JEOS RP ISSN03 | Página 25

18

J. Eur. Opt. Society-Rapid Publ. 22, 2( 2026)

Table 2. Bi 2-x Mn x O 3 thin-film optical parameter changes.

Sample Bi 2-x Mn x O 3

Direct

E Opt g( eV)

E u

( eV)

E d

( eV)

E 0

( eV)

k 0

( nm)

N / m * 10 55( m �3 kg �1)

e L

� n 2 �1 �

� 1 ¼ s 0 k 2 �1 1 � ðkÞ �2:

0 s 0 ð12Þ

Figure 8. Extinction Coefficients versus wavelength for Bi 2-x Mn x O 3 thin films.

� n 2 �1

� 1 ¼ E2 0

� ð hm Þ2 ¼ E 0

� ðhmÞ2 �

) n 2 � 1 E 0 E d E d E 0 E d

¼ E 0

� 1 ðhmÞ 2: ð10Þ E d E 0 E d

The parameters E 0 and E d were determined through a graphical analysis of( n 2 � 1) �1 vs( hv) 2, as presented in Figure 8b. Table 2 systematically lists the values of E 0 and E d. The lattice dielectric constant( e 1) and static refractive index( n 0) were calculated using a formula dependent on the constants E 0 and E d [ 37 ].

e 1 ¼ E d þ E 0

E 0 and n 0 ¼

�1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E d þ E 0

: ð11Þ

E 0

Table 2 reveals that the calculated values of both n 0 and e 1 increase with a higher Mn content, indicating enhanced optical density and dielectric polarization in the investigated thin films due to Mn incorporation. Additionally, the oscillator strength of the material, calculated as [ f ¼ E d E 0 ], was determined using the parameters E 0 and E d and is reported in Table 2. The Sellmeier dispersion model was utilized to calculate the resonance wavelength( k 0) and oscillator strength( s 0) at lower frequencies. This model is essential for elucidating the interaction of light with the material across various wavelengths. Its efficacy at lower frequencies highlights its suitability for analyzing optical properties [ 38 ].

Figure 9c illustrates the determination of k 0 and s 0 values through linear fitting of 1 /( n 2 �1) versus 1 / k 2, withthe resulting values reported in Table 2. The lattice dielectric constant( e L) quantifies the material’ s response to an applied electric field at low frequencies, reflecting its capacity to store electrical energy as polarization [ 39, 40 ]. The term( N / m *), representing the charge carrier contribution to the refractive index, indicates that the refractive index of the material changes with increasing carrier density or decreasing effective mass. Consequently, accurate measurement of e L and N is essential for characterizing the m

investigated thin films. Figure 8d depicts the relationship between n 2 and k 2, fromwhiche L and N are derived using m equation( 11) [ 41 ]. n 2 e 2 N ¼ e L � k 2: ð13Þ

4p 2 c 2 e 0 m The obtained values of e L and N are listed in Table 2. The m

observed increase in both the lattice dielectric constant( e L) and the charge carrier contribution to the refractive index( N) with higher Mn content indicates that the optical m

and electronic properties of Bi 2-x Mn x O 3 thin films are increasingly conducive to polarization and charge transport. This enhancement suggests potential for improved performance in optoelectronic applications.

The transport properties, grain structure, and grain boundary behavior of compounds can be analyzed through thedielectricloss( e 2) and dielectric constant( e₁). Additionally, the dielectric constant provides insight into a material’ s capacity to store electrical energy [ 42 – 47 ]. These parameters also characterize the response of the material to both optical and electric fields. The real component( e₁) governs the refractive index and phase velocity of light within the medium, whereas the imaginary component( e 2) relates to absorption, indicating the extent of light attenuation by the material. The real and imaginary components of the complex permittivity of the thin films can be derived using the following expressions, respectively [ 25 ]:

e 1 ¼ n 2 � k 2; ð14Þ e 2 ¼ 2nk: ð15Þ

Figures 10a and 10b present the spectral dependence of the real( e 1) and imaginary( e 2) dielectric constants,