diff --git a/.gitlab-ci.yml b/.gitlab-ci.yml
index ffcd70dfb5433e10b81507f59c63ad30576036a1..fb472ad21d186e799dbaf18de94c12bebce110c3 100644
--- a/.gitlab-ci.yml
+++ b/.gitlab-ci.yml
@@ -16,10 +16,10 @@ build and upload the contents:
     - echo "$WEBSITE_UPLOAD_KEY" | tr -d '\r' | ssh-add - > /dev/null
 
     ## Create the SSH directory and give it the right permissions
-    - mkdir -p /root/.ssh
-    - chmod 700 /root/.ssh
-    - ssh-keyscan tnw-tn1.tudelft.net >> /root/.ssh/known_hosts
-    - chmod 644 /root/.ssh/known_hosts
+    - mkdir -p ~/.ssh
+    - chmod 700 ~/.ssh
+    - ssh-keyscan tnw-tn1.tudelft.net >> ~/.ssh/known_hosts
+    - chmod 644 ~/.ssh/known_hosts
     - pip install -U mkdocs mkdocs-material python-markdown-math
 
   script:
@@ -28,3 +28,5 @@ build and upload the contents:
     - python code/bands_2d.py
     - mkdocs build
     - "rsync -rv site/* solidstate@tnw-tn1.tudelft.net:"
+  only:
+    - branches@solidstate/lectures
\ No newline at end of file
diff --git a/_layouts/website/summary.html b/_layouts/website/summary.html
deleted file mode 100644
index 958ec916c23b001239fdb369fcb4540a682b54fb..0000000000000000000000000000000000000000
--- a/_layouts/website/summary.html
+++ /dev/null
@@ -1,57 +0,0 @@
-{% macro articles(_articles) %}
-    {% for article in _articles %}
-        <li class="chapter {% if article.path == file.path and not article.anchor %}active{% endif %}" data-level="{{ article.level }}" {% if article.path %}data-path="{{ article.path|resolveFile }}"{% endif %}>
-            {% if article.path and getPageByPath(article.path) %}
-                <a href="{{ article.path|resolveFile }}{{ article.anchor }}">
-            {% elif article.url %}
-                <a target="_blank" href="{{ article.url }}">
-            {% else %}
-                <span>
-            {% endif %}
-                    {% if article.level != "0" and config.pluginsConfig['theme-default'].showLevel %}
-                        <b>{{ article.level }}.</b>
-                    {% endif %}
-                    {{ article.title }}
-            {% if article.path  or article.url %}
-                </a>
-            {% else %}
-                </span>
-            {% endif %}
-
-            {% if article.articles.length > 0 %}
-            <ul class="articles">
-                {{ articles(article.articles, file, config) }}
-            </ul>
-            {% endif %}
-        </li>
-    {% endfor %}
-{% endmacro %}
-
-<ul class="summary">
-    {% set _divider = false %}
-    {% if config.links.sidebar  %}
-    {% for linkTitle, link in config.links.sidebar  %}
-        {% set _divider = true %}
-        <li>
-            <a href="{{ link }}" target="_blank" class="custom-link">{{ linkTitle }}</a>
-        </li>
-    {% endfor %}
-    {% endif %}
-
-    {% if _divider %}
-    <li class="divider"></li>
-    {% endif %}
-
-    {% for part in summary.parts %}
-        {% if part.title %}
-        <li class="header">{{ part.title }}</li>
-        {% elif not loop.first %}
-        <li class="divider"></li>
-        {% endif %}
-        {{ articles(part.articles, file, config) }}
-    {% endfor %}
-    <li class="divider"></li>
-    <li>
-      <a href="https://gitlab.kwant-project.org/solidstate/lectures/issues/new?issuable_template=typo">Report a typo</a>
-    </li>
-</ul>
diff --git a/book.json b/book.json
deleted file mode 100644
index 592f658bbe5cb6a69c2b3fbe9138bcb4cc827e76..0000000000000000000000000000000000000000
--- a/book.json
+++ /dev/null
@@ -1,3 +0,0 @@
-{
-    "plugins": ["katex-plus", "-sharing"]
-}
diff --git a/code/band_structures.py b/code/band_structures.py
index 5b50b91fce5415aeae3f7cd9fbdfcfe7d9184d5d..54f9ec8eda5eb6f0705cce673d46cb8668cdfb50 100644
--- a/code/band_structures.py
+++ b/code/band_structures.py
@@ -76,7 +76,7 @@ def tight_binding_1d():
     k = np.linspace(-pi, pi, 300)
     pyplot.plot(k, -np.cos(k))
     pyplot.xlabel('$ka$'); pyplot.ylabel('$E$')
-    pyplot.xticks([-pi, 0, pi], ['$-\pi$', 0, '$\pi$'])
+    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')
 
@@ -90,7 +90,7 @@ def meff_1d_tb():
     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], ['$-\pi$', 0, '$\pi$']);
+    pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$']);
     pyplot.savefig('meff_1d_tb.svg')
 
 
@@ -110,7 +110,7 @@ def phonons_1d_2masses():
         ax.plot(k, dispersion_2m(k), label='acoustic')
         ax.set_xlabel('$ka$')
         ax.set_ylabel(r'$\omega$')
-        pyplot.xticks([-pi, 0, pi], ['$-\pi$', '$0$', '$\pi$'])
+        pyplot.xticks([-pi, 0, pi], [r'$-\pi$', '$0$', r'$\pi$'])
         pyplot.yticks([], [])
         pyplot.vlines([-pi, pi], 0, 2.2, linestyles='dashed')
         pyplot.legend()
diff --git a/docs/figures/pn_junction_bias.svg b/docs/figures/pn_junction_bias.svg
new file mode 100644
index 0000000000000000000000000000000000000000..73090ee1dfea96febf8b7c8f40496a5cbd74b262
--- /dev/null
+++ b/docs/figures/pn_junction_bias.svg
@@ -0,0 +1,712 @@
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+   xmlns:cc="http://creativecommons.org/ns#"
+   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+   xmlns:svg="http://www.w3.org/2000/svg"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
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+   id="svg5204"
+   version="1.1"
+   inkscape:version="0.92.2 (5c3e80d, 2017-08-06)"
+   sodipodi:docname="pn_junction_bias.svg">
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+     bordercolor="#666666"
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diff --git a/docs/index.md b/docs/index.md
index 767eb6d51dacb8ff67c6e6683921bd3f17560c5f..da0bca0eef3f5583035cab7420d8003a4bfe5964 100644
--- a/docs/index.md
+++ b/docs/index.md
@@ -1,5 +1,7 @@
 # Materials of the TN2844 course Solid State Physics
 
+If something is unclear, if you notice a typo, or if you have a suggestion, please [let us know](https://gitlab.kwant-project.org/solidstate/lectures/issues/new?issuable_template=typo)!
+
 ## Syllabus
 
 * Week 1: [Phonons and specific Heat](lecture_1.md)  
@@ -27,16 +29,12 @@
   Exercises 15.1, 15.3, 15.4, 16.1, 16.2
 
 * Week 7: [Semiconductors](lecture_7.md)  
-  (based on chapters 17–18)
+  (based on chapters 17–18)  
+  Exercises 17.1–17.4, 18.1, 18.2
 
 * 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
@@ -47,15 +45,19 @@
 
 [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
 
 ## Course discussion chat
 
-Whenever you have a question outside of regular hours, reach the course team and your fellow course participants in this chat:
-
-https://chat.quantumtinkerer.tudelft.nl/solidstate
+Whenever you have a question outside of regular hours, reach the course team and your fellow course participants in [course chat](https://chat.quantumtinkerer.tudelft.nl/solidstate)!
 
 ## Course team
 
diff --git a/docs/lecture_1.md b/docs/lecture_1.md
index 0fa82ab008cc6447395ba983778e3f7feda04010..e6d08d37f96b3a010db88f323225905536cbf6aa 100644
--- a/docs/lecture_1.md
+++ b/docs/lecture_1.md
@@ -3,6 +3,7 @@
 _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)_
diff --git a/docs/lecture_2.md b/docs/lecture_2.md
index 14c38d48a3ebdb082edc43b6d9301449d71251c2..8af060a5ff5b5d62ac8ec8d369c45388657eb521 100644
--- a/docs/lecture_2.md
+++ b/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: $\tau_0$
 
@@ -90,8 +91,9 @@ $$
 ![](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
+
+- 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_.
 
@@ -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.
 
diff --git a/docs/lecture_3.md b/docs/lecture_3.md
index 11b488eb7a5edb2010cf0f9ae77c6febc1971ec0..f0c7645d9d4893c85f805fd4ffb14eaa29dc66f6 100644
--- a/docs/lecture_3.md
+++ b/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, ...
diff --git a/docs/lecture_5.md b/docs/lecture_5.md
index b3631c2ee22bad9a6287e2bc48933104246fe725..4d40a8daf98359c5bb80c5a0baccd4463167fe8b 100644
--- a/docs/lecture_5.md
+++ b/docs/lecture_5.md
@@ -11,44 +11,43 @@ 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
-  - freedom of translation
-  - multiple lattices with different point densities possible
 
+- **_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
+    + freedom of translation
+    + 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
-
+    + 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
-
+    + 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
-
+    + 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!
+    + 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$^{\circ}$
+    - 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$\left(\frac{1}{2},\frac{5}{6}\right)$
+    - 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$\left(\frac{1}{2},\frac{5}{6}\right)$
 
 An alternative type of unit cell is the _Wigner-Seitz cell_: the collection of all points that are closer to one specific lattice point than to any other lattice point. You form this cell by taking all the perpendicular bisectrices or lines connecting a lattice point to its neighboring lattice points.
 
@@ -80,6 +79,7 @@ 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 +237,6 @@ $$
 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)
diff --git a/docs/lecture_6.md b/docs/lecture_6.md
index 09f548763cd782399a44527690bb799709827c11..26a065a97ae7a3c4a3ac6d7485f980cb122cef9c 100644
--- a/docs/lecture_6.md
+++ b/docs/lecture_6.md
@@ -10,8 +10,8 @@ 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.
+    - 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 $V(x)$:
 
@@ -20,9 +20,7 @@ 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 +64,21 @@ $$H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} =
 $$
 
 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
 
@@ -94,16 +93,10 @@ Now we expand the definition of $W$:
 
 $$W = \langle \psi_+ | V(x) | \psi_- \rangle = \int_0^{a} dx \left[e^{i\pi x/a}\right]^* V(x) \left[e^{-i\pi x/a}\right] = \int_0^a e^{2\pi i x /a} V(x) = V_1$$
 
-Here $V_1$ is the first Fourier component of $V(x)$ (using a *complex* Fourier transform).
+Here $V_1$ is the first Fourier component of $V(x)$ (using a complex Fourier transform).
 
 $$ V(x) = \sum_{n=-\infty}^{\infty} V_n e^{2\pi i n x/a}$$
 
-In the book the *real symmetric* Fourier transform is used:
-
-$$ V(x) = \sum_{n=0}^{\infty} \tilde{V}_n \cos(2\pi n x/a),$$
-
-which can be used *only* when $V(x) = V(-x)$. Then the relation between the two is $\tilde{V}_n = 2 V_n$.
-
 #### Crossings between the higher bands
 
 Everything we did applies to the crossings at higher energies, only there we would get higher Fourier components of $V(x)$: $V_2$ for the crossing between the second and third band, $V_3$ for the crossing between third and 4th, etc.
diff --git a/docs/lecture_7.md b/docs/lecture_7.md
index 9715197befb2d804927d5f700e78921a7b7a9693..e724ea61ed61e6f5e878059f331ea812f9cc3b9a 100644
--- a/docs/lecture_7.md
+++ b/docs/lecture_7.md
@@ -1,6 +1,7 @@
 # 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.
@@ -108,11 +109,12 @@ $$ E = E_G + {p^2}/{2m_e}$$
 $$ g(E) = (2m_e)^{3/2}\sqrt{E-E_G}/2\pi^2\hbar^3$$
 
 Holes
-$$ E = {p^2}/{2m_h}$$
+$$ E_h = {p^2}/{2m_h}$$
 
-$$ g(E) = (2m_h)^{3/2}\sqrt{E_h}/2\pi^2\hbar^3$$
+$$ g(E_h) = (2m_h)^{3/2}\sqrt{E_h}/2\pi^2\hbar^3$$
 
 **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!)
@@ -178,7 +180,7 @@ $n_i$ is the **intrinsic carrier concentration**, and for a pristine semiconduct
 Extra electron (or extra hole) is attracted to the extra charge of the nucleus.
 
 In H the energy levels are:
-$$ E_n = - \frac{me^4}{8\pi^2\hbar^3\varepsilon_0n^2} = -R_E /n^2= -\frac{13.6\text{eV}}{n^2}$$
+$$ E_n = - \frac{me^4}{8\pi^2\hbar^3\varepsilon^2_0n^2} = -R_E /n^2= -\frac{13.6\text{eV}}{n^2}$$
 
 Bohr radius (size of the ground state wave function): $4 \pi \varepsilon_0 \hbar^2/m_{\mathrm{e}} e^2$
 
@@ -199,12 +201,12 @@ Likewise an acceptor adds an extra state at $E_A$ (close to the top of the valen
 ![](figures/doping.svg)
 
 All donor/acceptor states at the same energy:
-$$g_A = N_A \delta(E_A),\quad g_D = N_D \delta(E_G - E_D)$$
+$$g_A(E) = N_A \delta(E-E_A),\quad g_D(E) = N_D \delta(E- (E_G - E_D))$$
 
 How large can $N_D/N_A$ be? The distance between donors should be such that the states don't overlap, so the distance must be much larger than 4 nm. Therefore **maximal** concentration of donors before donor band merges with conduction band is $N_D \lesssim (1Å/4\textrm{nm})^3 \sim 10^{-5}\ll N_C$.
 
 
-# Number of carriers
+## Number of carriers
 
 Charge conservation:
 $$n_e - n_h + n_D - n_A = N_D - N_A$$
@@ -233,6 +235,103 @@ $$E_F = kT\log[N_V/(N_A-N_D)], \textrm{ for } N_A > N_D$$
 
 **Answer:** The semiconductor is intrinsic when $|N_D-N_A| \ll n_i$, so $kT \gtrsim E_G/\log[N_C N_V/(N_D-N_A)^2]$.
 
+## Temperature dependence of the carrier density and Fermi level
+
+It is instructive to consider how $E_F$, $n_e$ and $n_h$ depend on carrier concentrations.
+
+![](figures/E_F_and_carrier_density.svg)
+
+Several noteworthy features:
+
+* At high temperature $n_e = n_h$ and $E_F$ has an upturn (if holes are heavier than electrons)
+* Once the temperature is sufficiently low, we expect the electrons to "freeze away" from the conduction band to the donor band, so that the donor band starts playing a role of the new valence band at $kT \ll E_G - E_D$.
+* At zero temperature $E_F$ should match the donor band since it has partially occupied states. If there are no acceptors, $E_F$ would be halfway between $E_D$ and $E_G$, and if there was no doping at all it would be at $E_G/2$.
+
+**Exercise:** check that you can reproduce all the relevant limits in a calculation.
+
+## Measuring band gaps
+
+### Conduction
+
+The change of $n_i$ is extremely rapid due to the factor $e^{-E_G/kT}$. Therefore the simplest way of determining the band gap is the temperature dependence of conductance
+
+The total current
+
+$$j = -n_e e v_e + n_h e v_h $$
+
+$$ -m_e v_e /\tau_e = -eE;\quad -m_h v_h /\tau_h = eE $$
+
+Combining the two we see that despite electron and hole velocities have opposite signs, they carry electric current in the same direction.
+
+$$ \sigma \equiv \frac{j}{E} = \left(\frac{n_e e^2 \tau_e}{m_e}+\frac{n_h e^2 \tau_h}{m_h}\right) = n_e e \mu_e + n_h e \mu_h.$$
+
+Since $n_e = n_h = n_i \propto e^{-E_G/kT}$, $E_G \approx d \log \sigma / d [kT]^{-1}$.
+
+Additional information can be obtained using Hall effect. However Hall effect is much more complex in semiconductors since only the current in the direction perpendicular to the applied electric field must vanish. This, however only means that the electron current is opposite of the hole current in that direction, not that the electrons and holes move parallel to the applied current.
+
+### Light adsorption
+
+See [previous lecture](lecture_6.md#light-adsorption)
+
+## Combining semiconductors: $pn$-junction
+
+Main idea: what happens if we bring two differently doped semiconductors together (one of $p$-type, one of $n$-type)?
+
+![](figures/pn-junction.svg)
+
+### Band diagram
+
+Previously we dealt with homogeneous materials, now coordinate (let's call it $x$) starts playing a role. We can represent the properties of inhomogeneous materials using the **band diagram**. The main idea is to plot dependence of various energies ($E_F$, bottom of conduction band, top of the valence band) as a function of position.
+
+So here is our problem for today:
+
+![](figures/band_diagram_question.svg)
+
+> An important remark is in order: before we counted all energies with respect to the top of the valence band. Now $E_F$ is the same everywhere if the sample is in equilibrium, and the bands are shifted due to an extra electrostatic potential.
+
+More specifically: How fast does the electrostatic potential change in the intermediate region? What is the charge density at the junction if we make the chemically-defined boundary between materials very precise?
+
+We may instead use a key bit of insight: **the density of electrons and holes drops exponentially fast as soon as potential deviates by $kT \ll \delta \varphi$ from its bulk value**.
+
+Using this information we can expect a formation of the **depletion region** with almost no electrons or holes (but donors and acceptors won't move anywhere from that region).
+
+So this is our expectation of the dependence of $\rho(x)$:
+
+![](figures/pn_charge_density.svg)
+
+The typical values of $w_n+w_p$ are $\sim 1 \mu \textrm{m}$ at $N_A,\,N_D \sim 10^{16} \textrm{cm}^{-3}$, and $\sim 0.1 \mu \textrm{m}$ at $N_A,\,N_D \sim 10^{18} \textrm{cm}^{-3}$, so it may be much larger than the distance between the dopant atoms.
+
+
+### $pn$-junction diode
+
+What happens if we apply voltage to a junction?
+
+Because the conductivity of the $p$-region and $n$-region is much larger than that of the depletion region, most of the voltage difference will appear in the depletion region:
+
+![](figures/pn_junction_bias.svg)
+
+The number of majority carriers moving across the junction is proportional to their concentration.
+Increasing the voltage bias "pushes" carriers up in energy, it depends exponentially on the voltage.
+
+We therefore get the Shockley diode equation:
+
+$$I = I_0 \left(\exp(eV/kT) -1\right)$$
+
+### Solar cell
+
+Light adsopbed in the $pn$-junction creates electron-hole pairs.
+The eletric field then moves electrons to the $n$-doped region, holes to the $p$-doped one, and therefore generates a voltage.
+
+![](figures/solar_cell.svg)
+
+### Semiconducting laser
+
+A heavily doped $pn$-junction so that the Fermi level is in the conduction/valence band produces an extremely high rate electron-hole recombination with an extremely high rate, and makes the $pn$-junction function like a laser.
+
+### MOSFET and quantum well
+
+See the book for details.
+
 ## Summary
 
 Density of states in a doped semiconductor:
@@ -243,4 +342,12 @@ 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).
+
+Temperature switches between intrinsic and extrinsic regimes, and controls the carrier density
+
+Conductance combines the contributions of electrons and holes, and allows to determine $E_G$.
+
+A $pn$-junction has a **depletion layer** in its middle with the potential in a $pn$-junction having the following shape (where the transition region is made out of two parabolas):
+
+![](figures/band_diagram_solution.svg)
\ No newline at end of file