1. Theory

Definition: A UV-Vis spectrophotometer measures the transmittance or reflectance of a sample from the ultraviolet to the near infrared, giving the absorption spectrum (NISER; Thermo Fisher).
Absorption coefficient: The Beer-Lambert law converts absorbance and thickness to the absorption coefficient (NISER).
Optical band gap: The Tauc relation describes absorption above the gap; plotting $(α h ν)^{1/ γ}$ against photon energy and extrapolating gives the gap (Thermo Fisher; NISER).
Powders: For scattering powders, diffuse reflectance with the Kubelka-Munk function replaces transmission (Thermo Fisher).

2. Working Principle

Optics: Deuterium and tungsten lamps, a monochromator or array spectrometer, and a detector measure sample and reference beams (background knowledge; NISER).
Baselines: Dark spectra with the lamp off and reference spectra with a bare substrate or white standard are recorded under the same settings (NISER; Thermo Fisher).
Diffuse reflectance: Integrating spheres or Praying Mantis optics collect light scattered by powders (Thermo Fisher).
Analysis: Absorbance is converted to absorption coefficient and analysed with Tauc or Urbach plots (NISER; background knowledge).

3. Applications

Thin films: Optical gaps of oxides, chalcogenides and organic semiconductors on glass (NISER).
Photocatalysts and pigments: Gaps of TiO2 and other powders (Thermo Fisher).
Transparent conductors: Transmission and gap of ITO and ZnO (background knowledge).
Process control: Absorption edges as a quick check of composition and disorder (background knowledge).

Absorption Measurement

Name	Formula	Related Formulae	Variables/Constants
Beer-Lambert law and absorbance	$I_{T} = I_{0} e^{- α d}, A = lo g_{10} \frac{I _{0}}{I _{T}}$	$α = \frac{2.303 A}{d}$	$I_{T}, I_{0} : t r an s mi tt e d an d in c i d e n t in t e n s i t i es$ $α : ab sor pt i o n coe f f i c i e n t$ $d : t hi c k n ess$ $A : ab sor ban ce; N I S E R : t h ese e q u a t i o n s; T h er m o F i s h er : ab sor ban ce p r o p or t i o na l t o α t h r o ug h B ee r^{'} s l a w, so i t c an s t an d in f or α in a T a u c pl o t$ $(co m p u t e d : A = 0.5 o n a 100 nm f i l m i s α = 1.15 \times 1 0^{5} c m^{- 1})$
Reflection-corrected absorption	$T = \frac{( 1 - R ) ^{2} e ^{- α d}}{1 - R ^{2} e ^{- 2 α d}}$	$α \approx \frac{1}{d} ln \frac{( 1 - R ) ^{2}}{T} (α d ≫ 1)$	$T : m e a s u r e d t r an s mi tt an ce$ $R : s u r f a ce r e f l ec t an ce; f or m s (ba c k g r o u n d k n o w l e d g e, P ank o v e); in t er f er e n ce f r in g es in t hin f i l m s m o d u l a t e T an d R (S p ec t r a l R e f l ec t o m e t er p a g e)$ $(a ss u m e d : T = 0.3, R = 0.2, d = 100 nm; co m p u t e d : α = 7.6 \times 1 0^{4} c m^{- 1} w i t h t h e cor r ec t i o n a g ain s t 1.20 \times 1 0^{5} c m^{- 1} w i t h o u t, 37 p er ce n t l o w er)$
Photon energy	$E = \frac{h c}{λ}, E [eV] \approx \frac{1239.84}{λ [ nm ]}$	$p = \frac{E}{c}$	$h = 6.63 \times 1 0^{- 34} J s, c = 3.0 \times 1 0^{8} m / s (T h er m o F i s h er v a l u es)$ $p : p h o t o n m o m e n t u m; T h er m o F i s h er : E = h c / λ co n v er t e d f r o m j o u l es t o e l ec t r o n v o l t s; N I S E R : p h o t o n m o m e n t u m n e g l i g ib l e, so in d i r ec t t r an s i t i o n s n ee d p h o n o n s; n u m er i c a l f or m (ba c k g r o u n d k n o w l e d g e)$ $(co m p u t e d : 3.23 eV i s 384 nm, 3.01 eV i s 412 nm, 3.35 eV i s 370 nm)$

Band-Gap Analysis

Name	Formula	Related Formulae	Variables/Constants
Tauc relation	$(α h ν)^{1/ γ} = B (h ν - E_{g})$	$γ = \frac{1}{2} (direct allowed), γ = 2 (indirect allowed), γ = \frac{3}{2}, 3 (forbidden)$	$h ν : p h o t o n e n er g y$ $B : co n s t an t$ $E_{g} : o pt i c a l g a p$ $γ : t r an s i t i o n e x p o n e n t; T h er m o F i s h er : t hi s f or m, T i O 2 t r e a t e d a s in d i r ec t w i t h γ = 2, T a u c a x i s 2.6 t o 4.0 eV; N I S E R : α = C (h ν - E_{g})^{n} w i t h n = 1/2 d i r ec t an d 2 in d i r ec t (see t h e Q / A); f or bi dd e n e x p o n e n t s (ba c k g r o u n d k n o w l e d g e)$ $(co m p u t e d : t h e g a p i s t h e in t er ce pt w h er e t h e l in e a r p a r t o f (α h ν)^{1/ γ} m ee t s z er o)$
Indirect absorption with phonons	$α = C (h ν - E_{g} \mp E_{p})^{2}$	$α (h ν) = α_{a} + α_{e}$	$E_{p} : p h o n o n e n er g y (min u s f or p h o n o n ab sor pt i o n, pl u s f or e mi ss i o n)$ $α_{a}, α_{e} : ab sor pt i o n an d e mi ss i o n b r an c h es; N I S E R : t hi s f or m; s u m o f b r an c h es (ba c k g r o u n d k n o w l e d g e)$ $(a ss u m e d : E_{p} = 58 meV f or s i l i co n; co m p u t e d : t h e tw o o n se t s l i e a t E_{g} - 58 an d E_{g} + 58 meV, 116 meV a p a r t)$
Urbach tail	$α (E) = α_{0} exp (\frac{E - E _{0}}{E _{U}}) (E < E_{g})$	$E_{U} = [\frac{d l n α}{d E}]^{- 1}$	$E_{U} : U r ba c h e n er g y (t ai l w i d t h)$ $α_{0}, E_{0} : co n s t an t s; f or m (ba c k g r o u n d k n o w l e d g e, P ank o v e); l a r g er E_{U} m e an s m or e d i sor d er$ $(a ss u m e d : E_{U} = 50 meV; co m p u t e d : α f a l l s b y e e v er y 50 meV, a f a c t or o f 10 e v er y 115 meV)$

Diffuse Reflectance and Reference Data

Name	Formula	Related Formulae	Variables/Constants
Kubelka-Munk function	$F (R) = \frac{( 1 - R ) ^{2}}{2 R} = \frac{k}{s}$	$(F (R) h ν)^{1/ γ} = B (h ν - E_{g})$	$F (R) : K u b e l k a - M u nk f u n c t i o n$ $R : d i f f u se r e f l ec t an ce (a s a f r a c t i o n)$ $k : ab sor pt i o n coe f f i c i e n t$ $s : sc a tt er in g coe f f i c i e n t; T h er m o F i s h er : t h ese e q u a t i o n s f or p o w d er s, I S A - 220 in t e g r a t in g s p h er e o n t h e E v o l u t i o n O n e P l u s w i t h S p ec t r a l o n ba c k g r o u n d an d P r a y in g M an t i s a ccessor y o n t h e E v o l u t i o n P r o w i t h P T F E ba c k g r o u n d, 250 t o 800 nm, 1.0 nm ban d w i d t h an d s t e p, 0.5 an d 0.25 s in t e g r a t i o n, h o l d er s o f 250 an d 31.6 m m^{3}$ $(co m p u t e d : R = 0.5 g i v es F = 0.25 an d R = 0.1 g i v es 4.05)$
TiO2 results	$E_{g}^{ana t a se} = 3.23 \pm 0.06 and 3.24 \pm 0.05 eV$	$E_{g}^{r u t i l e} = 3.01 \pm 0.03 (or \pm 0.02) and 3.04 \pm 0.02 eV$	$T h er m o F i s h er : in t e g r a t in g s p h er e an d P r a y in g M an t i s r es u l t s f or ana t a se an d r u t i l e, t h e r u t i l e s p h er e u n cer t ain t y g i v e n a s 0.03 in t h e t ab l e an d 0.02 in t h e t e x t; r u t i l e ha s a d i r ec t b u t d i p o l e - f or bi dd e n l o w es t t r an s i t i o n (ba c k g r o u n d k n o w l e d g e)$ $(co m p u t e d : t h e tw o a ccessor i es a g r ee w i t hin 0.01 eV f or ana t a se an d 0.03 eV f or r u t i l e, in s i d e t h e s t a t e d u n cer t ain t i es)$
Reference band gaps at 300 K	$E_{g} = 1.12 (Si), 0.66 (Ge), 1.42 (GaAs), 2.26 (GaP), 3.36 (GaN), 3.35 (ZnO) eV$	$E_{g} = 2.99 (α -SiC), 0.70 (InN), 5.47 (C), 1.70 (CdSe), 3.68 (ZnS) eV$	$N I S E R t ab l e a f t er H u mm e l : t h ese v a l u es w i t h d i r ec t or in d i r ec t t y p e (S i, G e, G a P, S i C, d iam o n d in d i r ec t; G a A s, G a N, I n N, Z n O, C d S e, Z n S d i r ec t); co m p a r e G a N w i t h t h e V a r s hni v a l u e o n t h e P h o t o l u min esce n ce S p ec t r o m e t er p a g e (see t h e Q / A)$ $(co m p u t e d : N e p a l e t a l .^{'} s V a r s hni p a r am e t er s g i v e 3.42 eV f or G a N a t 300 K, 0.06 eV ab o v e t h e t ab l e)$

Important Questions

Ques: How does UV-Vis spectroscopy give the absorption coefficient?

From the transmitted fraction and the sample thickness. The Beer-Lambert law $I_{T} = I_{0} e^{- α d}$ and the absorbance $A = lo g_{10} (I_{0} / I_{T})$ give $α = 2.303 A / d$ (NISER); an absorbance of 0.5 on a 100 $nm$ film is $1.15 \times 1 0^{5}$ $c m^{- 1}$ (computed). This simple form ignores reflection at the surfaces. With an assumed transmittance of 0.3 and reflectance of 0.2, correcting with $(1 - R)^{2} / T$ gives $7.6 \times 1 0^{4}$ $c m^{- 1}$ instead of $1.20 \times 1 0^{5}$ , 37 per cent lower (background knowledge; computed). Thin films also show interference fringes that modulate transmission, so a bare substrate reference and, where possible, reflectance measurements are needed. Thermo Fisher notes that absorbance itself can stand in for $α$ in a Tauc plot, since only the intercept matters. Reference: NISER (School of Physical Sciences) laboratory handout, ‘Ultraviolet-Visible Spectroscopy to determine the nature of band gap and estimation of the bandgap’ (author not stated; table after Hummel, Electronic Properties of Materials, 3rd ed.) (fetch-tool summary); Thermo Fisher Scientific, ‘Band Gap Analysis through UV-Visible Spectroscopy’, application note AN54685 (2023) (fetch-tool summary); J. I. Pankove, Optical Processes in Semiconductors, Dover 1971 (background knowledge; not opened).

Ques: How is the band gap found with a Tauc plot?

By plotting a power of the absorption against photon energy and extrapolating the straight part to zero. Above the gap, $(α h ν)^{1/ γ} = B (h ν - E_{g})$ , so the right exponent turns the edge into a line whose intercept is the gap (Thermo Fisher). For a direct allowed transition $γ = 1/2$ , so $(α h ν)^{2}$ is plotted; for an indirect allowed transition $γ = 2$ , so $(α h ν)^{1/2}$ ; forbidden transitions use 3/2 and 3 (background knowledge). Thermo Fisher treats TiO2 as indirect and plots from 2.6 to 4.0 $eV$ . The result depends on which range is taken as linear, so the fit range should be stated. Photon energies come from $E = h c / λ$ : its anatase gap of 3.23 $eV$ corresponds to 384 $nm$ (computed). Reference: Thermo Fisher Scientific, ‘Band Gap Analysis through UV-Visible Spectroscopy’, application note AN54685 (2023) (fetch-tool summary); J. I. Pankove, Optical Processes in Semiconductors, Dover 1971 (background knowledge; not opened).

Ques: Is the Tauc relation written with alpha or with alpha times h nu?

The two sources write it differently. NISER uses $α = C (h ν - E_{g})^{n}$ with n = 1/2 for direct and 2 for indirect transitions, while Thermo Fisher uses $(α h ν)^{1/ γ} = B (h ν - E_{g})$ , which corresponds to $α h ν \propto (h ν - E_{g})^{γ}$ . The exponents are the same, but the extra $h ν$ factor differs. Thermo Fisher’s form follows Tauc’s original derivation and is the more standard choice for amorphous and polycrystalline films (background knowledge; my reading). Over the narrow range used near the edge, $h ν$ changes by only a few per cent, so the two forms give similar gaps: for a direct gap at 3.2 $eV$ fitted over 0.25 $eV$ , using the wrong form shifts the intercept by 0.016 $eV$ (computed); the form used should still be stated, as it affects comparisons. Reference: NISER (School of Physical Sciences) laboratory handout, ‘Ultraviolet-Visible Spectroscopy to determine the nature of band gap and estimation of the bandgap’ (author not stated; table after Hummel, Electronic Properties of Materials, 3rd ed.) (fetch-tool summary); Thermo Fisher Scientific, ‘Band Gap Analysis through UV-Visible Spectroscopy’, application note AN54685 (2023) (fetch-tool summary).

Ques: How are band gaps of powders measured?

With diffuse reflectance and the Kubelka-Munk function. Powders scatter too strongly for transmission, so an integrating sphere or a Praying Mantis accessory collects diffusely reflected light, referenced to a white standard; Thermo Fisher used Spectralon for its ISA-220 sphere and PTFE for the Praying Mantis, scanning 250 to 800 $nm$ in 1 $nm$ steps. The reflectance converts to $F (R) = (1 - R)^{2} / (2 R)$ , proportional to the ratio of absorption to scattering coefficients; R = 0.5 gives 0.25 and R = 0.1 gives 4.05 (computed). $F (R)$ then replaces $α$ in the Tauc plot. Thermo Fisher found 3.23 and 3.24 $eV$ for anatase and 3.01 and 3.04 $eV$ for rutile with the two accessories, agreeing within 0.01 to 0.03 $eV$ (computed). Reference: Thermo Fisher Scientific, ‘Band Gap Analysis through UV-Visible Spectroscopy’, application note AN54685 (2023) (fetch-tool summary).

Ques: Is the GaN band gap at room temperature 3.36 or about 3.42 eV?

NISER’s table, after Hummel, gives 3.36 $eV$ for GaN, while the Varshni parameters of Nepal et al., on this vault’s Photoluminescence Spectrometer page, give 3.495 $eV$ minus 77.5 $meV$ , about 3.42 $eV$ at 300 $K$ (computed). The 0.06 $eV$ difference is larger than measurement error. Room-temperature values near 3.39 to 3.44 $eV$ are typical of modern high-quality wurtzite GaN (background knowledge), so 3.42 $eV$ is more likely correct for strain-free material, while 3.36 $eV$ probably reflects older data or a Tauc fit to a broadened absorption edge (my reading). The difference also shows a general point: absorption-edge gaps from Tauc plots on imperfect films often fall below the true gap because tails extend the edge. Reference: NISER (School of Physical Sciences) laboratory handout, ‘Ultraviolet-Visible Spectroscopy to determine the nature of band gap and estimation of the bandgap’ (author not stated; table after Hummel, Electronic Properties of Materials, 3rd ed.) (fetch-tool summary); N. Nepal et al., Appl. Phys. Lett. 87, 242104 (2005), via the Photoluminescence Spectrometer page of this vault.

Ques: What makes Tauc-plot gaps unreliable?

Tails, interference, the wrong exponent and the choice of fit range. Disorder and defects create an exponential Urbach tail below the gap, $α_{0} exp [(E - E_{0}) / E_{U}]$ ; with $E_{U} = 50$ $meV$ , absorption falls tenfold every 115 $meV$ (background knowledge; computed), and a fit range that includes the tail pulls the intercept down. Interference fringes in films a few hundred nanometres thick add oscillations that mimic slope changes. Choosing direct when the material is indirect, or ignoring that rutile’s lowest transition is direct but forbidden, changes the result (background knowledge). In indirect materials, phonon-assisted branches start at $E_{g} \mp E_{p}$ , 116 $meV$ apart for silicon (assumed phonon energy; computed). Thermo Fisher even reports two different uncertainties, 0.02 and 0.03 $eV$ , for one rutile result. Reference: J. I. Pankove, Optical Processes in Semiconductors, Dover 1971 (background knowledge; not opened); NISER (School of Physical Sciences) laboratory handout, ‘Ultraviolet-Visible Spectroscopy to determine the nature of band gap and estimation of the bandgap’ (author not stated; table after Hummel, Electronic Properties of Materials, 3rd ed.) (fetch-tool summary); Thermo Fisher Scientific, ‘Band Gap Analysis through UV-Visible Spectroscopy’, application note AN54685 (2023) (fetch-tool summary).

Ques: How is a good UV-Vis measurement made?

With stable light, correct baselines and an unsaturated detector. NISER’s procedure warms the lamp for about 5 minutes, starts with averaging and exposure settings of 5, keeps detector counts below 60000 of a 65500 maximum, records a dark spectrum with the lamp off without changing settings, and holds film samples by their edges, normal to the beam. A reference through the bare substrate removes its absorption and reflection. Thermo Fisher used 1.0 $nm$ bandwidth and step with 0.25 to 0.5 $s$ integration. High absorbances above about 3 leave too little light to measure accurately, so thick, strongly absorbing films must be thinned or measured in reflectance (background knowledge). Data are then converted to photon energy and absorption coefficient for analysis. Reference: NISER (School of Physical Sciences) laboratory handout, ‘Ultraviolet-Visible Spectroscopy to determine the nature of band gap and estimation of the bandgap’ (author not stated; table after Hummel, Electronic Properties of Materials, 3rd ed.) (fetch-tool summary); Thermo Fisher Scientific, ‘Band Gap Analysis through UV-Visible Spectroscopy’, application note AN54685 (2023) (fetch-tool summary).

References

NISER School of Physical Sciences, “Ultraviolet-Visible Spectroscopy to determine the nature of band gap and estimation of the bandgap,” M.Sc. laboratory handout (author not stated). Available: https://www.niser.ac.in/sps/assets/files/msc/Band_gap_estimation_with_Uv-Vis.pdf
Thermo Fisher Scientific, “Band Gap Analysis through UV-Visible Spectroscopy,” application note AN54685, 2023. Available: https://documents.thermofisher.com/TFS-Assets/CAD/Application-Notes/band-gap-analysis-uv-visible-spectroscopy-an54685-en.pdf
J. I. Pankove, Optical Processes in Semiconductors. New York: Dover, 1971 (cited from background knowledge).
N. Nepal, J. Li, M. L. Nakarmi, J. Y. Lin and H. X. Jiang, “Temperature and compositional dependence of the energy band gap of AlGaN alloys,” Applied Physics Letters, vol. 87, 242104, 2005 (as used on the Photoluminescence Spectrometer page).

Quartz 5

Explorer

UV-Vis Spectrophotometer