = N_D - N_A$ and $n_h = n_i^2/n_e$ (or vice versa).
\ No newline at end of file
+If dopant concentrations are high, then in $n$-doped semiconductor $n_e = N_D - N_A$ and $n_h = n_i^2/n_e$ (or vice versa).