Semiconductor Science and Light-Emitting Diodes

An interactive companion to Chapter 3 — Basics of Optoelectronics

Author

Philippe Velha

Published

October 16, 2026

Preface

This companion follows the slides of Chapter 3: Semiconductor Science and Light-Emitting Diodes (Basics of Optoelectronics, University of Trento) section by section. The slides tell the story; this book lets you compute it. Every figure is made by a few lines of Python that you can unfold (“Show the Python”), copy and modify. The live panels (marked with a blue bar on the left) recompute directly in your browser. They need no installation.

Interactive explorers in Google Colab (no installation, Google account needed).

Where this chapter sits in the course

Chapter 1 described light as a wave and Chapter 2 guided it in fibers. Chapter 3 builds the source. It starts from the band picture of a crystal, counts electrons and holes, assembles a pn junction, and ends with the light-emitting diode: its spectrum, its efficiency and how fast it can be switched. The same physics (bands, quasi-Fermi levels, injection, recombination) is the basis of laser diodes (Ch. 4) and photodetectors (Ch. 5), which are pn junctions run in the other direction.

How the book is organised

Chapter	Slides	What you can play with
Energy bands and holes	3–9, 33–39	Kronig–Penney bands, Fermi energy of metals, direct vs indirect gaps
Carrier statistics	10–17, 25	$g(E)f(E)$ in the bands, $N_c$, $N_v$, $n_i(T)$, intrinsic Fermi level
Doping, Fermi level and conductivity	18–32	doping explorer, compensation, degeneracy, mobility, freeze-out
The pn junction at equilibrium	40–47, 82–85	charge, field, potential and band diagram; exact Poisson solution
The diode: forward and reverse bias	48–69, 75–80	Shockley and recombination currents, ideality factor, GaAs example
Capacitance and small-signal model	70–74	$C_\text{dep}(V)$, $r_d$, $C_\text{diff}$, Mott–Schottky doping profiling
LED principle, heterojunctions and materials	81–97, 114–119	bandgap engineering, alloy colour picker, lattice matching
Emission spectrum	98–109	spectrum vs temperature, $3k_BT$ linewidth, Varshni shift
Quantum wells	110–113	finite-well solver, wavefunctions, 2D density of states
Efficiency, brightness and white LEDs	120–133, 139–143	escape cone, IQE/EQE/PCE chain, lumens, white-LED spectrum
LED types, drive circuits and speed	134–138, 144–151	fiber coupling, bias resistor, modulation bandwidth, rise time
Projects, toolbox and connections	—	Python packages, mini-projects, links to other chapters

Each chapter recomputes the worked examples of the slides, then offers Going deeper questions, Complements (reading), and Try it yourself exercises. A formula sheet is at the end.

Units in the code

The slides quote concentrations in cm$^{-3}$ and mobilities in cm$^2$ V$^{-1}$ s$^{-1}$, as most semiconductor texts do. The code works in SI units (m$^{-3}$, m$^2$ V$^{-1}$ s$^{-1}$) to avoid factor-of-$10^6$ mistakes in $\varepsilon$, $e$ and $h$. Energies are in eV. Two constants convert: sl.CM3 = 1e6 (cm$^{-3}$ → m$^{-3}$) and sl.CM2 = 1e-4 (cm$^2$ → m$^2$). At 300 K, $k_BT = 25.85$ meV.

Slide numbers

Slide numbers refer to Chapter3_2024pptx.pptx. The PDF export omits the empty slide 2, so PDF page = slide − 1 from slide 3 onwards.

Using the code

All functions live in a small, readable module, semilib.py, which you can import from your own notebooks:

import semilib as sl
sl.carriers(1e16*sl.CM3, 0, sl.MATERIALS["Si"]["ni"])   # n, p of n-type Si (m^-3)
sl.qw_finite(12e-9, 0.30, 0.067)                        # bound levels of a GaAs quantum well (eV)

See the appendix Running the notebooks for installation.

# Preface {.unnumbered} This companion follows the slides of **Chapter 3: Semiconductor Science and Light-Emitting Diodes** (Basics of Optoelectronics, University of Trento) section by section. The slides tell the story; this book lets you **compute it**. Every figure is made by a few lines of Python that you can unfold ("Show the Python"), copy and modify. The **live panels** (marked with a blue bar on the left) recompute directly in your browser. They need no installation. [![Open in Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/{{< var github_user >}}/{{< var repo >}}/blob/main/{{< var subdir >}}/notebooks/interactive_explorers.ipynb)  Interactive explorers in Google Colab (no installation, Google account needed). ## Where this chapter sits in the course Chapter 1 described light as a wave and Chapter 2 guided it in fibers. Chapter 3 builds the **source**. It starts from the band picture of a crystal, counts electrons and holes, assembles a pn junction, and ends with the light-emitting diode: its spectrum, its efficiency and how fast it can be switched. The same physics (bands, quasi-Fermi levels, injection, recombination) is the basis of laser diodes (Ch. 4) and photodetectors (Ch. 5), which are pn junctions run in the other direction. ## How the book is organised | Chapter | Slides | What you can play with | |---|---|---| | [Energy bands and holes](bands.qmd) | 3–9, 33–39 | Kronig–Penney bands, Fermi energy of metals, direct vs indirect gaps | | [Carrier statistics](statistics.qmd) | 10–17, 25 | $g(E)f(E)$ in the bands, $N_c$, $N_v$, $n_i(T)$, intrinsic Fermi level | | [Doping, Fermi level and conductivity](doping.qmd) | 18–32 | doping explorer, compensation, degeneracy, mobility, freeze-out | | [The pn junction at equilibrium](junction.qmd) | 40–47, 82–85 | charge, field, potential and band diagram; exact Poisson solution | | [The diode: forward and reverse bias](diode.qmd) | 48–69, 75–80 | Shockley and recombination currents, ideality factor, GaAs example | | [Capacitance and small-signal model](smallsignal.qmd) | 70–74 | $C_\text{dep}(V)$, $r_d$, $C_\text{diff}$, Mott–Schottky doping profiling | | [LED principle, heterojunctions and materials](led-principle.qmd) | 81–97, 114–119 | bandgap engineering, alloy colour picker, lattice matching | | [Emission spectrum](spectrum.qmd) | 98–109 | spectrum vs temperature, $3k_BT$ linewidth, Varshni shift | | [Quantum wells](quantum-wells.qmd) | 110–113 | finite-well solver, wavefunctions, 2D density of states | | [Efficiency, brightness and white LEDs](efficiency.qmd) | 120–133, 139–143 | escape cone, IQE/EQE/PCE chain, lumens, white-LED spectrum | | [LED types, drive circuits and speed](led-drive.qmd) | 134–138, 144–151 | fiber coupling, bias resistor, modulation bandwidth, rise time | | [Projects, toolbox and connections](ideas.qmd) | — | Python packages, mini-projects, links to other chapters | Each chapter recomputes the worked examples of the slides, then offers **Going deeper** questions, **Complements** (reading), and **Try it yourself** exercises. A **formula sheet** is at the end. ::: {.callout-note} ## Units in the code The slides quote concentrations in cm$^{-3}$ and mobilities in cm$^2$ V$^{-1}$ s$^{-1}$, as most semiconductor texts do. The code works in **SI units** (m$^{-3}$, m$^2$ V$^{-1}$ s$^{-1}$) to avoid factor-of-$10^6$ mistakes in $\varepsilon$, $e$ and $h$. Energies are in **eV**. Two constants convert: `sl.CM3 = 1e6` (cm$^{-3}$ → m$^{-3}$) and `sl.CM2 = 1e-4` (cm$^2$ → m$^2$). At 300 K, $k_BT = 25.85$ meV. ::: ::: {.callout-tip} ## Slide numbers Slide numbers refer to `Chapter3_2024pptx.pptx`. The PDF export omits the empty slide 2, so **PDF page = slide − 1** from slide 3 onwards. ::: ## Using the code All functions live in a small, readable module, [`semilib.py`](semilib.py), which you can import from your own notebooks: ```python import semilib as sl sl.carriers(1e16*sl.CM3, 0, sl.MATERIALS["Si"]["ni"]) # n, p of n-type Si (m^-3) sl.qw_finite(12e-9, 0.30, 0.067) # bound levels of a GaAs quantum well (eV) ``` See the appendix [Running the notebooks](running.qmd) for installation.