Merge branch 'q1' into main

01_fundamentals.tex

-\section{Fundamentals of semiconductor physics}
+\section{Basics of semiconductor physics}
+\subsection{Notation}
+We only care about free electrons and define
+\begin{equation}
+    \begin{split}
+        n &= \text{electron concentrations density}\ [cm^{-3}]\\
+        p &= \text{hole concentrations density}\ [cm^{-3}]\\
+    \end{split}
+\end{equation}
+
+\subsection{Band gap}
+If an electron energy $E$ is large enough to overcome the bandgap $E_g$,
+it can escape the valence band into the conduction band.
+It then leaves behind a hole.
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=.6\textwidth]{imgs/band_gap_electorn_holes.png}
+    \caption{Band-gap electrons/holes}
+\end{figure}
+
+\subsection{Generation and recombination}
+Generation is the breakup of covalent bonds,
+which creates free electrons and holes.
+This requires energy, either thermal or optical.
+Assuming the atom density far larger than n or p (and thus bonds can be split defacto infinitely),
+we have:
+\begin{align}
+    G & = G_{th} + G_{opt} \\
+    R & \sim  n\cdot p
+\end{align}
+
+
+In thermal equilibrium, the generation and recombination rates are equal:
+\begin{align}
+    G_0                & = f(T)=R_0 \\
+    \Rightarrow n_0p_0 & =n_i^2(T)
+\end{align}
+
+
+\subsection{Doping}
+Doners are in the 5th shell (V), usually As,P,Sb.
+Acceptors are in the 3rd shell (III), usually B,Al,Ga,In.
+For doners we have:
+\begin{align}
+    n_0 & = N_d               \\
+    p_0 & = \frac{n_i^2}{N_d}
+\end{align}
+
+\subsection{Charge neutrality}
+Every semiconductor is neutral,
+which imposes the following condition:
+\begin{equation}
+    P_0-n_0+N_d + N_a = 0
+\end{equation}
+where $p_0n_0=n_i^2$.
\ No newline at end of file

02_carrier_transport.tex

+\section{Carrier transport}
+\subsection{Fermi distribution}
+\label{label:sec:fermi}
+Fermions are weird particles, see QM II.
+Not sure if needed for this course,
+but heres the probability distribution:
+\begin{equation}
+    f(E) = \frac{1}{1+e^{(E-E_F)/kT}}
+\end{equation}
+
+Electron concentration in conductance band:
+\begin{equation}
+    n=N_ce^{-(E_c-E_f)/kT}
+\end{equation}
+Hole concentration in valence band:
+\begin{equation}
+    p=N_ve^{-(E_f-E_v)/kT}
+\end{equation}
+
+In intrinsic silicon ($n=p=n_i$) we have ($E_i$ somewhere in the middle of the bandgap)
+\begin{equation}
+    E_i=E_f=\frac{E_c+E_v}{2}-\frac{kT}{q}\ln{\frac{N_c}{N_v}}
+\end{equation}
+
+Which gives us the useful relation:
+\begin{equation}
+    n_i = \sqrt{N_cN_v}e^{-(E_c-E_v)/2kT} = \sqrt{N_cN_v}e^{-E_g/2kT}
+\end{equation}
+
+
+\subsection{Behaviour in thermal equilibrium}
+\begin{align}
+    \lambda & \equiv \text{mean free path} [cm]                   \\
+    \tau_c  & \equiv \text{mean time between collisions} [s^{-1}] \\
+    v_{th}  & \equiv \text{thermal velocity} [cm/s]               \\[1em]
+    \lambda & = v_{th}\cdot\tau_c
+\end{align}
+
+
+\subsection{Drift velocity}
+Quick electromag recap: (for holes use + and $m_p$)
+\begin{align}
+    F    & = -qE             \\
+    v(t) & =-\frac{qE}{m_n}t
+\end{align}
+
+Average drift velocity:
+\begin{equation}
+    v_d = \pm \frac{qE\tau_c}{2m_{n,p}}
+\end{equation}
+
+\subsection{Mobility}
+For the sake of simplicity, let's define mobility for both holes and electrons.
+(These values are usually found in diagrams.)
+\begin{align}
+    \mu_{n,p} & = \frac{q\tau_c}{2m_{n,p}} \equiv \text{mobility}\ [cm^2/Vs] \\
+    v_{dn}    & =-\mu_nE                                                     \\
+    v_{dp}    & = \mu_pE                                                     \\
+    \mu_n     & >_mu_p
+\end{align}
+
+\subsection{Drift current}
+For the net drift current density slap together velocity, density and charge.
+\begin{equation}
+    label{eq:drift_current}
+    J^{drift} = J_n^{drift}+J_p^{drift} = q(n\mu_n+p\mu_p)E
+\end{equation}
+From which we can  find Ohm's law:
+\begin{alignat}{2}
+    J    & =\sigma E         &  & = \frac{E}{\rho}                         \\
+    \rho & =\frac{1}{\sigma} &  & = \frac{1}{q \left(n\mu_n+p\mu_p\right)}
+\end{alignat}
+Which gives us different resistances for n and p type semiconductors.
+\begin{align}
+    \rho_n & \approx \frac{1}{qN_d\mu_n} \\
+    \rho_p & \approx \frac{1}{qN_a\mu_p}
+\end{align}
+
+
+\subsection{Diffusion current}
+If there is a concentration gradient,
+the carriers will diffuse to equalize the concentration.
+Here flux $F \ [cm^{-2}s^{-1}]$ is the number of electrons/holes per unit area per unit time.
+\begin{align}
+    F_n & = -D_n\frac{\mathrm{d} n}{\mathrm{d} x} \\
+    F_p & = -D_p\frac{\mathrm{d} p}{\mathrm{d} x}
+\end{align}
+
+Which gives us the diffusion current density:
+(Defined as density times charge,
+ergo the double negative for electron diffusion.)
+\begin{align}
+    J_n^{diff} & = qD_n\frac{\mathrm{d} n}{\mathrm{d} x}    \label{label:eq:diff_current_n} \\
+    J_p^{diff} & =- qD_p\frac{\mathrm{d} p}{\mathrm{d} x}   \label{label:eq:diff_current_p}
+\end{align}
+
+
+\subsection{Einstein relation between mobility and diffusion coefficient}
+\label{label:sss:einstein_rel_mob_diff}
+\begin{equation}
+    \frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q^2}
+\end{equation}
+
+\subsection{Total current}
+\begin{alignat}{2}
+    J_{total} & =J_n+J_p                &  &                                                                                          \\
+    J_n       & =J_n^{drift}+J_n^{diff} &  & =qn\mu_nE+qD_n\frac{\mathrm{d} n}{\mathrm{d} x}\label{label:eq:electron_current_density} \\
+    J_p       & =J_p^{drift}+J_p^{diff} &  & =qp\mu_pE-qD_p\frac{\mathrm{d} p}{\mathrm{d} x}
+\end{alignat}
\ No newline at end of file

03_pn_junction_basics.tex

+\section{PN junction basics}
+\subsection{Uniformly doped semiconductor}
+Assuming n type (i.e. lots of electrons, few holes) semiconductor with a uniform doping profile,
+we have the following volume charge density.
+\begin{align}
+    n_0  & = N_d                                             \\
+    \rho & = q\left(N_d-n_0\right)=0 \ \left[C/cm^{3}\right]
+\end{align}
+
+\subsection{Non-uniformly doped semiconductor}
+At thermal equilibrium, the total current must be 0.
+Because we have non-uniform doping,
+we need the drift current to balance the diffusion current.
+(For both electrons and holes.)
+\begin{equation}
+    J_n(x)=J_n^{drift}(x)+J_n^{diff}(x)=0
+\end{equation}
+Which implies
+\begin{equation}
+    n(x)\neq N_d(x)
+\end{equation}
+Which gives us the space charge density
+\begin{equation}
+    \label{label:eq:electron_space_charge_density}
+    \rho(x)=q\left[N_d(x)-n(x)\right]\neq 0
+\end{equation}
+
+This results in a potential difference (quick electromag recap):
+\begin{align}
+    \frac{\mathrm{d}E}{\mathrm{d}x} & = -\frac{\rho}{\varepsilon}   \label{label:eq:def_electric_field_differential}                        \\
+    E(x)-E(0)                       & =\frac{1}{\varepsilon}\int_{0}^{x}\rho(x')\,\mathrm{d}x' \label{label:eq:def_electric_field_integral}
+\end{align}
+
+Since there is an electric field, there's a potential.
+\begin{align}
+    \frac{\mathrm{d}\phi}{\mathrm{d}x} & = -E                              \\
+    \phi(x)-\phi(0)                    & =-\int_{0}^{x}E(x')\,\mathrm{d}x'
+\end{align}
+
+By combining eq \eqref{label:eq:electron_current_density} ,
+\eqref{label:eq:electron_space_charge_density},
+and \eqref{label:eq:def_electric_field_differential} we find
+\begin{equation}
+    \frac{\mathrm{d}^2}{\mathrm{d}x^2}\ln{n(x)} = \frac{q^2}{\varepsilon kT}\left(n(x)-N_d(x)\right)
+\end{equation}
+
+\subsection{Quasi-neutral approximation}
+If the doping changes slowly with x:
+\begin{equation}
+    n(x)\approx N_c(x)
+\end{equation}
+
+
+\subsection{Boltzman relation between $n$ and $\phi$}
+We saw in \autoref{label:sss:einstein_rel_mob_diff} the relation between mobility and diffusion coefficients.
+From this we find
+\begin{align}
+    n          & =n_{ref}e^{q(\phi-\phi_{ref})/kT} \\
+    \phi_{ref} & =0                                \\
+    n_{ref}    & =n_i
+\end{align}
+And by extension
+\begin{align}
+    n & = n_i e^{q\phi/kT} \\
+    p & =n_ie^{-q\phi/kT}
+\end{align}
+
+Rearranging the above, we find an expression for the potential:
+\begin{align}
+    \phi & =\frac{kT}{q}\ln\frac{n}{n_i}   \label{label:eq:boltzman:phi_n} \\
+    \phi & = -\frac{kT}{q}\ln\frac{p}{n_i} \label{label:eq:boltzman:phi_p}
+\end{align}
+For Si at room temperature this is an increase of 60 mV per decade in doping.
+\begin{equation}
+    \phi\approx(60\,\mathrm{mV})\log_{10}\frac{n}{10^{10}}
+\end{equation}

04_pn_junction.tex

+\section{PN junction}
+\subsection{What are we even doing}
+We stick together n and p doped regions, such that the doping effectively  becomes a step function.
+This causes majority carriers (electrons in n region, holes in p region) to diffuse the minority carrier side,
+resulting in a new equilibrium (\autoref{label:fig:pn_carrier_profile_equilibrium}).
+\begin{figure}[h]
+    \centering
+    \begin{subfigure}[b]{.45\textwidth}
+        \includegraphics[width=\textwidth]{imgs/pn_carrier_profile_equilibrium.png}
+        \caption{Resulting carrier profile in thermal equilibrium}
+        \label{label:fig:pn_carrier_profile_equilibrium}
+    \end{subfigure}
+    \hfill
+    \begin{subfigure}[b]{.45\textwidth}
+        \includegraphics[width=\textwidth]{imgs/pn_fermi_level_band_bending.png}
+        \caption{Resulting carrier profile in thermal equilibrium}
+        \label{label:fig:pn_fermi_level_band_bending}
+    \end{subfigure}
+\end{figure}
+
+As can be seen in \autoref{label:fig:pn_fermi_level_band_bending},
+the energy levels for conduction and valence bands bend, whereas the fermi level remains constant.
+
+
+
+\subsection{Depletion approximation}
+We assume p and n regions quasi-neutral,
+and the intermediate space charge region to be completely depleted of carriers.
+We further assume all transitions are expressed as step-functions.
+This allows the following simplified equations:
+\begin{align}
+    \rho(x) & = \begin{dcases}
+                    0     & x<-x_p   \\
+                    -qN_a & -x_p<x<0 \\
+                    qN_d  & 0<x<x_n  \\
+                    0     & x_n<x
+                \end{dcases}                            \\
+    E(x)    & =\begin{dcases}
+                   0                                & x<-x_p    \\
+                   -\frac{qN_a}{\varepsilon}(x+x_p) & - x_p<x<0 \\
+                   \frac{qN_d}{\varepsilon}(x-x_n)  & 0<x<x_n   \\
+                   0                                & x_n<x
+               \end{dcases}
+\end{align}
+Where $E$ is found using \eqref{label:eq:def_electric_field_integral}.
+
+\subsection{Electrostatic potential - (Width of the depletion zone)}
+Because of overall charge neutrality
+\begin{equation}
+    qN_ax_p = qN_cx_n
+\end{equation}
+and continuity of the potential at the junction interface
+\begin{equation}
+    \phi_p+\frac{qN_a}{2\varepsilon}x_p^2 = \phi_n-\frac{qN_d}{2\varepsilon}x_n^2
+\end{equation}
+we can find $x_n$ and $x_p$:
+\begin{align}
+    x_n & = \sqrt{\frac{2\varepsilon\phi_BN_a}{qN_d(N_a+N_d)}} \\
+    x_p & = \sqrt{\frac{2\varepsilon\phi_BN_d}{qN_a(N_a+N_d)}}
+\end{align}
+Where $\phi_B$ is the built-in potential, which is the potential over the junction.
+Also, it is the less heavily doped region that defines the junction width.
+It is also in the less heavily doped region that the depletion zone extends farther.
+
+\subsection{Contact potential}
+Although there is a potential accross the diode, it cannot be measured because there are the metal semi-conductor junctions for both p and n regions.
+\begin{equation}
+    \phi_B = \phi_mn+\phi_mp
+\end{equation}
\ No newline at end of file

05_pn_junction_bias.tex

+\section{PN junction bias}
+\subsection{Model}
+
+We can see the junction as a series of resistors as follows.
+(Junction $V_{pn}>0$ is a forward bias, QNR is a quasi-neutral region.)
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,0) to[R, l=$R_{mp}$] ++(2,0)
+        to [R, l=\textnormal{p-QNR}] ++(2,0)
+        to [R, l=\textnormal{SCR}] ++(2,0)
+        to [R, l=\textnormal{n-QNR}] ++(2,0)
+        to [R, l=$R_{mn}$] (10,0);
+        \draw (0,0) to [short] (0,2)
+        to [V, v=$V_{pn}$] (10,2)
+        to [short] (10,0);
+        \begin{scope}[opacity=.5]
+            \draw  (1,-1) rectangle (9,1);
+            \draw (4,-1) -- (4,1);
+            \draw (6,-1) -- (6,1);
+            \draw[dotted] (5,-1) -- (5,1);
+        \end{scope}
+        \node at (2.5,-1.5) {p-QNR};
+        \node at (5,-1.5) {SCR};
+        \node at (7.5,-1.5) {n-QNR};
+        \node at (4.5,-0.5) {$-$};
+        \node at (5.5,-0.5) {$+$};
+    \end{circuitikz}
+\end{center}
+
+Importantly, the SCR resistance ist the most important one and others can be neglected.
+
+\subsection{Space charge region (SCR)}
+In essence, applying a forward/reverse bias effects the depletion region:
+\begin{align}
+    \phi_B            & \rightarrow \phi_B-V_{pn}                                 \\
+    x_n(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_a}{q(N_a+N_d)N_d}}   \\
+    x_p(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_d}{q(N_a+N_d)N_a}}   \\
+    x_d(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_dN_a}{q(N_a+N_d)}}   \\
+    \left|E(V)\right| & =\sqrt{\frac{2q(\phi_B-V)(N_aN_d)}{\varepsilon(N_a+N_d)}}
+\end{align}
+
+In the case of a strongly doped $p^+n$ junction,
+we can approximate the SCR since it exists only in the lesser doped region.
+\begin{equation}
+    x_n(V)=x_{n0}\sqrt{a-\frac{V}{\phi_B}}
+\end{equation}
+
+
+\subsection{PN small-signal capacitance}
+In reverse bias, the PN junction acts as a capacitor.
+\begin{equation}
+    C_{j0} = \frac{\varepsilon}{W_{dep}}
+\end{equation}
+
+So as a function of the bias voltage, we get
+\begin{equation}
+    \begin{split}
+        C_j(V) &= \frac{\varepsilon}{x_c(V)}\\
+        &=\sqrt{\frac{q\varepsilon N_aN_d}{2q(\phi_B-V)(N_a+N_d)}}\\
+        &=\frac{C_{j0}}{\sqrt{1-\frac{V}{\phi_B}}}
+    \end{split}
+\end{equation}
+
+In a strongly asymmetric junction $p^+n$
+\begin{equation}
+    \frac{1}{C_j^2} \approx \frac{2(\phi_B-V)}{q\varepsilon N_d}
+\end{equation}
+
+

06_pn_junction_diode.tex

+\section{PN junction diode}
+\subsection{Carrier concentration under bias}
+Under forward bias,  the net current is no longer zero.
+\begin{equation}
+    \left| J_{drift} \right|<\left| J_{diff} \right|
+\end{equation}
+Which causes injection of minority carriers into the QNR regions giving rise to `high' currents.
+
+
+\subsection{Diode current}
+To calculate the current, we \begin{enumerate}
+    \item Calculate concentration of minority carriers at the edges of SCR
+    \item Calculate minority carrier diffusion current in each QNR for $I_n$ and $I_p$
+    \item Sum the currents $I_n$ and $I_p$
+\end{enumerate}
+
+\subsubsection{Minority carrier conditions}
+We use the quasi-equilibrium equation to misuse equations for equilibrium.
+\begin{align}
+    \frac{n(x_1)}{n(x_2)} & \approx \exp{\frac{q(\phi(x_1)-\phi(x_2))}{kT}}  \\
+    \frac{p(x_1)}{p(x_2)} & \approx \exp{\frac{-q(\phi(x_1)-\phi(x_2))}{kT}}
+\end{align}
+
+So by using $x_n$ and $x_p$ in the above equation we have the following:
+\begin{align}
+    \frac{n(x_n)}{n(-x_p)} & \approx \exp{\frac{q(\phi_B-V)}{kT}}  \\
+    \frac{p(x_n)}{p(-x_p)} & \approx \exp{\frac{-q(\phi_B-V)}{kT}} \\
+    p(-x_p)                & =N_a                                  \\
+    n(x_n)                 & =N_d
+\end{align}
+
+And so we find what we needed:
+\begin{align}
+    n(-x_p) & \approx N_d\exp\frac{q(V-\phi_B)}{kT} \\
+    p(x_n)  & \approx N_a\exp\frac{q(V-\phi_B)}{kT}
+\end{align}
+
+
+Then by using the Boltzman relations \eqref{label:eq:boltzman:phi_n} and \eqref{label:eq:boltzman:phi_p} we find
+\begin{align}
+    \phi_B              & = \frac{kT}{q}\ln\frac{N_dN_a}{n_i^2}      \\
+    \Rightarrow n(-x_p) & \approx \frac{n_i^2}{N_a}\exp\frac{qV}{kT} \\
+    \Rightarrow p(x_n)  & \approx \frac{n_i^2}{N_d}\exp\frac{qV}{kT}
+\end{align}
+
+\subsubsection{Diffusion current in QNR}
+We assume a linear gradient between $n(-W_p)$ and $n(-x_p)$ to easily use \eqref{label:eq:diff_current_n} to find
+\begin{equation}
+    \begin{split}
+        J_n^{diff} &= qD_n\frac{n_p(-x_p)-n_p(-W_p)}{W_p-x_p}\\
+        &= qD_n \frac{\left(\frac{n_i^2}{N_a}\exp{\frac{qV}{kT}}\right)-\frac{n_i^2}{N_a}}{W_p-x_p}\\
+        &= q\frac{n_i^2}{N_a}\frac{D_n}{W_p-x_p}\left(\exp{\frac{qV}{kT}}-1\right)
+    \end{split}
+\end{equation}
+
+\subsubsection{Total diode current}
+\begin{equation}
+    \begin{split}
+        J & = J_n+J_p                                                                                             \\
+        & =q n_i^2 \left( \frac{1}{N_A}\frac{D_n}{W_p-x_p} + \frac{1}{N_D}\frac{D_p}{W_n-x_n} \right)\left(\exp
+        \frac{qV}{kT} - 1\right)
+    \end{split}
+\end{equation}
+Or so simplify
+\begin{align}
+    I   & =I_0\left(\exp\frac{qV}{kT}-1\right)                                 \\
+    I_0 & = A q n_i^2 \left( \frac{D_n}{L_n N_A} + \frac{D_p}{L_p N_D} \right)
+\end{align}
+
+Note that sometimes a non-ideality factor $n$ is used:
+\begin{equation}
+    I  = I_0\left(\exp\frac{qV}{nkT}-1\right)
+\end{equation}
+
+
+\subsection{PN junction reverse bias}
+When applying a reverse bias, the depletion region gets wider and the electric field increases.
+There comes a point when the diode breaks down and destroys itself.
+\begin{equation}
+    W_{dep} = \sqrt{\frac{2\varepsilon}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)\left(V_0+V_R\right)}
+\end{equation}
\ No newline at end of file

07_diode_applications.tex

+\section{Diode applications}
+\subsection{Small signal}
+If $i\ll I$ and $v\ll V$, then we can use the small signal approximation.
+\begin{equation}
+    \begin{split}
+        I+i &= I_0\left(\exp\frac{q(V+v)}{kT}-1\right) \\
+        &=I_0\left(\exp\frac{q(V)}{kT}\exp\frac{q(v)}{kT}-1\right)\\
+        &\approx I_0\left(\exp\frac{q(V)}{kT}\left(1+\frac{qv}{kT}\right)-1\right)\\
+        &= I_0\left( \exp\frac{qV}{kT} - 1 \right) + I_0 \left( \exp\left(\frac{qV}{kT}\right)\frac{qv}{kT} \right)
+    \end{split}
+\end{equation}
+
+Which gives us the smallsignal current
+\begin{align}
+    i   & = \frac{q\left(I+I_0\right)}{kT}v = g_d v \\
+    g_d & = \frac{q\left(I+I_0\right)}{kT}
+\end{align}
+
+
+\subsubsection{Capacitances of small-signal model}
+\begin{figure}[h]
+    \centering
+    \caption*{Small signal model for diode}
+    \begin{circuitikz}
+        \draw (-3,2) to [Do] (-3,0);
+        \draw[->] (-2,1) -- (-1,1);
+        \draw (0,0) to [R,l=$g_d$] (0,2);
+        \draw (2,0) to [C,l=$C_j$] (2,2);
+        \draw (4,0) to [C,l=$C_d$] (4,2);
+        \draw (0,0) to [short] (4,0);
+        \draw (0,2) to [short] (4,2);
+        \draw (2,2) to [short] (2,2.5);
+        \draw (2,-0.5) to [short] (2,0);
+    \end{circuitikz}
+\end{figure}
+\begin{align}
+    C_j    & = \frac{C_{j0}}{\sqrt{1-\frac{V}{\phi_B}}}        \\
+    C_d    & = \frac{q}{kT}\tau_T I                            \\
+    \tau_T & \equiv \text{equivalent transit time of carriers}
+\end{align}
+Where $C_j$ dominates in reverse bias and small forward bias,
+and $C_d$ dominates in large forward bias ($V>\frac{\phi_B}{2}$).
+
+
+\subsection{Large signal}
+\subsubsection{Rectifier}
+This is pretty much trivial.
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,-2) to [sV,l=$V_{in}$] ++(0,4)
+        to [short] ++(2,0)
+        to [Do,v=$V_D$] ++(0,-2)
+        to [R,l=$R_L$,v=$V_o$] ++(2,0)
+        to [Do] ++(0,-2)
+        to [short] (0,-2);
+        draw (4,0) to [Do] ++(0,2)
+        to [short] ++(-2,0);
+        \draw (2,-2) to [Do] ++(0,2);
+        \draw (4,0) to [Do] ++(0,2)
+        to [short] ++(-2,0);
+    \end{circuitikz}
+\end{center}
+\begin{equation}
+    V_o = V_{in}-2V_D
+\end{equation}
+
+\subsubsection{Voltage regulator}
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,0) to [V,v<=$10\pm1\ V$] (0,8)
+        to [short] ++(2,0)
+        to [R,l=$R_1$] ++(0,-2)
+        to [short] ++(0,-2)
+        to [Do] ++(0,-1)
+        to [Do] ++(0,-1)
+        to [Do] ++(0,-1)
+        to [short] ++(0,-1)
+        to [short] (0,0);
+        \draw (2,5) to [nos] ++(2,0)
+        to [R,l=$R_2$] ++(0,-5)
+        to [short] (0,0);
+    \end{circuitikz}
+\end{center}
+
+How to go about it:
+\begin{align}
+    I          & = \frac{V_{in}-\sum V_D}{R_1}                                            \\
+    r_d        & =\left[\frac{\mathrm{d}I_D}{\mathrm{d}V_d}\right]^{-1} =\frac{nV_t}{I_0} \\
+    r          & = \sum r_d                                                               \\
+    \Delta V_o & = \Delta V_{in}\frac{r}{r+R_2}                                           \\[1em]
+    n          & \equiv\text{non-ideality factor}                                         \\
+    V_t        & = \frac{kT}{q}
+\end{align}
+
+Once the load is connected and draws current, we have a further small variation:
+\begin{align}
+    I_{load}   & = \frac{\sum V_D}{R_2} \\
+    \Delta V_o & = I_{load}r
+\end{align}
+
+\subsection{Special diode types}
+\subsubsection{Zener}
+Is heavily doped making the depletion layer extremely thing, and thus allowing for QM tunneling in reverse biased diode.
+(In this case known as band-to-band tunneling.)
+
+\begin{figure}[h]
+    \centering
+    \caption[short]{Band-bending zener diode}
+    \includegraphics[width=.75\textwidth]{imgs/zener_diode_band_bending.png}
+\end{figure}
+The voltage at which the diode starts conducting is called the zener voltage $V_Z$.
+The diode then has a low resistance $R_Z$.
+
+\subsubsection{Esaki}
+Heavily doped with the tunneling effect in forward bias.
+\begin{figure}[h]
+    \centering
+    caption{Esaki tunnel diode}
+    \includegraphics[width=.75\textwidth]{imgs/esaki_tunnel_diode.png}
+\end{figure}
+
+\subsubsection{Schottky}
+A schottky diode has $I_0$ $10^3$ to $10^8$ times bigger than a PN diode.
+Preferred in low voltage high current applications.
+
+\subsubsection{Photodiodes}
+\begin{equation}
+    I = I_0\left( e^{\frac{qV}{kT}} - 1\right)-I_{photo}
+\end{equation}
\ No newline at end of file

08_bjt.tex

+\section{Bipolar junction transistor (BJT)}
+\begin{figure}[h]
+    \centering
+    \caption{BJT}
+    \includegraphics[width=.75\textwidth]{imgs/bjt_terminals_and_functioning.png}
+\end{figure}
+
+But what's going on?
+If $V_{BE}>0$ injection of electrons from E to B, of holes from B to E.
+If  $V_{BC}<0$  extraction of electrons from B to C, of holes from C to B.
+
+
+\subsection{BJT characteristics}
+\begin{align}
+    I_E & = -I_C-I_B \\
+    \begin{split}
+        \beta &= \frac{I_C}{I_B}
+        =\frac{n_{pB_0}\frac{D_n}{W_B}}{p_{nE_0}\frac{D_p}{W_E}}\\
+        &=  \frac{N_{dE} D_n W_E}{N_{aB} D_p W_B}
+    \end{split}
+\end{align}
+
+Collector current,
+focus on electron diffusion in base:
+\begin{align}
+    n_{pB}(0) & =n_{pB_0}e^{\frac{qV_{BE}}{kT}}                                                     \\
+    n_{pB}(x) & =n_{pB}(0)(1-\frac{x}{W_B})                                                         \\[1em]
+    \begin{split}
+        J_{nB} &= qD_n\frac{\mathrm{d} n_{pB}}{\mathrm{d}x}\\
+        &= -qD_n\frac{n_{pB}(0)}{W_B}
+    \end{split} \\
+    \begin{split}
+        I_C &=-J_{nB}A_E\\
+        &=qA_E\frac{E_n}{W_B}n_{pB_0}e^{\frac{qV_{BE}}{kT}}
+    \end{split}                        \\
+    I_C       & = I_Se^{\frac{qV_{BE}}{kT}}
+\end{align}
+Base current,
+focus on hole injection and recombination in emitter:
+\begin{align}
+    p_{nE}(-x_{BE})     & =p_{nE_0}e^{-\frac{qV_{BE}}{kT}}                                                                                        \\
+    p_{nE}(-W_E-x_{BE}) & =p_{nE_0}                                                                                                               \\
+    p_{nE}(x)           & =\left[ p_{nE}(-x_{BE}-p_{nE_0}) \right]\left( 1+\frac{x+x_{BE}}{W_E} \right)+P_{nE_0} & \leftarrow \text{Hole Profile} \\[1em]
+    \begin{split}
+        J_{pE}&=-qD_p\frac{\mathrm{d}p_{nE}}{\mathrm{d}x}\\
+        &=-qD_p\frac{p_{nE(-x_{BE})-p_{nE_0}}}{W_E}
+    \end{split}                                                                                              \\
+    \begin{split}
+        I_B&=-J_{pE}A_E\\
+        &=qA_E\frac{D_p}{W_E}p_{nE_0}\left( e^{\frac{qV_{VE}}{kT}} -1 \right)
+    \end{split}                     \\
+    I_B                 & =\frac{I_S}{\beta}\left(e^{\frac{qV_{BE}}{kT}}-1\right)                                                                 \\
+    I_B\approx\frac{I_C}{\beta}
+\end{align}
+
+\subsubsection{`Good' transistor}
+We want  collector and emitter current to be identical and so we define $\alpha$ as measurement of how close we are:
+\begin{align}
+    I_C   & =-\alpha I_E                \\
+          & =\alpha\left(I_B+I_C\right) \\
+          & =\frac{\alpha}{1-\alpha}I_B \\
+          & =\beta I_B                  \\
+    \beta & =\frac{\alpha}{1-\alpha}
+\end{align}
+
+\subsection{Summary forward active}
+\begin{align}
+    I_C & = I_Se^{\frac{qV_{BE}}{kT}}                              \\
+    I_B & = \frac{I_S}{\beta}\left(e^{\frac{qV_{BE}}{kT}}-1\right) \\
+    I_E & = -I_C-I_B
+\end{align}
+
+For reverse, it is the same but $\beta_R\approx [0.1,5]\ll\beta$.
+
+\subsection{Summary cut-off}
+\begin{alignat}{2}
+    I_{B1} & = -\frac{I_S}{\beta}  &  & =-I_E \\
+    I_{B2} & =-\frac{I_S}{\beta_R} &  & =-I_C
+\end{alignat}
+
+\subsection{Summary saturation}
+\begin{align}
+    I_C & =I_S\left(e^{\frac{qV_{BE}}{kT}} - e^{\frac{qV_{BC}}{kT}}\right)-\frac{I_S}{\beta_R}\left( e^\frac{qV_{BC}}{kT} - 1 \right)  \\
+    I_B & =\frac{I_S}{\beta}\left( e^{\frac{qV_{BE}}{kT}}-1 \right)+\frac{I_S}{\beta_R}\left( e^{\frac{qV_{BC}}{kT}} -1 \right)        \\
+    I_E & =\frac{I_S}{\beta}\left(e^{\frac{qV_{BE}}{kT}} - 1\right) - I_S\left( e^{\frac{qV_{BE}}{kT}} -e^{\frac{qV_{BC}}{kT}} \right)
+\end{align}
+
+\subsection{Ebers-Moll model}
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,0) node[left] {B} to [short,*-] ++(1,0)
+        to [Do,l=$\frac{I_S}{\beta_R}\left( e^{\frac{qV_{BC}}{kT}} -1 \right)$] ++(0,2)
+        to [short] ++(2,0)
+        to [I,l=$I_S\left( e^{\frac{qV_{BE}}{kT}} - e^{\frac{qV_{BC}}{kT}} \right)$,i=$$] ++(0,-4)
+        to [short] ++(-1,0);
+        \draw (1,0) to [Do,l_=$\frac{I_S}{\beta}\left( e^{\frac{qV_{BE}}{kT}}-1 \right)$] ++(0,-2)
+        to [short] ++(1,0)
+        to [short,-*] ++(0,-1) node [below] {E};
+        \draw (2,2) to [short,-*] ++(0,1) node[above] {C};
+    \end{circuitikz}
+\end{center}
+
+\subsection{Early effect}
+With increasing $V_{CE}$, the depletion region inceases.
+To not have to deal with that, we introduce a correction factor
+\begin{equation}
+    I_C = I_S e^{\frac{V_{BE}}{V_{th}}}\left(1+\frac{V_{CE}}{V_A}\right)
+\end{equation}
+
+\subsection{Transfer characteristics}
+We evaluate the transistor at its operating point ($OP$ or $Q=(V_{BE},V_{CE})$) to find the transconductance $g_m$.
+\begin{equation}
+    \label{label:eq:bjt_transconductance}
+    g_m = \left. \frac{\partial i_C}{\partial V_{BE}} \right|_{OP} = \frac{qI_C}{kT}
+\end{equation}
+

09_bjt_small_signal.tex

+\section{BJT small signal}
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,2) node[left]{B} to [short,i=$i_b$ , *-] ++(1,0)
+        to [R,l=$r_{be}$,v=$v_{be}$] ++(0,-2)
+        to [short] ++(2,0)
+        to [cI,l_=$g_mv_{be}$,i<=$$] ++(0,2)
+        to [short] ++(2,0)
+        to [R,l=$r_{o}$] ++(0,-2)
+        to [short] ++(-2,0)
+        to [short,i=$i_e$,-*] ++(0,-1)node[below]{E}
+        ++(2,3)
+        to [short,i<=$i_c$,-*] ++(1,0)  node[right]{C};
+        % to [open,v_=$v_{ce}$] ++(0,-2);
+    \end{circuitikz}
+\end{center}
+
+\subsection{Transistor amplifiers, 2 step analysis}
+
+
+\begin{enumerate}
+    \item DC analysis
+          \begin{enumerate}
+              \item Get DC equivalent circuit (rm C, cc L)
+              \item Find OP using large signal model of transistor
+          \end{enumerate}
+    \item AC analysis
+          \begin{enumerate}
+              \item Get AC equivalent circuit (cc C, rm L)
+              \item Replace transistor with small signal model
+              \item Use small-signal AC model to find characteristics
+              \item Combine AC and DC analysis
+          \end{enumerate}
+\end{enumerate}

format.tex

@@ -27,6 +27,11 @@
 \usepackage{subcaption}
 \usepackage{graphicx, xcolor}
 
+
+
+\usepackage[european,straightvoltages]{circuitikz}
+\usepackage{tikz}
+
 \usepackage{url}
 \usepackage[pdfusetitle]{hyperref}
 \hypersetup{
@@ -50,3 +55,7 @@
 \rhead{\today}
 \lhead{}
 \cfoot{\thepage}
+
+
+
+\numberwithin{equation}{subsection}

semiconductor_summary.tex

@@ -2,7 +2,7 @@
 
 
 
-\title{Micro and nanoelectronic devices \\ PHYS-201(d)}
+\title{Micro and nanoelectronic devices \\ EE-333}
 \author{Simon Thür}
 \date{ \today \\ \small{ Version 0.1 } }
 
@@ -13,24 +13,38 @@
     \thispagestyle{empty}
 
     This summary for
-    \href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{Micro and nanoelectronic devices}
-    © 2021 by
+    \href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{micro and nanoelectronic devices}
+    © 2023 by
     \href{https://gitlab.epfl.ch/sthuer}{Simon Thür}
     is licensed under
     \href{http://creativecommons.org/licenses/by/4.0/}{CC BY 4.0}.
     To view a copy of this license, visit
     \url{http://creativecommons.org/licenses/by/4.0/}
 
-
+    To see the source code, visit \href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{https://gitlab.epfl.ch/sthuer/semiconductors\_summary}
 
 
     \section*{Introductory remarks}
     This is a summary of Prof Ionescu's course on semiconductor devices.
+    It follows the course structure but does not retain the same enumeration.
+    Since the reference material is in english, this summary is also in english.
+
+    The aim of this summary is to provide a denser overview of the course material.
+    Specifically, it is intended to be used as a reference for the exam,
+    which is why it has only few remarks and focuses mainly on equations with little regard for how they were derived.
+
 
 \end{titlepage}
 
 \tableofcontents
 
-\include{01_fundamentals}
-
+\include{01_fundamentals.tex}
+\include{02_carrier_transport.tex}
+\include{03_pn_junction_basics.tex}
+\include{04_pn_junction.tex}
+\include{05_pn_junction_bias.tex}
+\include{06_pn_junction_diode.tex}
+\include{07_diode_applications.tex}
+\include{08_bjt.tex}
+\include{09_bjt_small_signal.tex}
 \end{document}