Nonlinear optical parameters and electronic properties of plasma polymerized methyl acrylate thin films

Abstract

Plasma polymerized methyl acrylate (PPMA) thin films were fabricated on a borosilicate glass substrate at a plasma power of 28 W to study nonlinear optical parameters and electronic properties. X-ray Diffraction analysis confirmed the amorphous nature of the PPMA films, while Attenuated total reflectance Fourier transform infrared spectroscopy indicated monomer fragmentation due to plasma polymerization. Field emission scanning electron microscope images of the films display a water wave-like structure. PPMA films demonstrated thermal stability up to approximately 574 K in both N₂ and air environments. The values of direct band gap energies increase from 3.30 to 3.41 eV with increasing film thicknesses, whereas the Urbach energy values show in an opposite manner. The Wemple-DiDomenico model was employed to analyze both the oscillator energies ranging from 5.54 to 6.44 eV and the dispersion energies ranging from 10.56 to 12.89 eV. The static linear refractive index showed typical dispersion behavior, with film thickness conforming to Moss's rule. The nonlinear refractive index and 2nd and 3rd order nonlinear susceptibilities decrease with increasing the studied films. The lattice dielectric constant values exceeded the high-frequency dielectric constant, indicating the presence of lattice vibrations and free carriers. Electronic parameters are fluctuating with increasing film thickness. The observed changes in nonlinear optical parameters and electronic properties highlight the potential of PPMA films for applications in photovoltaic and optoelectronics.

Highlights

• •Plasma polymerized methyl acrylate thin films were synthesized onto glass substrates.
• •X-ray diffraction analysis confirmed the amorphous nature of the studied films.
• •Thermal stability of the studied films is found up to 574 K in both the N₂ and air environments.
• •Direct band gap increases from 3.30 to 3.41 eV with increasing film thickness.
• •Nonlinear refractive index decreased with increasing film thickness.
• •Electronic parameters are fluctuating with increasing film thickness.

1. Introduction

The non-interacting nature of photons in linear optics makes it a highly valuable tool in many cutting edge communication technologies. However, nonlinear optics describes the interacting property of photons that result from the changing of characteristics of various optical materials in the presence of extremely high light intensity [1]. As a result, the nonlinear optical characteristics of amorphous thin films have attracted a lot of attention in the manufacturing of optical switching, photonic devices, and other optical limiting applications [2,3]. Due to the increased interest in optical switching and signal processing devices, the development of optimal non-linear optical materials is becoming increasingly important. On top of this, amorphous materials are a great option for infrared sensors and ultrafast optical devices due to their high refractive index (n₂) and high nonlinear optical susceptibility (χ⁽³⁾) [4,5]. Furthermore, a variety of significant techniques such as chemical vapor deposition, plasma polymerization (PP), spray/spin coating, atomic layer deposition, sol gel, and plasma-enhanced chemical vapor deposition for the fabrication of functional polymer thin films [[6], [7], [8], [9], [10], [11]]. However, the PP technique offers excellent substrate adhesion, chemical viability, and confirmative ultra-thin film production. Almost any substrate can be utilized to apply thin films with it. Additionally, PP finishes all of the steps in essentially one step, whereas traditional polymer coating of a substrate requires several steps [12]. The PP technique is highly recommended since it can create polymer thin films from nearly any organic vapor, making it feasible to create thin films that would be impossible to create with other methods [13]. A review of the literature reveals that methyl acrylate (MA) derivatives are crucial in the state-of-the-art technology. Therefore, MA has been chosen as the monomer precursor for the deposition of plasma polymerized methyl acrylate (PPMA) thin films. MA is very harmful if breathed in, swallowed, or absorbed through the skin. If the lab isn't well-ventilated, researchers need to wear special gear like respirators, gloves, coats, and goggles for safety. However, after plasma polymerization, MA becomes safe, eco-friendly, and can break down naturally. The produced films exhibit desirable characteristics for a range of promising state-of-the-art applications, including biomedical and microelectronic devices, optoelectronics, rechargeable batteries, sensors, photovoltaics, light-emitting diodes [14]. Yaseen et al. [15] calculated the (χ⁽³⁾) value ranges near about 1.37 × 10⁻¹² esu of an octahedral metalloporphyrin thin film. They also observed that the high frequency dielectric constants (ε_∞ = 4.76) is smaller than the lattice dielectric constants (ε_L = 5.79) which confirmed the existence of free carriers and lattice vibrations. Abdullah et al. [16] obtained the first-order linear susceptibility (χ⁽¹⁾), χ⁽³⁾ and n₂ values reached near about 2.14, 3.5 × 10⁻⁹ esu, and 5.8 × 10⁻⁸ esu, respectively. Hassanein and Sharma [17] determined the values of oscillator resonance energy (E_o) and the oscillator dispersion energy (E_d) of Cu_25-x(ZnGe)_25-xSe_50+2x thin films ranging from 2.94 eV to 3.42 eV and 24.34–30.83 eV, respectively. Hassanein and Sharma [18] found that the electronic polarizability (α_p) slightly increased from 3.941 × 10⁻²⁴ to 3.981 × 10⁻²⁴ cm³ of CZGSe thin films. Priyadarshini et al. [19] observed the large nonlinearity of Bi_xIn_35-xSe₆₅ thin films where the obtained χ⁽³⁾ value ranges from 1.54 × 10⁻¹¹ to 7.09 × 10⁻¹¹ esu and n₂ value ranges from 2.07 × 10⁻¹⁰ to 8.02 × 10⁻¹⁰ esu for direct transition. They also observed that the value of optical density (OD) increases with increasing wavelength which showed the same trends as the absorption coefficient. Taleb et al. [20] found the values of E_o and E_d of the Ge₂₆In_xSe₆₉ (x = 1, 2, 3 and 5) amorphous thin films ranging from 3.38 to 3.97 eV and 18.06–19.55 eV, respectively. They also found the values χ⁽³⁾ ranging from 2.91 × 10⁻¹² to 7.60 × 10⁻¹² esu and n₂ ranges between 4.65 × 10⁻¹¹ and 1.10 × 10⁻¹⁰ esu and plasma frequency ranging from 3.46 × 10¹⁴ to 3.61 × 10¹⁴ Hz. Hassan et al. [21] calculated the values E_o and E_d which ranges from 4.22 eV to 5.20 eV and 1.98–2.14 eV, respectively of the Cu₂O thin film. Previous research based on the derivatives of methyl acrylate has largely overlooked the electronic parameters, such as plasma frequency, valence electron plasmon energy, Penn energy gap, Fermi energy, and electronic polarizability. This manuscript seeks to address this gap by systematically examining these parameters across varying plasma polymerized methyl acrylate (PPMA) thin films film thicknesses, thereby contributing valuable insights into the electronic characteristics of PPMA films. The authors studied in the previously published few papers based on optical, electrical and others properties of the PPMA thin film samples at different plasma power which were successfully synthesized by using the plasma polymerization technique.

The aim of the present work is to produce thin films of plasma polymerized methyl acrylate (PPMA) of various thicknesses using the PP method at a plasma power of 28 W. The authors estimated the thickness-dependent linear optical constants of the films ranging in wavelength from 200 to 1100 nm from the absorbance data using ultraviolet visible (UV–VIS) spectroscopy. They also calculated the dispersion of the refractive index and discussed based on the Wemple Di-Domenico single oscillator model [22]. Furthermore, they also calculated the value of the optical nonlinear susceptibility (χ⁽³⁾) and nonlinear refractive index (n₂) utilizing empirical relations according to Miller's rule [23]. The authors studied the electronic properties such as electronic polarizability in three different methods of the PPMA thin films. Furthermore, the authors also investigated the structural, chemical and thermal properties of PPMA thin films by X-ray diffraction (XRD), attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR), and thermogravimetric analysis (TGA) respectively. As PPMA film thickness increases, both linear and nonlinear parameters can be changed to regulate the thin film's optical and electrical characteristics. These qualities could show whether PPMA is appropriate for usage in different kinds of electronic and optoelectronic non-linear devices.

2. Materials and experimental characterizations

2.1. Sample preparation

Methyl acrylate (MA) is a chemical bought from VWR Inter. Ltd. It's used as is, with its formula being C₄H₆O₂, density 0.95 g/cm³, weight 86.1 g/mol, purity ≥99.5 % and boiling point 80⁰ C. Fig. 1 shows the chemical composition of MA. Borosilicate glass substrates, known for their exceptional chemical stability, come in dimensions of 25.4 mm × 76.2 mm × 1.2 mm (Sail Brand, China) were used for the deposition of PPMA thin films. Their durability and resistance to thermal shock make them ideal for electronics, optics, and laboratory applications where reliability is key. After the cleaning process, substrates were positioned on the bottom electrode within a bell jar-type plasma chamber, equipped with a pair of stainless steel electrodes, each with a thickness of 0.001 m and a diameter of 0.09 m, spaced 0.039 m apart. The reaction chamber was evacuated using a rotary pump and maintained at a pressure of approximately 13.3 Pa. An AC power supply (50 Hz) was connected to the electrodes via the bottom flange. A monomer container, connected to a flow meter with a flow rate of 20 cm³/min, was attached to the top flange. The monomer entered the system due to the pressure differential between the container and the chamber. PPMA thin films were deposited at 28 W for 45–90 min in 15-min intervals to achieve substantial thickness.

Fig. 1.

Chemical composition of Methyl Acrylate.

Numerous techniques, including ellipsometric, interferometric, intensity measurement, and guided wave approaches, have been used to determine the thickness of nominally transparent films. To measure the thickness of the film, the Fizeau interference technique was utilized because in this method there is no need for polarizing optics, changes in wavelength or angle of incidence, or accurate comparisons of light intensities using this approach [24]. Thickness of the film was calculated using equation (1) [25]:

$$d=\frac{\lambda y}{2x}$$ (1)

Here, λ stands for the wavelength of Na light, which is 589.3 nm; x signifies the fringe width; and y represents the step height between the fringe's neighboring test surfaces.

2.2. Experimental techniques

An X-ray diffractometer (Philips PW3040 X′ Pert PRO) was used to analyze the amorphous nature of PPMA films. The instrument operated at 30 mA and 40 kV, with a scanning rate of 1.2°/min and Cu-K_α radiation, over a 2θ range of 20°–80°. A field emission scanning electron microscope (FESEM) (Gemini Sigma 300, Zeiss, Germany) with a variable accelerating voltage of 0.02–30 kV was utilized for a detailed analysis of the ribbon surfaces. A smart SEM software (Zeiss, Germany) was used to collect digitized images. Attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR) (Bruker Alpha, FTIR, Germany) was performed on PPMA powder and liquid MA samples, covering the wavenumber range between 650 and 4000 cm⁻¹. In ambient conditions, the spectra were recorded as a percentage of T % mode. A potassium bromide measuring cell was utilized for measuring the liquid MA's FTIR spectra. TGA analysis of powder PPMA samples collected from the substrate's surface was conducted by TGA instrument (Seiko EXSTAR 6000 station, Japan), employing a heating rate of 10 K/min, under both nitrogen (N₂) and air atmospheres. Alumina was utilized as the reference material for comparison during the analysis. Utilizing a UV–vis spectrometer (UV-1601, SHIMADZU, Japan), optical reflectance R(λ) and transmittance T(λ) spectra were recorded at room temperature across the 200–1100 nm wavelength range.

3. Results and discussion

3.1. XRD analysis

The XRD data for PPMA thin films of various thicknesses ranging from 136 to 245 nm, are depicted in Fig. 2. The absence of clear Bragg peaks in the XRD patterns suggests that the PPMA thin films are amorphous in nature. This lack of crystalline structure implies that the continuous interaction between ions and electrons at the film surface during plasma polymerization inhibits crystallization [26,27].

Fig. 2.

XRD spectra of the studied thin films.

3.2. Surface morphology analyses

Fig. 3 shows FESEM images of 180 nm thick PPMA thin films at magnifications of × 30 k, × 50 k, × 100 k and × 150 k. The lower magnification images reveal a smooth surface without fractures, while higher magnification uncovers a wave-like structure, suggesting the presence of aggregates due to prolonged polymerization and cross-linking [28].

Fig. 3.

FESEM micrographs of thickness 180 nm were taken at (a) × 30 k, (b) × 50 k, (c) × 100 k, and (d) × 150 k magnifications.

3.3. ATR-FTIR analyses

Fig. 4 displays the ATR FTIR spectra of MA and PPMA at 28 W, revealing changes in chemical structure due to plasma polymerization. In the PPMA spectrum, absorption bands appeared between 3380 and 3395 cm⁻¹ (A), indicating OH stretching vibrations. The C-H stretching is observed between 2856 and 3018 cm⁻¹ (B) for both MA and PPMA, important for recognizing the compound as organic [29]. The absorption peak at 1727 cm⁻¹ (C) in MA and 1700 cm⁻¹ in PPMA are attributed to carbonyl compounds. Absorption bands between 1617 and 1637 cm⁻¹ (D) indicate the presence of C=C stretching vibrations in both MA and PPMA. Both MA and PPMA display absorption bands at approximately 1438 cm⁻¹ (E), which correspond to the asymmetric bending of CH₃ groups. In the PPMA, these peaks shift towards lower wavenumbers because of CH₂ bending vibrations (F), resulting from hydrogen loss in the CH₃ groups. Within range 800–1200 cm⁻¹, PPMA exhibits chemical structural changes that are ascribed to bending vibrations in the C-O and C=C complexes [[30], [31], [32]]. Table 1 provides a summary of the assigned positions for different bands of MA and PPMA, facilitating comparison. Analysis of the FTIR spectra indicates that the proportion of hydrophilic groups (C-O) is lower than that of hydrophobic groups (C=C and C-H), suggesting that both surfaces become enriched in carbon content [33].

Fig. 4.

ATR-FTIR spectra of the studied thin films.

Table 1.

Assignments of ATR-FTIR absorption bands for MA and PPMA at 28 W.

Assignments	Absorption Bands	Wavenumber (cm−1)		Band intensity
		MA (Monomer)	PPMA films
OH stretching	A	–	3380	Broad peak
CH3 stretching	B	2957	3018	Merged
CH2 stretching		2917	2926
C-H stretching		2856	2864
C=O stretching	C	1727	1700	Broad peak
C=C stretching	D	1635	1617	Very low
CH3 bending	E	1439	1438	Merged
CH2 bending	F	1403	1380
C-O	G	1276	1306	Low
	H	1203	–	Merged
	I	1181	1170
	J	1069	1052
C=C bending	K	987	1026	Broad peak
	L	854	886	Very low

3.4. Thermal analysis

The thermal stability of PPMA powder samples collected from glass substrates was investigated using a TG instrument, with thermograms captured in temperature ranges of 309–850 K under both air and N₂ environments. In Fig. 5, different stages of weight loss because of heating are illustrated. The thermal degradation behavior of studied films is characterized by three distinct stages, denoted as A, B, and C, corresponding to increasing temperature ranges. In the initial stage (region A, 309–366 K), there is a minimal weight loss (1.92 % for N₂ and 2.49 % for air environments), likely attributed to the volatilization of low molecular mass species such as H₂O and CO₂ formed during thermal stabilization [34]. This observed low weight loss suggests the hydrophobic nature of the PPMA films. Moving to region B (366–574 K), weight loss increases to 4.18 % for N₂ and 6.37 % for air environments, indicative of the removal of inactive oligomers or monomers [35]. Finally, in region C (574–794 K), a significant weight loss is observed (39.87 % for N₂ and 64.22 % for air environments), attributed to polymer chain breakdown [36]. Despite this, a considerable portion of the original weight remains intact (54.03 % for N₂ and 26.92 % for air environments). Consequently, it can be declared that PPMA exhibits thermal stability up to 574 K in both N₂ and air environments, as illustrated in Fig. 5.

Fig. 5.

Comparison of weight loss of PPMA in air and N₂ environments.

3.5. Linear optical properties

3.5.1. Absorbance (A)

In the UV–visible range (200–1100 nm), Fig. 6 show a pronounced absorbance peak at 300 nm, followed by a rapid decrease until approximately 400 nm, after which the absorbance levels off and approaches a constant value near 500 nm. The absorption peak intensity of PPMA films remains relatively stable in the visible region and approaches zero at higher wavelengths, indicating high transparency, while maximum absorbance occurs in the UV range due to electronic transitions within the organic molecules or conjugated systems; the position and intensity of these peaks suggest the presence of functional groups in the material [37].

Fig. 6.

Variations of absorbance as a function of λ for studied films.

3.5.2. Transmittance (T) and reflectance (R)

Fig. 7 illustrates the spectral distribution of T and R concerning λ for PPMA thin films. Notably, T decreases with escalating film thickness, whereas R exhibits an inverse relationship with T. When the T spectra were examined, the films showed excellent optical transparency, which was roughly measured to be 80 %. This exceptional optical transparency is very beneficial in optoelectronic devices where the device's overall performance and energy conversion efficiency depend heavily on the effective capture of sunlight. In particular, there was a noticeable decrease in the film's transmittance as the thin film thickness rose. This phenomenon may be systematically explained by the intrinsically absorbing properties of PPMA, whereby an increase in film thickness probably amplifies the absorption of received light, hence reducing its transmittance [38].

Fig. 7.

Transmittance and reflectance spectra as a function λ of the studied films.

3.5.3. Absorption coefficient (α)

In amorphous materials, the absorption coefficient (α) spectra can be delineated into three distinct regions [39]: (i) α > 10⁴ cm⁻¹, governing the band gap energy (E_g) (ii) 1 < α < 10⁴ cm⁻¹, strongly correlated with the structural randomness, and (iii) α < 1 cm⁻¹, stemming from defects and impurities.

The absorption power of the studied films, as assessed through the α, is computed using the following expression (2) [40]:

$$\alpha \left(\lambda \right)=\frac{1}{d}\mathit{ln}\left[\left(\frac{{\left(1-R\right)}^2}{T}\right)+\sqrt{\left(\frac{{\left(1-R\right)}^4}{{4T}^2}+R^2\right)}\right]$$ (2)

;where d represents film thickness, T (λ) denotes the values of transmission, and R (λ) is the values of Reflection of the PPMA thin films, As the film thickness increased, Fig. 8 demonstrates a rise in the E_g. The observed α, roughly of the order of 10⁴ cm⁻¹. Furthermore, it diminishes with increasing λ, suggesting enhanced transparency of the film. Consequently, light waves can traverse the film more swiftly and effortlessly.

Fig. 8.

Variation of absorption and extinction coefficient (Inset) as function of λ the studied films.

3.5.4. Extinction coefficient (k)

The extinction coefficient (k) characterizes how much incident light interacts with a material, reflecting its absorbing capability. It's determined by equation (3) [41]:

$$k=\frac{\alpha \lambda }{4\pi }$$ (3)

This constant, k, also indicates the material's ability to polarize light, as depicted in the inset of Fig. 8. The increasing light loss from electromagnetic wave scattering is indicated by the rise in k value with PPMA thin film thickness. Stronger nonlinear optical characteristics of the material are indicated by a larger k value. Additionally, surface defects and structural irregularities are connected to it.

3.5.5. Optical band gap (E_g)

In the strong absorption area (α > 10⁴ cm⁻¹), the E_g is connected by the following equation (4) [42], which follows Tauc relation:

$$\alpha =\frac{C{\left(E-E_g\right)}^p}{E}$$ (4)

;where C is a Tauc parameter which depends on the transition probability, and E = hυ is the photon energy. The parameter, p, indicates the nature of the optical transition: 2 for indirect allowed, 1/2 for direct allowed, 3 for indirect forbidden, and 3/2 for direct forbidden transitions [43]. At first introducing the natural logarithm in both side of equation (4) and then differentiate both sides, we get equation (5):

$$\frac{d\left[\mathit{ln}\left(\alpha h\nu \right)\right]}{d\left(h\nu \right)}=\frac{p}{h\nu -E_g}\text{.}$$ (5)

In this study, a significant observation was made regarding the behavior of the d[ln(αhv)]/d(hv) versus hv curves in Fig. 9 (a) at the band gap energy (hv = E_g). This discontinuity at E_g signifies the band gap. To further analyze this, ln(αhv) versus ln(hv- E_g) curves were plotted in Fig. 9 (b) using the E_g value to determine the p value [44]. It was found that the value of p is approximately equal to 1/2. This p value confirms the transition is direct allowed transitions ($E_g^d$). Direct transitions across the band gap occur between the valence and conduction band edges in k-space, where it's imperative to uphold the conservation of both total energy and momentum within the electron-photon system. Fig. 10 shows the plots of (αhv)² versus h $\nu$ and the intercept cuts the h $\nu$ axis at y = 0 gives the E_g for different d of PPMA. The values of $E_g^d$ are given in Table 2. It is noticed from Table 2 that $E_g^d$ is systematically increased from 3.30 to 3.41 eV with increasing d. The obtained $E_g^d$ values are suitable for many technological applications [45].

Fig. 9.

Fig. 9 (a) d[ln(αhv)]/d(hv) against hv for the studied films. Fig. 9 (b) ln(α_lcohv) versus ln(hv- Eg) for the studied films.

Fig. 10.

Variation of (αhν)² with hν for the studied films.

Table 2.

Linear and nonlinear optical parameters of the PPMA thin films at 28 W.

Optical parameters	Thickness, d
	136 nm	180 nm	210 nm	245 nm
$E_g^d$ (eV)	3.30	3.32	3.37	3.41
Eu (eV)	1.00	0.94	1.05	0.68
n	2.596	2.603	2.619	2.638
E0 (eV)	6.32	5.54	6.44	6.35
Ed (eV)	11.92	12.89	10.56	11.76
E0/$E_g^d\approx 2$	1.91	1.67	1.91	1.86
n0	2.32	2.31	2.30	2.29
Egn04∼ Constant (95)	95.7	95.50	94.96	94.54
$x^{\left(1\right)}$	0.349	0.347	0.343	0.339
$x^{\left(3\right)}$ (esu)	2.521 × 10−12	2.471 × 10−12	2.349 × 10−12	2.257 × 10−12
$n_2($ esu)	4.096 × 10−11	4.022 × 10−11	3.843 × 10−11	3.707 × 10−11
ε∞	5.385	5.363	5.308	5.265
${\epsilon }_L$	6.75	6.87	6.41	7.09
$\frac{N}{m^{\ast }}$ (m−3 kg−1)	3.22 × 1055	3.29 × 1055	4.98 × 1055	4.23 × 1055
ωp (Hz)	1.175 × 1014	1.177 × 1014	1.50 × 1014	1.31 × 1014
$\psi$ (eV)	0.077	0.077	0.099	0.087
$E_p$ (eV)	0.037	0.037	0.047	0.042
$E_F$ (eV)	0.009	0.009	0.013	0.011

3.5.6. Urbach energy (E_u)

The presence of defect levels within the forbidden gap is indicated by the Urbach tail. The Urbach energy (E_u) can be determined using formula (6) [46]:

$$\alpha ={\alpha }_0\mathit{exp}\left(\frac{h\nu }{E_u}\right)$$ (6)

;where α₀ is a constant, h is Planck's constant. Plotting ln(α/α₀) against hν and fitting a straight line to the curve allows the slope to be determined as (1/E_u). The E_u value, which indicates the level of disorder in amorphous materials, can be obtained by taking the reciprocal of this slope [47]. The calculated values are tabulated in Table 2 where E_u values ranges from 0.68 to 1.05 eV. This decline suggests a decrease in defect states within the specific area and a reduction in disorder, ultimately resulting in a rise in the $E_g^d$ [48]. This trend of increasing $E_g^d$ and decreasing E_u with d is illustrated in Fig. 11.

Fig. 11.

Variation of ln(α/α₀) with hν for the studied films.

3.5.7. Optical density (OD)

The concentration of absorbing material and the d are connected to the optical density (OD), also known as absorbance. One may estimate the OD of the films described using formula (7) [49]:

$$OD=\alpha \times d$$ (7)

The fluctuation in OD with λ is depicted in Fig. 12, and it exhibits the same variation as the $\alpha$. High $\alpha$ values are the cause of increasing tendency of the OD values with respect to d. This pattern demonstrates unequivocally how the materials' capacity to absorb radiation increases when subjected to incoming radiation.

Fig. 12.

Optical density as a function of λ for the studied films.

3.5.8. Linear refractive index (n)

Light transmission through a substance can be determined using the linear refractive index (n). To calculate n, the following Fresnel's formula (8) in terms of R and k is used [50]:

$$n=\left(\frac{1+R}{1-R}\right)+\sqrt{\frac{4R}{{\left(1-R\right)}^2}-k^2}$$ (8)

The calculated n values obtained through the Fresnel method can be adjusted using Cauchy's formula (9) [51]:

$$n=a+\frac{b}{{\lambda }^2}$$ (9)

;where a and b are constants. This formula is employed to correlate the n values across all λ, as depicted in Fig. 13. It's observed that the n decreases rapidly with increasing λ, demonstrating normal dispersion behavior. Additionally, the n increases with d, ranging from 2.596 to 2.638. The high n value of the material renders it suitable for photonic applications [52].

Fig. 13.

Variation of the n as a function of λ for the studied films.

3.6. Nonlinear optical properties

3.6.1. Oscillator energy (E_o) and dispersion energy (E_d)

Dispersion is crucial in optical material research, especially for optical communication and designing devices for spectral dispersion. The optical spectra of n can be evaluated using the Wemple-DiDomenico model for single oscillator (10) [53]:

$$n^2=1+\frac{E_0{E}_d}{{E_0}^2-{\left(h\nu \right)}^2}$$ (10)

;where E_o signifies the oscillator energy that estimates the typical energy difference, while E_d represents dispersion energy, quantifying the average intensity of interband optical transitions. The plot of 1/($n$ ² – 1) against $\left(h\nu \right)$² for PPMA films of various d is shown in Fig. 14. From the slope [=(E_oE_d)⁻¹] and its intersection $\left[=\frac{E_0}{E_d}\right]$, the E_o and E_d values are calculated. The obtained oscillator energies ranging from 5.54 to 6.44 eV and the dispersion energies ranging from 10.56 to 12.89 eV. Furthermore, the values of $Eo\approx 2E_g^d$ as was found by Tanaka [54].

Fig. 14.

Plots of 1/($n$²– 1) against $\left(h\nu \right)$² for the studied films.

The static linear refractive index (n₀) and high-frequency dielectric constant (ε_∞) of the PPMA thin films were calculated utilizing the Dimirov and Sakka relation (11) [55]:

$$n_0=\sqrt{{\left(\frac{180}{E_g^d}\right)}^{1/2}-2}$$ (11)

$${and\epsilon }_{\infty }=n_0^2$$ (12)

;where $E_g^d$ represents the optical band gap estimated using the Tauc relation. All values of E_d, E_o, ε_∞, and n₀ are listed in Table 2. It's observed that the n₀ values decreases as d increases. This behavior also adheres to Moss's rule [56], indicating that $E_g^d{n}_0^4$ remains approximately constant. Interestingly, the variation in $E_g^d$ shows an opposite trend compared to the n₀.

3.6.2. Third order nonlinear susceptibility (χ⁽³⁾)

The nonlinearity of polymeric materials hinges on the strength of the electric field, which drives nonlinear effects within the system. Interactions among nuclei, spurred by electronic polarization, alongside their influence on bond lengths, contribute to the optical nonlinearities observed in these materials [57]. The nonlinear portion of the refractive index, n₂, correlates with nonlinear electronic polarizability, P_NL. Consequently, the total electronic polarizability (P) stemming from such interactions is expressed as equation (13) [58]:

$$P=x^{\left(1\right)}E+P_{NL},\text{where}{P}_{NL}=x^{\left(2\right)}{E}^2+x^{\left(3\right)}{E}^3$$ (13)

;where χ⁽¹⁾ signifies the linear susceptibility, while χ⁽²⁾ and χ⁽³⁾ are the 2nd and 3rd order nonlinear susceptibilities, respectively indicative of the material's electronic and nuclear structure and E is the electric field.

In optically isolated amorphous materials, the value of χ⁽²⁾ is 0. Hence, within these materials, χ⁽³⁾ arises as the primary nonlinearity, notably triggered by excitation within the transparent frequency range situated below $E_g^d$. Consequently, as per Miller's rule, one can determine the values of χ⁽¹⁾ and χ⁽³⁾ for PPMA thin films using the relationship outlined in equation (14) [59]:

$${\chi }^{\left(1\right)}=\frac{n_0^2-1}{4\pi }{and\chi }^{\left(3\right)}=C{\left(\frac{n_0^2-1}{4\pi }\right)}^4$$ (14)

;where n₀ is the static refractive index for hv→0 and C is a constant = 1.7 × 10⁻¹⁰ e.s.u. The obtained χ⁽¹⁾, χ⁽²⁾ and χ⁽³⁾ values for various $E_g^d$ and d are tabulated in Table 2. It is observed that the χ⁽³⁾ value for direct band transitions decreased with increasing d. The variation of χ⁽³⁾ and $E_g^d$ as a function of d are clearly shown in Fig. 15. The susceptibility decreases with increasing d due to changes in the material structure resulting from phase transformation [60].

Fig. 15.

Variation of $E_g^d$ and χ⁽³⁾ with d for the studied films.

3.6.3. Nonlinear refractive index (n₂)

The total refractive index, ‘n’ can be written as follows: n = n₀ + n₂ <E²>, where photon energy is zero, and ‘n₂’ for non-linear refractive index, which is independent of light intensity and proportional to the mean square of the electric field applied (<E²>) to the medium, (in this case, thin films) and n₀≫ n₂ [61]. Ticha, Tichy, and Miller's rule [23,62] states that the n₂ and χ⁽³⁾ can be related by the use of the following relation (15):

$$n_2=\frac{12\pi {\chi }^{\left(3\right)}}{n_0}$$ (15)

Table 2 lists the value of the obtained n₂. The n₂ drops with d, which reduces local polarizabilities, as a result of changes in defect states within the films [63]. The significant values recorded for 'n₂' in the films suggest their potential suitability for nonlinear optical applications, such as rapid optical switching devices and high-speed signal communication [64,65].

3.7. Electronic properties

The high frequency lattice dielectric constant, ε_L, is derived from the n, through the relationship given in (16) [66]:

$${\epsilon }_r=n^2-k^2={\epsilon }_L-A{\lambda }^2$$ (16)

where, ε_r represents the real part of dielectric constant, and A denotes a parameter that correlates with carrier concentration (N) according to equation (17) [67]:

$$A=\left(\frac{e^2}{4{\pi }^2{\epsilon }_0{c}^2}\right)\left(\frac{N}{m^{\ast }}\right)$$ (17)

;where e = electronic charge, ε₀ represents the permittivity of free space (=8.854 × 10⁻¹² F/m), c = the speed of light and N/m∗ = the ratio of the carrier concentration to the effective mass.

The graphs of ε_r versus λ² for PPMA thin films, which are linear at high λ, are displayed in Fig. 16. The values of ε_L and N/m∗ can be obtained from the linear part's intercept and slope, respectively, and are presented in Table 2. It has been noted that the values of ε_∞ is lower than the values of ε_L, which might be attributed to the polarization process occurring inside the materials and a growth in the concentration of free charge carriers [68].

Fig. 16.

Plots of ε_r vs. λ² for the studied films.

The following features of the engineered films can be derived using equation (18) to equation (22) with respect to the superposition between the hypothesis of the single oscillator model and the classical free electron model: (i) plasma frequency (ω_p), (ii) valence electron plasmon energy (ψ), (iii) Penn or average energy gap (E_p), (iv) Fermi energy (E_F), and (v) electronic polarizability (α_p) of the PPMA thin films [69,70]:

$${\omega }_p=\sqrt{\frac{e^2}{{\epsilon }_0{\epsilon }_L}\left(\frac{N}{m^{\ast }}\right)}$$ (18)

$$and\psi =\hslash {\omega }_p$$ (19)

The Penn model [71] provides a relationship between the E_p and the ψ for the studied films is given by

$$E_p=\frac{\psi }{\sqrt{\left({\epsilon }_{\infty }-1\right)}}$$ (20)

Additionally, the computation of E_F based on Phillip's concepts [72] follows this equation (21):

$$E_F=0.2948\times {\left(\psi \right)}^{\frac{4}{3}}$$ (21)

In relation to film thickness, there was a notable difference in the computed values of the parameters specified, which are tabulated in Table 2. This verifies that the electronic structure of the film is efficiently tuned for tailored optical and electrical characteristics as the film thickness increases. Inside the amorphous materials during electrical polarization, the polarization is thought to be the primary source of wave creation. The electron gas cloud is obviously pushed from the nucleus due to internal charge distribution, producing a net dipole moment. At net displacement, mechanical restoring forces become nonlinear functions. Owing to the vast array of applications, polarizabilities in solids have been continuously optimized via research and development [73,74].

Theoretical methods have greatly helped predict accurate values for α_p, which is crucial in the quest for new nonlinear optical materials, adding fresh energy into these dynamic fields. In this study, three distinct approaches are used to find the values of α_p which are explained below: (i) Using formula (22) [75], one may ascertain the value of α_p for a material (methyl acrylate in my case) in terms of molecular weight (M = 86.1 g/mol) and density (ρ = 0.95 g/cm³), based on the traditional theory of dielectric constant:

$${\alpha }_p=\left(3.9\times {10}^{-25}\right)\times \frac{M}{\rho }\left[\frac{{\left(\psi \right)}^2{y}_0}{{\left(\psi \right)}^2{y}_0+3E_p^2}\right]$$ (22)

$$;\text{Where}{y}_0=\left[1-\left(\frac{E_p}{{4E}_F}\right)+\frac{1}{3}{\left(\frac{E_p}{{4E}_F}\right)}^2\right]$$

• (ii)In scientific study, determining a material's ε_∞ value is essential for interpreting its electrical and physical properties. In the optical frequency range, ε_∞ mainly reflects electronic polarization. Using the Clausius-Mossotti (CM) local-field polarizability model, α_p can be estimated for studied films with the following formula (23) [76]:

$${\alpha }_p=\frac{{10}^{-24}}{2.53}\times \frac{M}{\rho }\left[\frac{{\epsilon }_{\infty }-1}{{\epsilon }_{\infty }+2}\right]$$ (23)

• (iii)By plugging values into this CM equation, the theoretical α_p can be found. Furthermore, because α_p is strongly influenced by the Eg, Reddy [77] proposed an empirical formula (24) to calculate α_p:

$${\alpha }_p=\left(3.9\times {10}^{-25}\right)\times \frac{M}{\rho }\left[1-\frac{\sqrt{E_g^d}}{4.06}\right]\left({cm}^3\right)$$ (24)

The computed values of α_p, listed in Table 3, decreases with increasing d for the studied films.

Table 3.

The values of the electronic polarizability (α_p) for PPMA thin films.

Thickness, d (nm)	Electronic polarizability, αp (cm3)
	Based on Eq. (22)	Based on Eq. (23)	Based on Eq. (24)
136	1.159 × 10−23	2.127 × 10−23	1.953 × 10−23
180	1.111 × 10−23	2.123 × 10−23	1.948 × 10−23
210	1.058 × 10−23	2.112 × 10−23	1.936 × 10−23
245	1.117 × 10−23	2.103 × 10−23	1.927 × 10−23

4. Conclusions

The transmittance (T) and reflectance (R) data of amorphous PPMA thin films were utilized to determine most studied properties and parameters. The XRD results confirm the amorphous nature of the studied films. FESEM images of the films display a water wave-like structure. PPMA demonstrates thermal stability up to 574 K in both the N₂ and air environments. The monomer fragmentation has been caused by plasma polymerization which is confirmed by FTIR. When the thickness increases, the optical band gap increases from 3.30 to 3.41 eV whereas the Urbach energy values, the third-order nonlinear optical susceptibility and the nonlinear refractive index shows the opposite manner. The WDD model was used to analyze oscillator energies between 5.54 and 6.44 eV and dispersion energies from 10.56 to 12.89 eV. It is also found that WDD energy gap is consistent with $E_g^d$, where E₀ $\approx 2E_g^d$.The lattice dielectric constant exceeded the high-frequency dielectric constant indicating the presence of lattice vibrations and free carriers. The electronic parameters including plasma frequency, valence electron plasmon energy, Penn energy gap, and Fermi energy and electronic polarizability parameter (it calculated in three different methods) of the PPMA thin films are decreasing with increasing d. This confirms that with increases d, its electronic structure is effectively tuned for tailored nonlinear optical and electrical properties. Based on the obtained results it may be inferred that the studies system may be used for optoelectronics, electronics, and nonlinear devices.

CRediT authorship contribution statement

S.D. Nath: Writing – original draft, Visualization, Software, Investigation, Conceptualization. A.H. Bhuiyan: Writing – review & editing, Validation, Supervision, Resources, Conceptualization.

Data availability statement

Data will be made available on request.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.