Explore band dispersion across k using selectable models. Tune lattice spacing, effective mass, and coupling. Get clear tables, plots, and exports for quick decisions.
| Model | a (nm) | m* | Parameter | k Range | Points |
|---|---|---|---|---|---|
| Tight binding | 0.50 | 1.00 | t = 1.00 eV | First Brillouin zone | 41 |
| Gap model | 0.40 | 0.20 | |VG| = 0.60 eV | First Brillouin zone | 61 |
| Free electron | 0.55 | 1.00 | E₀ = 0.00 eV | Custom: −5 to +5 (1/nm) | 81 |
Band structure relates electron energy to wavevector k in a periodic lattice. This calculator offers three classic 1D approximations that are widely used for intuition, teaching, and early-stage parameter studies before heavier simulations.
Bloch theory makes energy a function of k. A standard plotting domain is the first Brillouin zone, from −π/a to +π/a. Here, a is the lattice constant; many crystals fall near 0.2–0.6 nm. Use the Brillouin-zone mode for canonical dispersion plots, or custom ranges for targeted scans.
The free-electron model is parabolic: E(k)=E₀+ħ²k²/(2m*). With m*=mₑ, k=1 nm−1 produces about 0.038 eV above E₀. Smaller m* increases curvature, raising energy faster with k and representing lighter carriers.
Nearest-neighbor tight binding uses E(k)=E₀−2t cos(ka). The extrema occur at k=0 and k=π/a, so the bandwidth is 4t. If t=1.0 eV, the band spans 4.0 eV; if t=2.5 eV, it spans 10.0 eV. Larger t generally implies stronger orbital overlap and higher group velocities.
A weak periodic potential mixes states near the zone boundary and opens a gap. In the coupled-state form used here, the minimum separation near k≈π/a is approximately 2|VG|. For |VG|=0.6 eV, the gap is about 1.2 eV, comparable to common semiconductor scales.
Effective masses vary widely: light bands can be 0.05–0.2 mₑ, while heavier bands may approach 1 mₑ. Use E₀ to shift the reference level (for example, to align with a conduction-band edge). For tight binding, typical t values from 0.5–3 eV cover many simple bonding strengths.
The table reports k in either π/a or 1/nm, matching your selection. For the gap model, two branches (E₁ and E₂) appear; they separate most near the zone boundary. Export CSV and plot E versus k to visualize curvature, bandwidth, and gap features.
You can also estimate transport intuition from the slope and curvature of the curve. The group velocity is proportional to dE/dk, so steeper bands indicate faster carriers. The effective mass is related to the second derivative d²E/dk²; flatter curvature corresponds to heavier carriers. When comparing scenarios, keep the same k range and point count for consistent charts.
Use this tool to compare trends, estimate band widths, and demonstrate Brillouin-zone concepts. The models are one-dimensional and omit multi-band coupling, anisotropy, and spin-orbit effects. Treat results as first-pass estimates rather than final device-level predictions.
It automatically sets k from −π/a to +π/a using your lattice constant a, which is the standard domain for 1D band plots.
Use a literature or datasheet value when possible. Smaller m* makes the free-electron parabola steeper and increases energy at the same k.
t measures how strongly electrons move between neighboring sites. Larger t widens the band; the bandwidth in this model is 4t.
State mixing near the zone boundary splits the dispersion into a lower and upper branch. The separation is largest near k≈π/a.
Near k≈π/a in this simplified 1D coupling picture, the minimum separation is approximately 2|VG|. Away from the boundary, the gap varies.
Yes. Choose “Custom range” and set the unit to 1/nm. The calculator converts those values internally to 1/m for energy evaluation.
This page focuses on reliable tables and exports. Download CSV and plot E versus k in your preferred tool for full control over axes and styling.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.