From 85e216cad78b8bfd176d9038abf1386e7ed85b64 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Simon=20Th=C3=BCr?= <thuer.simon@hotmail.com> Date: Sat, 8 Apr 2023 01:26:38 +0200 Subject: [PATCH] add intro to PN junction --- 01_fundamentals.tex | 1 + 02_carrier_transport.tex | 5 +- 03_pn_junction_basics.tex | 76 ++++++++++++++++++++++++ 04_pn_junction.tex | 46 ++++++++++++++ imgs/pn_carrier_profile_equilibrium.png | Bin 0 -> 32866 bytes imgs/pn_fermi_level_band_bending.png | Bin 0 -> 104280 bytes semiconductor_summary.tex | 16 +++-- 7 files changed, 138 insertions(+), 6 deletions(-) create mode 100644 03_pn_junction_basics.tex create mode 100644 04_pn_junction.tex create mode 100644 imgs/pn_carrier_profile_equilibrium.png create mode 100644 imgs/pn_fermi_level_band_bending.png diff --git a/01_fundamentals.tex b/01_fundamentals.tex index 1587fd9..0952adb 100644 --- a/01_fundamentals.tex +++ b/01_fundamentals.tex @@ -15,6 +15,7 @@ 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} diff --git a/02_carrier_transport.tex b/02_carrier_transport.tex index cccd15b..87ee148 100644 --- a/02_carrier_transport.tex +++ b/02_carrier_transport.tex @@ -96,13 +96,14 @@ ergo the double negative for electron diffusion.) \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} \\ + 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 diff --git a/03_pn_junction_basics.tex b/03_pn_junction_basics.tex new file mode 100644 index 0000000..e4647a0 --- /dev/null +++ b/03_pn_junction_basics.tex @@ -0,0 +1,76 @@ +\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 a rule of thumb for the potential: +\begin{align} + \phi & =\frac{kT}{q}\ln\frac{n}{n_i} \\ + \phi & = -\frac{kT}{q}\ln\frac{p}{n_i} +\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} diff --git a/04_pn_junction.tex b/04_pn_junction.tex new file mode 100644 index 0000000..b84b162 --- /dev/null +++ b/04_pn_junction.tex @@ -0,0 +1,46 @@ +\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}. + diff --git a/imgs/pn_carrier_profile_equilibrium.png b/imgs/pn_carrier_profile_equilibrium.png new file mode 100644 index 0000000000000000000000000000000000000000..37ef88669ee31d961b4a7aa0da5b01689036801d GIT binary patch literal 32866 zcmeFadpy+n`#=1)YT35MMA%Ixbh1S{k77m{X)A|T(K?rH3n`P5VvG)(nTXmBgp5?C 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devices} + \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 @@ -21,11 +21,18 @@ 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} @@ -33,5 +40,6 @@ \include{01_fundamentals} \include{02_carrier_transport} - +\include{03_pn_junction_basics} +\include{04_pn_junction} \end{document} -- GitLab