Tauc plot

A Tauc plot is a graphical analysis technique used to estimate the optical band‑gap energy of a material, most commonly disordered, amorphous, or poorly crystalline semiconductors. The method plots $(\alpha h u)^{1/n}$ (where $\alpha$ is the absorption coefficient, $h u$ is the photon energy, and $n$ depends on the nature of the electronic transition) against the photon energy $h u$. The linear portion of the resulting curve is extrapolated to intersect the energy axis; the intercept is taken as the optical band‑gap (often called the “Tauc gap”).

Historical background
The technique was introduced by Jan Tauc in the 1960s while studying amorphous germanium and silicon. It was developed to provide a practical way of extracting band‑gap information from UV‑visible absorption spectra of materials that lack long‑range order.

Mathematical form
For a direct allowed transition, the commonly used expression is

$$ (\alpha h u)^{2} = A(h u - E_{\text{g}}) $$

where $A$ is a constant and $E_{\text{g}}$ is the optical band‑gap. For indirect transitions the exponent $n$ is set to $1/2$; for direct forbidden transitions $n = 3/2$; and for indirect forbidden transitions $n = 3$. By selecting the appropriate exponent, the plot linearizes the absorption edge for the transition type under investigation.

Procedure

1. Measure the absorption spectrum of the sample, typically using UV‑visible spectroscopy.
2. Convert the measured absorbance to the absorption coefficient $\alpha$.
3. Choose the exponent $n$ based on the expected transition type.
4. Plot $(\alpha h u)^{1/n}$ versus $h u$.
5. Identify the linear region near the absorption edge and extrapolate the line to the horizontal axis; the intercept gives $E_{\text{g}}$.

Applications
The Tauc plot is widely employed in material science and solid‑state physics for:

• Determining band‑gap energies of amorphous semiconductors such as a‑Si, a‑Ge, and various metal‑oxide thin films (e.g., TiO₂).
• Characterizing nanostructured and hybrid materials where disorder broadens the absorption edge.
• Comparing the effects of dopants, annealing, or compositional changes on the electronic structure of a material.

Limitations and considerations

• The method assumes a parabolic band structure and may not be appropriate for highly crystalline or strongly excitonic materials.
• Accurate determination of $\alpha$ requires careful correction for scattering and reflectance, especially for powders or rough films.
• Misapplication of the exponent $n$ or incorrect selection of the linear region can lead to systematic errors in the extracted band‑gap value. Recent literature highlights that unconventional variations of the Tauc plot (e.g., plotting $\alpha^{1/r}$ versus $h u$) may produce inconsistent results if not justified by the underlying physics.

References

• Wikipedia, “Tauc plot”.
• Jan Tauc, original work on amorphous semiconductor optical absorption (1960s).
• NumberAnalytics, “Understanding Tauc Plot for Optical Properties”.
• Advanced Materials literature on limitations of the Tauc method (2023).