Choosing a quantum-well model
Learning objectives. After reading this page you should be able to:
- Match each OghmaNano model to the material systems it suits.
- Compare the conduction and valence treatments across models.
- Weigh accuracy against computational cost.
- Choose a model for a given physical question.
- Understand that the best model depends on the system, not on generality alone.
Prerequisites. A reading of the individual model pages; this page ties them together.
Estimated reading time. 20 minutes.
OghmaNano offers several quantum-well models because no single model is best for every system. A scalar parabolic model is ideal for a quick estimate; an 8-band k·p model is needed for accurate valence physics and gain; a silicon-specific model is required for indirect-gap valleys. This page compares the models so that you can choose deliberately rather than defaulting to the most complex one.
1. Comparison at a glance
| Model | Materials | Conduction | Valence | Non-parabolic | Strain | Band mixing | Cost |
|---|---|---|---|---|---|---|---|
| Parabolic | Any (estimate) | single mass | single mass | no | offsets only* | none | very low |
| 8-band zincblende | GaAs, InGaAs, InP, InAs, GaSb... | non-parabolic | HH+LH+SO coupled | yes | full Bir–Pikus | c–v + valence | high |
| 10-band zincblende | dilute nitrides (GaInNAs) | BAC + non-parabolic | HH+LH+SO coupled | yes | full Bir–Pikus | c–v + N + valence | high |
| 8-band wurtzite | GaN, AlN, InN, alloys | anisotropic | HH+LH+CH coupled | yes | full (\(a_i,D_i\)) | c–v + valence | high |
| Si Δ-valley | Si, SiGe (conduction) | 6 anisotropic valleys | — | valley masses | \(\Xi_d,\Xi_u\) split | valley (via strain) | low |
| Luttinger–Kohn 6-band | Si, Ge, SiGe (valence) | — | HH+LH+SO coupled | yes | full Bir–Pikus | valence | medium |
*The parabolic solver includes band offsets and (optionally) a rigid strain shift of the edges, but not the strain-induced valence splitting.
2. Conduction treatment
For direct-gap III–V materials the conduction band is well described by the 8-band model, which makes it non-parabolic through the explicit Kane coupling. For dilute nitrides the conduction band additionally splits by band anticrossing, requiring the 10-band model. For silicon and SiGe the conduction minima are the six Δ valleys, which are indirect and anisotropic and are not described by any zone-centre k·p model — the valley solver is essential. The parabolic solver treats the conduction band as a single fixed mass, adequate for a rough level estimate but not for higher subbands or narrow gaps.
3. Valence treatment
The valence band is where a single mass fails most severely, because the heavy, light and split-off holes are always strongly coupled. Any quantitative valence calculation — hole subband spacings, gain, or the TE/TM ratio — needs a multiband model: the 8-band zincblende or wurtzite model for III–Vs, or the six-band Luttinger–Kohn model for group-IV materials. The parabolic hole model should be used only for a first estimate.
4. Cost and practicality
Computational cost scales with the basis size and the number of \(k_\parallel\) points. The scalar models (parabolic, silicon valley) solve tridiagonal matrices and are inexpensive. The multiband models solve complex Hermitian matrices of dimension \(6N\)–\(10N\) at every \(k_\parallel\) point, so a full dispersion is substantially more costly, though the shift-invert eigensolver keeps this tractable by computing only the few states that matter. A sensible workflow is to use a scalar model to locate the levels and set up the structure, then switch to the appropriate multiband model for the final, quantitative calculation.
5. Choosing a model
- GaAs / InGaAs / InP / InAs / GaSb quantum wells and lasers: 8-band zincblende.
- GaInNAs and other dilute nitrides: 10-band zincblende.
- GaN / InGaN / AlGaN (nitride LEDs and lasers): 8-band wurtzite, using the TE/TM gain outputs.
- Silicon / SiGe conduction (transport, quantum wells): Δ-valley solver.
- Si / Ge / SiGe valence (holes): six-band Luttinger–Kohn.
- Quick level estimates or teaching: parabolic solver.
The right model is the simplest one that contains the physics of the question being asked. A more complex model is not automatically more correct if it is fed poorer parameters or applied to a system outside its assumptions; conversely, a simple model applied within its range can be both accurate and fast.
Key points
- No model is universally best; each is matched to a material system and a question.
- Multiband models are essential for any quantitative valence-band or gain calculation.
- Silicon requires the dedicated valley (conduction) and Luttinger–Kohn (valence) solvers.
- Use a scalar model to set up and estimate, then a multiband model for the final result.