Same equation, different exponent. What n = 2 and n = 1/2 actually mean physically, how each curve looks, and how to choose the right Tauc exponent for your material.
A direct band gap Tauc plot and anindirect band gap Tauc plot use the same equation but a different exponent: n = 2 for a direct allowed transition and n = 1/2 for an indirect allowed one. The exponent is not a fitting preference — it reflects whether the electron can cross the gap without a phonon.
Choosing the wrong n changes the shape of the curve, moves the linear onset, and shifts the extrapolated band gap. This page explains the physics behind the two models and how to decide which exponent belongs to your material — before you fit anything.
| Property | Direct allowed | Indirect allowed |
|---|---|---|
| Exponent n | 2 | 1/2 |
| Plotted y-value | (αhν)² | (αhν)^(1/2) |
| Momentum rule | Δk = 0 — vertical transition | Δk ≠ 0 — phonon required |
| Band structure | Conduction band minimum above valence band maximum at the same k | Minimum and maximum at different k points |
| Edge sharpness | Usually steeper, well-defined onset | Broader onset, phonon-assisted tail |
| Typical materials | ZnO, GaAs, CdTe, MAPbI₃, GaN | Si, Ge, anatase TiO₂, GaP, AlAs |
| Typical Eg | Often 1.5–3.5 eV for semiconductors | Si ≈ 1.1–1.2 eV, Ge ≈ 0.66 eV |
Raising αhν to the power of 2 stretches the upper part of the absorption edge and sharpens the onset; taking the square root compresses it and softens the edge. In practice:
When the dipole matrix element vanishes at k = 0, the transition is forbidden and the exponent changes again: n = 3/2 for direct forbidden (Cu₂O, SnO) and n = 2/3 for indirect forbidden (SnO₂, some rare-earth oxides). These models are far less common — reach for them only when the allowed model clearly fails and the literature for your material supports it.
Yes, different transition models use different Tauc exponents. The appropriate model depends on the material and transition being investigated.
They use different exponents in the Tauc relationship. A direct allowed transition uses n = 2, while an indirect allowed transition uses n = 1/2.
The exponent should be selected according to the electronic transition model appropriate for the material. It should not simply be chosen because it produces the highest R².
For a direct allowed transition the exponent is n = 2, so the plotted quantity is (αhν)² against photon energy.
For an indirect allowed transition the exponent is n = 1/2, so the plotted quantity is the square root (αhν)^(1/2) against photon energy.
Yes. Because each exponent reshapes the curve, the linear onset moves and the extrapolated intercept changes. Using the wrong model is one of the most common sources of disagreement with literature values.
Switch exponents in the calculator and watch the fitted band gap update — a quick way to see how much the transition model matters.