Tauc Plot Band Gap Calculator | Expert Analysis & Formula

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Tauc Plot Band Gap Calculator

Determine the optical band gap (E_g) of a semiconductor by extrapolating the linear region of a Tauc plot. This tool simplifies the calculation of band gap using the Tauc plot method by finding the x-intercept from two points on your linearized data.

Nature of Electronic Transition

Select the Tauc exponent ‘n’ that linearizes your (αhν)^1/n vs. hν plot.

Enter two points from the linear-fit region of your Tauc plot:

Point 1: Photon Energy (x₁)

Unit: electron-volts (eV)

Point 1: (αhν)^1/n Value (y₁)

Unit: (cm^-1eV)^1/n or arbitrary units

Point 2: Photon Energy (x₂)

Unit: electron-volts (eV)

Point 2: (αhν)^1/n Value (y₂)

Unit: (cm^-1eV)^1/n or arbitrary units

Calculated Optical Band Gap (E_g)

2.25 eV

Line Slope (m)

60.00

Y-Intercept (b)

-135.00

Formula

E_g = x₁ – y₁/m

Dynamic Tauc Plot Visualization

Live visualization of the entered data points and the extrapolated line intersecting the energy axis to find the optical band gap.

What is the Calculation of Band Gap Using Tauc Plot?

The calculation of a band gap using a Tauc plot is a widely used method in materials science and physics to determine the optical band gap of semiconductors. The method involves transforming UV-Vis absorption data to find the energy threshold for photon absorption. A Tauc plot graphs the quantity (αhν)^1/n on the y-axis against the photon energy (hν) on the x-axis.

Here, ‘α’ is the absorption coefficient, ‘h’ is Planck’s constant, ‘ν’ is the photon frequency, and ‘n’ is an exponent that depends on the nature of the electronic transition. By identifying a linear region in this plot and extrapolating it to the energy axis (where the y-value is zero), we can find the material’s optical band gap (E_g). This calculator automates the final extrapolation step for a more accurate UV-Vis spectroscopy analysis.

The Tauc Plot Formula and Explanation

The relationship described by Jan Tauc is given by the formula:

(αhν)^1/n = A(hν – E_g)

This equation forms the basis for the entire Tauc plot analysis. It shows a linear relationship between the Tauc variable (αhν)^1/n and the photon energy (hν) near the absorption edge.

Description of variables in the Tauc relation.
Variable	Meaning	Unit	Typical Range
E_g	Optical Band Gap Energy	electron-Volts (eV)	0.5 – 4.0 eV for most semiconductors
α	Absorption Coefficient	cm^-1	10³ – 10⁶ cm^-1
hν	Photon Energy	electron-Volts (eV)	1.5 – 5.0 eV (Visible/UV range)
n	Tauc Exponent	Unitless	0.5, 1.5, 2, or 3
A	Proportionality Constant	Varies	Dependent on material properties

The key to a successful calculation of band gap using Tauc plot is selecting the correct value of ‘n’. The choice depends on whether the electron transition is direct or indirect, and allowed or forbidden. For a deeper dive, check out our article on direct vs indirect band gaps.

Practical Examples

Example 1: Direct Band Gap Semiconductor

A researcher analyzes a thin film of Gallium Arsenide (GaAs), a direct band gap material. They choose n = 0.5. After plotting their data, they pick two points from the linear section of their Tauc plot.

Inputs:
- Transition Type: Direct Allowed (n = 0.5)
- Point 1 (x₁, y₁): (1.45 eV, 50 arb. units)
- Point 2 (x₂, y₂): (1.55 eV, 150 arb. units)
Results:
- The calculator finds the slope and extrapolates to the x-axis.
- Calculated Band Gap (E_g): 1.40 eV

Example 2: Indirect Band Gap Semiconductor

Another scientist is working with amorphous Silicon (a-Si), an indirect band gap material, and selects n = 2 for their Tauc plot analysis online.

Inputs:
- Transition Type: Indirect Allowed (n = 2)
- Point 1 (x₁, y₁): (1.8 eV, 20 arb. units)
- Point 2 (x₂, y₂): (2.0 eV, 80 arb. units)
Results:
- Using the two points from the linear fit.
- Calculated Band Gap (E_g): 1.73 eV

How to Use This Tauc Plot Band Gap Calculator

Collect Data: Obtain absorption or transmission data from a UV-Vis spectrophotometer.
Process Data: Convert your raw data (e.g., wavelength vs. absorbance) into Photon Energy (hν) vs. Absorption Coefficient (α). Our photon energy calculator can help with the first part.
Select Transition Type: Choose the appropriate ‘n’ value (0.5 for direct allowed, 2 for indirect allowed, etc.) from the dropdown menu. This should be the value that results in the longest, most linear segment on your plot.
Plot Your Data: Create a graph of (αhν)^1/n vs. hν using spreadsheet software.
Identify Linear Region: Find the straight-line portion of your plot, typically just above the absorption onset.
Enter Two Points: Select two distinct points from this linear region and enter their coordinates (x₁, y₁) and (x₂, y₂) into the calculator.
Interpret Results: The calculator automatically performs the extrapolation, providing the Optical Band Gap (E_g), the slope of the line, and a visual representation in the chart.

Key Factors That Affect Band Gap Calculation

Material Purity: Impurities can create states within the band gap, leading to “sub-bandgap” absorption and making the linear extrapolation difficult.
Crystallinity: Amorphous materials often have less defined band edges (known as Urbach tails) compared to crystalline materials, which can complicate the Tauc plot analysis.
Film Thickness: An accurate measurement of the sample’s thickness is critical for calculating the absorption coefficient ‘α’ correctly. Errors here directly impact the Tauc plot’s y-values.
Data Range Selection: Choosing points from a region that is not truly linear is the most common source of error. The linear fit should only be applied to the steep absorption onset.
Choice of ‘n’: Using the wrong exponent ‘n’ will result in a curved plot instead of a straight line, making a valid extrapolation impossible. The semiconductor band gap formula relies on the correct ‘n’ value.
Surface Roughness: In thin films, surface scattering can affect the measured absorbance, introducing noise into the thin film characterization data.

Frequently Asked Questions (FAQ)

1. What is a Tauc plot used for?: A Tauc plot is primarily used to determine the optical band gap of a semiconductor material from its absorption spectrum.
2. How do I choose between n=0.5 and n=2?: Choose n=0.5 (for direct transitions) or n=2 (for indirect transitions) and see which one gives you a clearer, longer straight-line segment in your plot of (αhν)^1/n vs hν. The correct ‘n’ value will linearize the data near the band edge.
3. What if my plot has no linear region?: This can happen if the material is not a classic semiconductor, is highly disordered, or if there are significant defect states. It may indicate that the Tauc model is not appropriate for your material.
4. Can I use absorbance instead of absorption coefficient?: For a quick estimate, yes, as absorbance is proportional to the absorption coefficient for a fixed sample thickness. However, for an accurate calculation of band gap using Tauc plot, you must calculate ‘α’ using the Beer-Lambert law (α = 2.303 * Absorbance / thickness).
5. Why is my calculated band gap different from the literature value?: Discrepancies can arise from experimental conditions, sample quality (purity, defects), measurement errors (especially film thickness), and the subjective selection of the “linear” region for extrapolation.
6. What does the x-intercept in a Tauc plot represent?: The x-intercept, found by extrapolating the linear part of the plot to the x-axis, represents the optical band gap energy (E_g) of the material.
7. What is an Urbach tail?: The Urbach tail or Urbach energy refers to an exponential tail of absorption just below the main band edge, often seen in amorphous or disordered materials. It can make identifying the linear Tauc region more challenging.
8. Does this calculator work for quantum dots?: Yes, the Tauc method is frequently used for quantum dots and other nanomaterials, which often show a blue-shift in their band gap compared to bulk material due to quantum confinement effects. For more on thin films, see our thin film reflectance calculator.

Related Tools and Internal Resources

Explore other tools and resources for materials science and optical analysis:

Photon Energy to Wavelength Calculator: Quickly convert between energy (eV) and wavelength (nm).
Article: Direct vs. Indirect Band Gaps: A detailed explanation of the physics behind different electronic transitions.
Semiconductor Materials Database: Look up properties of common semiconductor materials.
A Guide to UV-Vis Spectroscopy: Learn the fundamentals of absorption spectroscopy.
Thin Film Reflectance Calculator: Model and analyze the reflectance of multilayer thin films.
Contact Us: Have questions or need a custom tool? Get in touch with our experts.