13 Projects, toolbox and connections

13.1 How Chapter 3 connects to the rest of the course

Table 13.1: Where Chapter 3 concepts come from and where they reappear.

Chapter 3 concept	Where it comes from / where it goes
Photon energy $E_g=h\nu$, absorption edge	Ch. 1: complex index and absorption; Ch. 5: photodetector cut-off wavelength
Escape cone, Fresnel transmittance at the chip surface	Ch. 1: TIR and Fresnel equations; extraction is Snell’s law in disguise
Double heterostructure, optical confinement factor	Ch. 2: planar dielectric waveguide ($V$-number, confinement)
DBR LED, resonant-cavity LED	Ch. 1: Bragg mirrors and Fabry–Perot cavities
LED linewidth $\Delta\lambda\approx3k_BT\lambda^2/hc$	Ch. 1: coherence length; Ch. 2: chromatic dispersion limits the bit rate of LED links
Coupling into fibers, NA²	Ch. 2: numerical aperture, multimode and plastic fibers
Quasi-Fermi levels, injection, QWs	Ch. 4: population inversion and optical gain in laser diodes ($E_{Fn}-E_{Fp}>h\nu$)
pn junction under reverse bias, generation current, $C_\text{dep}$	Ch. 5: pin and avalanche photodiodes, dark current, RC bandwidth
Carrier lifetime and modulation bandwidth	Ch. 4: relaxation oscillations of lasers; Ch. 6: modulators
Quantum-confined Stark effect	Ch. 6: electro-absorption modulators

13.2 Mini-projects

For students who want to go further. Each reuses semilib.py and can be done in one to three weeks.

#	project	skills	difficulty
1	Characterise a real LED: measure I–V (source-meter), fit $I_o$, $\eta$, $R_s$ with `sl.diode_iv_rs`; measure the spectrum at two currents and estimate the junction temperature from the peak shift (Chapter 9)	experiment, fitting	★
2	1D drift–diffusion simulator: extend `poisson_pn` (Section 4.5) with quasi-Fermi levels and Scharfetter–Gummel currents; compute the I–V curve and compare with the Shockley equation	numerics, device physics	★★★
3	Schrödinger solver for heterostructures: finite differences for arbitrary potentials (graded wells, coupled wells, superlattices); compare with `sl.qw_finite` and the Kronig–Penney model	quantum mechanics, linear algebra	★★
4	Monte Carlo light extraction: ray-trace photons in a cuboid chip with absorption, Fresnel reflection, textured top surface and photon recycling; reproduce the 1.6 % bare-chip value and find how texturing improves it	ray optics, statistics	★★
5	White-LED designer: optimise blue fraction and phosphor peak for maximum efficacy at a given CRI and CCT with `colour-science`	photometry, optimisation	★★
6	Efficiency droop: fit the ABC model to published EQE–current curves of blue LEDs; discuss Auger recombination versus carrier leakage	data analysis, physics	★★
7	Visible-light communication link: simulate an LED (first-order response), a photodiode and noise; compare OOK with OFDM bit loading, then test with a lab LED and an oscilloscope	signal processing	★★★
8	Band structure from first principles: download Si and GaAs band structures from the Materials Project (`mp-api`) and extract $E_g$, $k_0$ and the effective masses	databases, materials science	★★

13.3 Python toolbox

All worked examples run with the scientific core only. The packages below extend them.

Table 13.2: Optional Python packages for Chapter 3.

Package	Install	What it is good for	Sections
`numpy`, `scipy`, `matplotlib`, `pandas`	`pip install ...`	core numerics, root finding (`brentq`), sparse Newton solvers, Lambert W	all
`colour-science`	`pip install colour-science`	CIE colour matching, chromaticity, CCT and CRI of LED spectra	11, 8
`solcore`	`pip install solcore`	semiconductor materials database, Poisson–drift–diffusion and QW solvers (built for solar cells, works for LEDs)	4, 5, 10
`devsim`	`pip install devsim`	open-source TCAD: drift–diffusion device simulation in 1D/2D/3D	4, 5
`aestimo`	from GitHub	self-consistent Schrödinger–Poisson solver for heterostructures	10
`lmfit`	`pip install lmfit`	robust curve fitting with parameter bounds (diode I–V, Arrhenius plots, spectra)	5, 9
`pint`	`pip install pint`	unit-aware calculations (cm⁻³ vs m⁻³ mistakes)	all
`mp-api`, `pymatgen`	`pip install mp-api pymatgen`	band structures and properties from the Materials Project	1
`PySpice`	`pip install PySpice`	SPICE simulation of LED driver circuits and speed-up capacitors	12
`rayflare`	`pip install rayflare`	ray tracing and TMM for textured surfaces (light extraction)	11
Meep	`conda install -c conda-forge pymeep`	FDTD simulation of extraction from photonic-crystal and textured LEDs	11
`tmm` (or `optolib.tmm` of Ch. 1)	`pip install tmm`	DBR mirrors and resonant-cavity LEDs	11

13.4 Improvements to the course

From band diagram to device in one model. The exact Poisson solver (Section 4.5) shows students that the depletion approximation, the law of the junction and the built-in voltage are the same physics. It is worth five minutes of class time.
“Predict, then compute.” Before moving a slider, ask: “If I double the lifetime, does the crossover voltage between η = 2 and η = 1 go up or down?”, “Which side of a p⁺n junction holds the depletion region?”, “Does a thinner quantum well emit bluer or redder light?”.
One running device. Follow a single GaAs LED through the chapter: doping and Fermi levels → junction → I–V (slides 75–80) → spectrum (slides 107–108) → extraction (slides 122–123) → bandwidth. All the numbers are already in the companion.
Lab connection. A cheap lab (an LED, a resistor, a USB spectrometer and a multimeter) covers I–V fitting, ideality factor, spectrum versus temperature, Varshni shift and luminous flux.
Modern content. The slides stop at 2014-era devices. Possible updates: microLED displays, efficiency droop, UV-C LEDs for disinfection, Li-Fi (IEEE 802.11bb, 2023), perovskite and quantum-dot LEDs.

13.5 Notes for the instructor: checks on the slides

While recomputing the examples, everything matched the slides except these points:

Slide numbering: the PDF export (Chapter3_2024pptx.pdf) omits the empty slide 2, so PDF page = slide − 1 from slide 3 on. The companion uses the .pptx numbers.
Slide 22: titled “Extrinsic Semiconductors: n-Type” and writes “p ≈ Nd”; it describes p-type doping with $p\approx N_a$.
Slide 25 (Table 3.1): $N_v$ of Ge is printed as $6.0\times10^{19}$ cm⁻³. The mass action law with $n_i=2.3\times10^{13}$ cm⁻³ requires $6.0\times10^{18}$ cm⁻³ (the value of Kasap’s book).
Slides 65–66: the ideality factor symbol η is rendered as a missing glyph (“□ is an ideality factor”, “e/(□k_BT)”).
Slide 69: the model with Varshni $E_g(T)$ gives Arrhenius slopes of about 0.86 and 0.40 eV, higher than the measured 0.63 and 0.33 eV. The factor of two is reproduced. A good discussion point (Chapter 5).
Slide 99: “Fermi-Direct function” should be “Fermi–Dirac”.
Slide 107: the absolute wavelengths (864.2 and 852.4 nm) are 0.9 nm too long because of rounded constants ($hc/e=1.2409$ instead of 1.2398 eV µm). The shift of 11.8 nm is correct.
Slide 112: “d = 12×10⁻⁹ nm” should read 12×10⁻⁹ m. The finite-well values quoted (0.022, 0.088, 0.186 eV) are about 15 % below a direct solution with $m^*=0.067m_e$ (0.026, 0.101, 0.216 eV); they correspond to $m^*\approx0.08m_e$ (Chapter 10).
Slide 133: the formula is written with “633 lm W⁻¹”; the numbers use the correct 683 lm W⁻¹.
Slide 146: the caption sentence “At $f_c$, $[I_\text{ph}(f)/I_\text{ph}(0)]^2$ is 0.5. However, it is 0.707 at a lower frequency $f_c$” is confusing. Suggested wording: “At $f_c$ the optical power is 0.707 of its low-frequency value and the detected electrical power is 0.5; the optical power reaches 0.5 at the higher frequency $f_\text{op}=\sqrt3f_c$.”

# Projects, toolbox and connections {#sec-ideas} ## How Chapter 3 connects to the rest of the course | Chapter 3 concept | Where it comes from / where it goes | |---|---| | Photon energy $E_g=h\nu$, absorption edge | Ch. 1: complex index and absorption; Ch. 5: photodetector cut-off wavelength | | Escape cone, Fresnel transmittance at the chip surface | Ch. 1: TIR and Fresnel equations; extraction is Snell's law in disguise | | Double heterostructure, optical confinement factor | Ch. 2: planar dielectric waveguide ($V$-number, confinement) | | DBR LED, resonant-cavity LED | Ch. 1: Bragg mirrors and Fabry–Perot cavities | | LED linewidth $\Delta\lambda\approx3k_BT\lambda^2/hc$ | Ch. 1: coherence length; Ch. 2: chromatic dispersion limits the bit rate of LED links | | Coupling into fibers, NA² | Ch. 2: numerical aperture, multimode and plastic fibers | | Quasi-Fermi levels, injection, QWs | Ch. 4: population inversion and optical gain in laser diodes ($E_{Fn}-E_{Fp}>h\nu$) | | pn junction under reverse bias, generation current, $C_\text{dep}$ | Ch. 5: pin and avalanche photodiodes, dark current, RC bandwidth | | Carrier lifetime and modulation bandwidth | Ch. 4: relaxation oscillations of lasers; Ch. 6: modulators | | Quantum-confined Stark effect | Ch. 6: electro-absorption modulators | : Where Chapter 3 concepts come from and where they reappear. {#tbl-links tbl-colwidths="[45,55]"} ## Mini-projects For students who want to go further. Each reuses `semilib.py` and can be done in one to three weeks. | # | project | skills | difficulty | |---|---|---|---| | 1 | **Characterise a real LED**: measure I–V (source-meter), fit $I_o$, $\eta$, $R_s$ with `sl.diode_iv_rs`; measure the spectrum at two currents and estimate the junction temperature from the peak shift (@sec-spectrum) | experiment, fitting | ★ | | 2 | **1D drift–diffusion simulator**: extend `poisson_pn` (@sec-poisson) with quasi-Fermi levels and Scharfetter–Gummel currents; compute the I–V curve and compare with the Shockley equation | numerics, device physics | ★★★ | | 3 | **Schrödinger solver for heterostructures**: finite differences for arbitrary potentials (graded wells, coupled wells, superlattices); compare with `sl.qw_finite` and the Kronig–Penney model | quantum mechanics, linear algebra | ★★ | | 4 | **Monte Carlo light extraction**: ray-trace photons in a cuboid chip with absorption, Fresnel reflection, textured top surface and photon recycling; reproduce the 1.6 % bare-chip value and find how texturing improves it | ray optics, statistics | ★★ | | 5 | **White-LED designer**: optimise blue fraction and phosphor peak for maximum efficacy at a given CRI and CCT with `colour-science` | photometry, optimisation | ★★ | | 6 | **Efficiency droop**: fit the ABC model to published EQE–current curves of blue LEDs; discuss Auger recombination versus carrier leakage | data analysis, physics | ★★ | | 7 | **Visible-light communication link**: simulate an LED (first-order response), a photodiode and noise; compare OOK with OFDM bit loading, then test with a lab LED and an oscilloscope | signal processing | ★★★ | | 8 | **Band structure from first principles**: download Si and GaAs band structures from the Materials Project (`mp-api`) and extract $E_g$, $k_0$ and the effective masses | databases, materials science | ★★ | ## Python toolbox All worked examples run with the scientific core only. The packages below extend them. | Package | Install | What it is good for | Sections | |---|---|---|---| | `numpy`, `scipy`, `matplotlib`, `pandas` | `pip install ...` | core numerics, root finding (`brentq`), sparse Newton solvers, Lambert W | all | | `colour-science` | `pip install colour-science` | CIE colour matching, chromaticity, CCT and CRI of LED spectra | [-@sec-efficiency], [-@sec-led] | | `solcore` | `pip install solcore` | semiconductor materials database, Poisson–drift–diffusion and QW solvers (built for solar cells, works for LEDs) | [-@sec-junction], [-@sec-diode], [-@sec-qw] | | `devsim` | `pip install devsim` | open-source TCAD: drift–diffusion device simulation in 1D/2D/3D | [-@sec-junction], [-@sec-diode] | | `aestimo` | from GitHub | self-consistent Schrödinger–Poisson solver for heterostructures | [-@sec-qw] | | `lmfit` | `pip install lmfit` | robust curve fitting with parameter bounds (diode I–V, Arrhenius plots, spectra) | [-@sec-diode], [-@sec-spectrum] | | `pint` | `pip install pint` | unit-aware calculations (cm⁻³ vs m⁻³ mistakes) | all | | `mp-api`, `pymatgen` | `pip install mp-api pymatgen` | band structures and properties from the Materials Project | [-@sec-bands] | | `PySpice` | `pip install PySpice` | SPICE simulation of LED driver circuits and speed-up capacitors | [-@sec-drive] | | `rayflare` | `pip install rayflare` | ray tracing and TMM for textured surfaces (light extraction) | [-@sec-efficiency] | | Meep | `conda install -c conda-forge pymeep` | FDTD simulation of extraction from photonic-crystal and textured LEDs | [-@sec-efficiency] | | `tmm` (or `optolib.tmm` of Ch. 1) | `pip install tmm` | DBR mirrors and resonant-cavity LEDs | [-@sec-efficiency] | : Optional Python packages for Chapter 3. {#tbl-toolbox tbl-colwidths="[18,24,44,14]"} ## Improvements to the course 1. **From band diagram to device in one model.** The exact Poisson solver (@sec-poisson) shows students that the depletion approximation, the law of the junction and the built-in voltage are the same physics. It is worth five minutes of class time. 2. **"Predict, then compute."** Before moving a slider, ask: *"If I double the lifetime, does the crossover voltage between η = 2 and η = 1 go up or down?"*, *"Which side of a p⁺n junction holds the depletion region?"*, *"Does a thinner quantum well emit bluer or redder light?"*. 3. **One running device.** Follow a single GaAs LED through the chapter: doping and Fermi levels → junction → I–V (slides 75–80) → spectrum (slides 107–108) → extraction (slides 122–123) → bandwidth. All the numbers are already in the companion. 4. **Lab connection.** A cheap lab (an LED, a resistor, a USB spectrometer and a multimeter) covers I–V fitting, ideality factor, spectrum versus temperature, Varshni shift and luminous flux. 5. **Modern content.** The slides stop at 2014-era devices. Possible updates: microLED displays, efficiency droop, UV-C LEDs for disinfection, Li-Fi (IEEE 802.11bb, 2023), perovskite and quantum-dot LEDs. ## Notes for the instructor: checks on the slides While recomputing the examples, everything matched the slides except these points: - **Slide numbering:** the PDF export (`Chapter3_2024pptx.pdf`) omits the empty slide 2, so PDF page = slide − 1 from slide 3 on. The companion uses the `.pptx` numbers. - **Slide 22:** titled "Extrinsic Semiconductors: n-Type" and writes "p ≈ Nd"; it describes p-type doping with $p\approx N_a$. - **Slide 25 (Table 3.1):** $N_v$ of Ge is printed as $6.0\times10^{19}$ cm⁻³. The mass action law with $n_i=2.3\times10^{13}$ cm⁻³ requires $6.0\times10^{18}$ cm⁻³ (the value of Kasap's book). - **Slides 65–66:** the ideality factor symbol η is rendered as a missing glyph ("□ is an ideality factor", "e/(□k_BT)"). - **Slide 69:** the model with Varshni $E_g(T)$ gives Arrhenius slopes of about 0.86 and 0.40 eV, higher than the measured 0.63 and 0.33 eV. The factor of two is reproduced. A good discussion point (@sec-diode). - **Slide 99:** "Fermi-Direct function" should be "Fermi–Dirac". - **Slide 107:** the absolute wavelengths (864.2 and 852.4 nm) are 0.9 nm too long because of rounded constants ($hc/e=1.2409$ instead of 1.2398 eV µm). The shift of 11.8 nm is correct. - **Slide 112:** "d = 12×10⁻⁹ nm" should read 12×10⁻⁹ m. The finite-well values quoted (0.022, 0.088, 0.186 eV) are about 15 % below a direct solution with $m^*=0.067m_e$ (0.026, 0.101, 0.216 eV); they correspond to $m^*\approx0.08m_e$ (@sec-qw). - **Slide 133:** the formula is written with "633 lm W⁻¹"; the numbers use the correct 683 lm W⁻¹. - **Slide 146:** the caption sentence "At $f_c$, $[I_\text{ph}(f)/I_\text{ph}(0)]^2$ is 0.5. However, it is 0.707 at a lower frequency $f_c$" is confusing. Suggested wording: "At $f_c$ the optical power is 0.707 of its low-frequency value and the detected electrical power is 0.5; the optical power reaches 0.5 at the higher frequency $f_\text{op}=\sqrt3f_c$."