16
J. Eur. Opt. Society-Rapid Publ. 22, 2( 2026)
Figure 4.( a) Transmittance and( b) reflectance as a function of wavelength for the Bi 2-x Mn x O 3 thin films.
Figure 5. Absorbance coefficient for the Bi 2-x Mn x O 3.
the creation of localized energy states that modify the band structure, thereby reducing the energy required for particular electronic transitions to occur.
The Urbach tail, corresponds to photon energies below the bandgap, indicates structural disorder in thin-film semiconductors, related to localized states [ 28 ]. The established empirical approach can be applied to calculate the Urbach energy in the low energy region, as detailed in references [ 29 – 31 ].
am ðÞ¼a 0 exp hm E u
: ð4Þ
The symbols m, h, a 0, anda denote the frequency, Planck’ s |
constant, a material-specific constant, and the absorption |
coefficient, |
respectively. Equation |
( 2) |
is |
expressed |
as |
follows: |
|
|
|
|
|
|
ln ðam ðÞÞ ¼ ln ða 0
Þþ 1 hm:
E u
|
ð5Þ |
By analyzing the slope of the relationship between photon energy and ln( a)( Fig. 6a), the Urbach energy values were calculated. As exhibited in Figure 6b, the Urbach energy rises with higher Mn concentrations. This can be attributed to significant alterations in the electronic properties due to increased Mn content, which modifies the band structure and broadens the band tail, elevating the Urbach energy. The Williamson – Hall analysis( Table 1) indicates that the x = 0.1 sample exhibits the highest microstrain( e = 0.0093) in the series, reflecting a pronounced lattice distortion that generates band-tail states through the formation of localized energy levels within the band gap. This effect is further intensified by the considerable reduction in crystallite size from approximately 1633 nm( x = 0) to about 541 nm( x = 0.1), which significantly increases the grain boundary area. These boundaries represent disordered regions rich in suspended bonds and defects, leading to a higher density of intra-gap states and a consequent broadening of the band tails. Morphological observations from the FESEM images( Fig. 2c) support these findings, as the x = 0.1 sample displays a rough, porous surface with evident structural flaws, directly illustrating the increased disorder at this doping level. Moreover, the incorporation of Mn ions at high concentrations introduces substantial electronic disorder through charge compensation mechanisms – such as the formation of oxygen vacancies – and localized lattice distortions around the dopant sites, collectively contributing to the pronounced tailing of the band edges. Similar behavior was observed in S x WO 3, where higher S content correlated with increased Urbach energy [ 29 ].
Additionally, the optical energy gap( E opt g
) for Bi 2-x Mn x O 3 thin films at the high absorption edge was calculated by the Tauc formula [ 30 ] as follows:
A: ð6Þ ahm ¼ B hm � E opt g
Here, B represents the Tauc parameter, and A is a constant. Semiconductors can facilitate optical transitions either directly or indirectly [ 31 ]. Equation( 4) can be written as:
Ap ffiffiffiffiffiffiffi ahm ¼ ffiffiffiffi
Ap B hm � E opt
Ap ffiffiffiffiffiffiffi g
Or; ahm
Ap ¼ ffiffiffiffi
Ap ffiffiffiffi B hm � B E opt g
: ð7Þ ffiffiffiffiffiffiffi
Equation( 5) shows the straight-line relation between Ap ahm and hm. The optical band gap( E opt g
) isdeterminedfromthe x-axis intercept of the extrapolated line as follows: At Ap ffiffiffiffiffiffiffi
Ap ahm ¼ 0) ffiffiffiffi
Ap ffiffiffiffi B hm ¼ B E opt g
) hm ¼ E opt g
: ð8Þ
Now, to determine the direct and the indirect allowed transition, this can be obtained using the constant A. The value of“ A” depends on the nature of the electronic transition