Optical Absorption and Band-Gap Measurement

Introduction

An absorption spectrum can reveal when photons have enough energy to excite electrons into accessible states. That threshold is related to a semiconductor's electronic structure, but reading a gap directly from the first nonzero absorbance is hazardous. Defects, excitons, film thickness, reflection and transition type all shape the edge. Tauc analysis is a structured way to estimate an optical threshold, provided its assumptions match the material and the plotted quantity is measured correctly.

Core explanation

Photon energy is E = hν = hc/λ. A convenient numerical conversion is E(eV) ≈ 1240/λ(nm). Shorter wavelengths correspond to higher energies. For a homogeneous absorbing slab under simple conditions, transmitted intensity follows I = I₀ exp(−αd), where α is absorption coefficient and d is thickness. The measured transmission also includes reflection at surfaces, scattering and sometimes interference. If these are appreciable, using −ln(T)/d as though it were a pure α gives a biased result. An absorbance spectrum is therefore an experimental starting point, not automatically a calibrated absorption coefficient.

Near an ideal allowed direct edge in a simple parabolic-band model, (αhν)² may vary approximately linearly with hν over a suitable range. Extrapolating that linear segment to zero estimates an optical transition energy. For an allowed indirect transition, a different power—often a plot of (αhν)^(1/2) against hν—is used in the simplest model, with phonon absorption and emission introducing separate thresholds. These powers depend on selection rules and the model; a visually straight line chosen after trying every exponent is not proof of transition type. Tauc's original treatment is especially associated with disordered semiconductors, and conventions vary across material classes. MIT's lecture on optical properties of amorphous materials discusses the Tauc gap and absorption of disordered solids; MIT's semiconductor optics lecture distinguishes direct and indirect absorption.

An absorption edge is not a perfect step. Thermal vibrations, disorder and defects can produce tails below a nominal band-edge transition. Excitons may absorb just below the free electron–hole continuum. Additional impurity or charge-transfer transitions can create separate visible bands. The optical gap extracted from one fit can differ from a transport activation energy or a quasiparticle gap because the experiments probe different processes. Report method, fit range, temperature and transition assumption with the number.

Colour is a perceptual response to the spectrum of reflected or transmitted visible light under illumination. A material can appear coloured because it absorbs one region strongly, but a dark sample may simply absorb broadly or scatter little. A thin film may show interference colour unrelated to the band gap. A transparent wide-gap oxide can carry electronic current when doped without becoming strongly coloured in the visible, although free-carrier absorption may occur at longer wavelengths. Thus “the solid is blue, therefore E g is exactly the energy of orange light” is an overinterpretation.

Step-by-step reasoning

1. Convert wavelength to photon energy and establish the spectral region of the rise. 2. Determine whether measured attenuation has been corrected for reflection, scattering and thickness. 3. Choose a transition model from independent structural or band evidence. 4. Transform αhν with the matching exponent and fit only a justified linear range. 5. Report the extrapolated value as a model-dependent optical threshold, with uncertainty.

Visual explanation

Draw α versus photon energy: a gradual tail rises into a steep edge. Draw a separate Tauc-style transformed plot with a highlighted linear region and a dashed line extrapolated to the energy axis. Put an impurity-absorption bump below the main edge to show why the first visible absorption need not equal the fundamental band gap.

Real-world analogy

A door that opens only above a push threshold resembles an absorption onset, but real measurements include people slipping through side doors and friction that blurs the threshold. Fitting the main door's behaviour requires deciding which entries count and measuring the push accurately. The analogy illustrates why a graph's intercept depends on the chosen mechanism and data range.

Real-world example

A researcher compares two oxide powders intended for photocatalysis. Diffuse reflectance spectra, converted with an appropriate scattering model, show different apparent edges. An edge shift might reflect a changed electronic gap, but altered particle size, phase fraction or defect absorption can also affect the spectrum. X-ray diffraction, composition analysis and photoluminescence can help decide which interpretation is supported.

Why?

Why include hν in a Tauc transform rather than plot α alone? The near-edge form of the joint electronic density of states and optical transition probability gives a power-law relation involving αhν in the simplified model. The transformation can linearise a particular class of transitions. Its usefulness is conditional on the assumed band geometry and selection rules, not a general mathematical trick that always exposes a unique gap.

Common misconception

Any straight segment on a Tauc plot is not necessarily the fundamental gap. Small ranges of noisy or mixed-mechanism data can look linear. Another error is choosing the exponent that produces the most appealing intercept and then claiming that the gap type is proven. Independent evidence and sensitivity to fit range are needed.

Worked example

A measured absorption rise occurs near λ = 620 nm. The corresponding photon energy is approximately 1240/620 = 2.00 eV . This conversion identifies the energy scale only. If an allowed-direct transform of corrected α data has a reliable linear segment extrapolating to 1.92 eV, report an approximate 1.92 eV optical threshold under that model, rather than insisting the first visible rise at 2.00 eV is exact. The difference could arise from how the edge and tail are defined.

Quick check

1. Which has greater photon energy, light at 400 nm or at 800 nm? Answer: The 400 nm photon has twice the energy because E is inversely proportional to wavelength.

Exam focus

Use E(eV) ≈ 1240/λ(nm) with units. Distinguish absorbance, transmission and absorption coefficient. State whether a proposed Tauc transform assumes a direct, indirect or disordered-solid transition. Treat colour as supporting evidence only when the spectral mechanism is established.

Advanced insight

For thin films, ellipsometry can recover optical constants while accounting for reflection and interference, though it uses an optical model. In strongly excitonic semiconductors, the optical onset can lie below the quasiparticle gap by the exciton binding energy. In disordered solids, an Urbach-like tail reflects a distribution of local environments and can change with temperature or processing.

Summary

Optical absorption rises when photons access electronic transitions, but the edge is shaped by transition type, disorder, excitons and measurement geometry. Tauc-style extrapolation estimates a model-dependent optical threshold. A material's colour alone cannot uniquely determine its fundamental gap.

Practice questions

1. Estimate the energy of a 500 nm photon. Answer: About 1240/500 = 2.48 eV. 2. Why must film thickness be known to estimate α from simple exponential attenuation? Answer: The exponent is αd; without d, the same transmission could represent different α values. 3. What extra issue affects an indirect absorption edge? Answer: A phonon participates in near-edge transitions, producing different thresholds and energy dependence. 4. Does a blue appearance alone prove a particular Tauc gap? Answer: No. Impurities, charge transfer, scattering and thin-film interference can also determine visible colour.