Theoretical Analysis of the Solid State and Optical Characteristics of Calcium Sulphide [CaS] Thin Film

Abstract

The analysis of the calcium sulphide thin film material which is one of the families of chalcogenide groups of thin film materials was carried out in this work using a theoretical approach for which the propagated wave through the medium of the thin film that is deposited on a glass substrate is considered to be a scalar wave in nature. The thin film material is sectioned into twos, first section is termed homogeneous reference dielectric constant, εref where no thin film is deposited on the substrate and the second part is termed perturbed dielectric function, Δεpz containing the deposited thin film on the glass substrate. These two terms were substituted on the defined scalar wave equation that was subsequently solved using the method of separation of variable which invariably utilized in the transformation of the equation into the second type of Volterra equation. On the other hand, Green’s function approach was also introduced in order to arrive at the model equation that culminates in an expression showcasing the wave propagated through the thin film material medium. This was subsequently applied in the computation of waves, ψz that is propagating through the material medium for various wavelengths within the ultraviolet, visible, and near-infrared region of the electromagnetic wave spectrum for which the influence of the aforementioned dielectric constant and function were invoked. The computed values from this mechanism were in turn utilized in the analysis of the band gap, optical, and solid-state properties of the calcium sulphide (CaS) thin film materials.

1. Introduction

The Chalcogenide family of thin films to which calcium Sulphide belong is one of the sulphide-based thin film that has a wide range of applications and based on that, a lot of researchers have shown so much interest in its study and as a result, both experimental and theoretical techniques are being utilized to get into deep analysis to unravel more uniqueness the thin film. Based on this various growth techniques have been utilized to develop the thin film including the CBT growth technique with an emphasis on the study of its optical and structural properties [1]. Theoretically, mathematical tools have been used for the analysis of thin films of similar types by making use of their various properties in conjunction with wave propagating through the film medium. The first of its type was the use of beam propagation technique whereby the dielectric properties were employed in studying and computing beam or field propagation through a medium with variation in small refractive index [2, 3]. The beam propagation method based on diagonalization of the Hermetician operator that generates the solution of the Helmholtz equation in media with real refractive indices [4], has been utilized in this study by some researchers while others had somehow used 2x2 propagation matrix formalism for finding the obliquely propagated electromagnetic fields in layered inhomogeneous un-axial structure which also involved bean propagation [5]. Structures such as optical fibers and optical wave guides in the presence of electro-optical perturbation have been well understood by the application of this method [6, 7, 8]. Although earlier before then, work had been going on veraciously on the study of wave propagation in a stratified media, plasma and ionosphere that gave a more clear picture of atmospherics behavior as regards wave propagating through its medium [6, 9, 10]. Van Roey in his work derived a general beam propagation relation in a number of specific cases along with the extensive simulation of wave propagation in a variety of material mediums.

Scientists have also looked at the propagation of electromagnetic field through a conducting surface [3] where the behavior of wave propagated through such material coupled with the influence of the dielectric function of the medium on such material coupled with the effect of variation of the refractive index on some species of the thin film had been analyzed as well using the same approach [11]. And a close look at the concept made it clear that recognition of the importance of the effect of the refractive index of the medium and dielectric function culminated in the reality of the creation of two velocity components that normally give rise to phase and group refractive indices as considered in the study of wave propagation [12, 13, 14, 15]. Recently more complicated work had been embarked upon on the study of wave propagation through a modeled thin with dielectric perturbation in which WKB approximation in conjunction with numerical approach were used [16, 17, 18] along with beam propagation technique to unravel the mechanism of theoretical analysis of wave propagation through materials and based on that, lots of work had been veraciously carried out in term of the influence of dielectric constants and refractive index on beam propagating through materials [3, 5, 12, 17, 19] in other to ascertain their roles and efficacy on the use of beam propagation method and again the effect of the refractive index of the medium in the reality of the two velocity components that normally give rise to phase and group refractive indices as generally considered when it comes to the study of wave propagation because their influence on the propagated beam [13]. This had gone a long way to add and to reveal the efficacy of the theoretical approach in understanding the beam propagation mechanism in wave propagating through thin film material. This is seen to be achieved due to the flexibility of Green’s function as a tool because this is what facilitated the use of the iterative process that will be involved in the computational technique in this work that has enabled one to embark on this theoretical frame work [18, 19].

2. Theoretical procedures

General wave equation

$${\nabla }^2\psi \left(z\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.\frac{{\partial }^2\psi \left(z\right)}{\partial t^2}$$E1

$${\nabla }^2\psi \left(z\right)+{\omega }^2{\mu }_o{\epsilon }_o\psi \left(z\right)=0$$E2

from which we obtain the Helmholtz form of it using the separation of variable technique which is one of the methods being used for the solution of the wave propagating through a medium in which a dielectric function as defined in our model in Figure 1 which shown in Eq. (3).

Figure 1.

Scalar wave impinging upon the substrate with reference medium ε_ref and the thin film medium $\mathrm{\Delta }{\epsilon }_p\left(z\right)$.

$$\epsilon \left(z\right)={\epsilon }_{\mathit{ref}}+\mathit{\Delta \epsilon }\left(z\right)$$E3

consisting of two parts is imposed on the wave equation. The dielectric function consists of a perturbed part, $\mathit{\Delta \epsilon }\left(z\right)$ representing the part where the thin film is deposited and the reference section where no film is deposited, ${\epsilon }_{\mathit{ref}}$. Substituting the dielectric function in Eq. (2), we obtain [18]

$${\nabla }^2\psi \left(z\right)+{\mu }_0{\epsilon }_0{\omega }^2\psi \left(z\right)=-V\left(z\right)$$E4

where

$$V\left(z\right)={\gamma }^2\epsilon \left(z\right)$$E5

Green’s function technique is used to obtain an expression for a wave propagating through the film as.

$$\psi \left(z\right)={\int }_0^{z}G\left(z,z^{\prime }\right)V\left(z^{\prime }\right)\psi \left(z^{\prime }\right)$$E6

The Integral

$$\psi \left(z,z^{\prime }\right)=I\left(z,z^{\prime }\right)={\int }_o^{z^{\prime }}G\left(z,z^{\prime }\right)V\left(z^{\prime }\right)\psi (z,z^{\prime }d{z}^{\prime }$$E7

This when critically considered is a homogeneous Volterra equation of the second type with the kernel.

$$k\left(z,z^{\prime }\right)=G\left(z,z^{\prime }\right)V\left(z^{\prime }\right)$$E8

In this equation, the Neumann series method is not applicable to this type of equation; hence we apply Born approximation method that enables us to rewrite the equation as

$$\psi \left(z,z^{\prime }\right)\equiv I\left(z,z^{\prime }\right)={\int }_o^{z^{\prime }}\mathrm{\Phi }\left(z,x\right)V\left(x\right)\psi \left(z,x\right)\mathit{dx}$$E9

where we put x in place of $z^{\prime }$.

However, according to Born approximation procedure, we replace the unknown function $\psi \left(z,x\right)$ in the integral with a known function $\tilde{\mathrm{\Phi }}\left(z,x\right)$ and use it to get an approximate solution of $\psi \left(z,z^{\prime }\right)$.

That is

$$\psi \left(z,z^{\prime }\right)=I\left(z,z^{\prime }\right)={\int }_o^{z^{\prime }}\mathrm{\Phi }\left(z,x\right)V\left(x\right)\tilde{\mathrm{\Phi }}\left(x\right)$$E10

The sign on $\mathrm{\Phi }\left(z,x\right)$ is signified as an indication of the possible phase difference between the incoming wave and the outgoing wave. Thus neglecting the influence of the reflected wave in the system, we then use Green’s function as in this case, we have.

$$G\left(z,z^{\prime }\right)=\mathrm{\Phi }\left(z,z^{\prime }\right)=\frac{2}{z}\sum _{n=1}^{\infty }\frac{sin\mathit{n\pi }\raisebox{1ex}{$z^{\prime }$}\!\left/ \!\raisebox{-1ex}{$z$}\right.}{{\gamma }^2-\frac{n^2{\pi }^2}{z^2}}$$E11

This results in the solution as given

$$\begin{array}{rcl}\psi \left(z,z^{\prime }\right)& =& I\left(z,z^{\prime }\right)\\ & =& -\sum _{n=1}^{\infty }\frac{{\gamma }^2}{{\gamma }^2-\frac{n^{2{\pi }^2}}{z^2}}[\frac{\left({\epsilon }^{\prime }+c{z}^{\prime }\right)cos\left(\frac{\mathit{n\pi }}{z}+\frac{2\pi }{\lambda }\right)z^{\prime }}{\frac{\mathit{n\pi }}{z}+\frac{2\pi }{\lambda }}\\ & & +\frac{c}{{\left(\frac{\mathit{n\pi }}{z}+\frac{2\pi }{\lambda }\right)}^2}sin\left(\left(\frac{\mathit{n\pi }}{z}+\frac{2\pi }{\lambda }\right)z^{\prime }\right)-\left(\frac{{\epsilon }^{\prime }+c{z}^{\prime }}{\frac{\mathit{n\pi }}{z}+\frac{2\pi }{\lambda }}\frac{c}{{\left(\frac{\mathit{n\pi }}{z}+\frac{2\pi }{\lambda }\right)}^2}\right)]\end{array}$$E12

$\psi \left(\lambda \right)$ vs.$\lambda$ when z and $z^{\prime }$ are fixed.

In all use, we considered in the analysis the wavelength within λ = 250 nm to 1200 nm.

From the equation, it is obvious that ($\psi \left(z\right)$ tends to 0 as the dielectric constant ${\epsilon }^{\prime }\to \infty$,

Therefore, for fixed values of other parameters, the resultant solution can be approximated to any number of terms as may be required in relation to wave propagation terms.

This is used to obtain the absorption co-efficient using [20]

$$\begin{array}{l}I=I_{o}exp\mathit{\alpha z}\\ \alpha =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$z$}\right.\mathit{\text{In}}\left(\frac{1}{T}\right)\end{array}$$E13

known as Lamber-Beer- Bouguer law

The absorption coefficient is used in Eq. [21] to obtain the expression below [22].

$${\left(\mathit{\alpha h\nu }\right)}^2=A\left[\mathit{h\nu }-E_g\right]$$E14

In the case of dielectric, further deduction was involved since it is known that the refractive index and dielectric function which appears in both real and imaginary parts characterize the optical properties of any material because they are related to refractive index, n, and extinction coefficient, k.

$${\epsilon }_r=n^2+k^2$$E15

$${\epsilon }_i=2\mathit{nk}$$E16

3. Results/discussion

Figure 1 represents a diagram showing the deposited thin film on the glass substrate which depicts a model of the thin film that is considered to represent a dielectric function with perturbed and the other part where there is no thin film we assume to be the termed reference section. This concept is mathematically represented in Eq. (3). And this equation on the other hand was used in conjunction with the general scalar wave equation to formulate a second-order differential equation in terms of dielectric function that was solved using the method of separation of variables and invariably applied to formulate Helmholtz equation as in Eq. (4). Eq. (3) was substituted in (2) in order to come up with an expression that signifies a propagating was through the thin film medium in terms of Green’s function. The further deduction was made to harmonize the equations in agreement with Green’s function which depicts a function that is in line with the homogeneous Volterra equation of the second type as in Eq. (7). The equation was invariably solved using the Born approximation approach that culminated in Eq. (10) that indicated the phase difference between the incident wave on the thin film and the outgoing wave of which the reflected part of the wave was ignored for the entire system in order to enable the analysis to be carried out in terms of transmitted waves as it propagates through the material thin film. The result of the solution is given in Eq. (12) which depicts $\psi \left(\lambda \right)$ as function of wavelength. The concept was finally made use of in the computation and deduction of absorption co-efficient as in Eq. (13) which formed the backbone of deduction as it concerns all the required optical and solid state properties of the material thin film as required to be analyzed in this work.

Secondly, this also was utilized in obtaining the bad gap as shown in Figure 2 of the thin film based on the model that was presented in Figure 1 which was presented separately in terms of all the wavelengths within the UV, Visible and near-infrared regions of electromagnetic wave spectra. The graph which depicts a plot of ${\left(\mathit{\alpha h\nu }\right)}^2$ as a function of wavelength within the three regions showcased the extrapolation that indicates the positions of their respective band gaps within the wavelength of the respective spectrum.

Figure 2.

A graph of ${\left(\mathit{\alpha h\nu }\right)}^2$ as a function of wavelength for UV, visible and infrared of electromagnetic wave region (CaS) thin film.

In determining the optical properties, the same process was carried out in determining the percentage transmittance, reflectance and absorbance as in Figures 3–5, which were also presented in terms of the wavelength spectrum of the considered region respectively. In the case of transmittance, percentage transmittance was used and it was discovered that within the visible and near-infrared it is zero with fluctuation up to a point at the UV region where it rose sharply while the reflectance appeared to be negative except at a particular point within the visible where it grazes the axis at zero at the point on the visible region. From observation as depicted in the graph, the absorbance seems to be negative as well.

Figure 3.

Transmittance VS wavelength, $\lambda$nm for UV, visible and infrared of electromagnetic wave region, calcium sulphide (CaS) thin film.

Figure 4.

Reflectance VS wavelength, $\lambda$ for UV, visible and infrared of electromagnetic wave region; calcium sulphide (CaS) thin film.

Figure 5.

Absorbance VS wavelength, $\lambda$_nm for UV, visible and infrared of electromagnetic wave region; calcium sulphide (CaS) thin film.

The dielectric constants of the film were obtained by considering the fundamental electron excitation spectrum of thin films is described by means of a frequency-dependent dielectric constant that is related to n and k as shown in Eqs. (15) and (16). However, the graph showing the plot dielectric constant for both real and imaginary parts respectively were plotted as a function of wavelength as considered in this work and were shown in Figures 6 and 7 while the extinction co-efficient is depicted in Figure 8 was also plotted in the same manner.

Figure 6.

Real dielectric constant VS wavelength, $\lambda$ for UV, visible and infrared of electromagnetic wave region; calcium sulphide (CaS) thin film.

Figure 7.

Imaginary dielectric constant VS wavelength, $\lambda$ for UV, visible and infrared of electromagnetic wave region; calcium sulphide (CaS) thin film.

Figure 8.

Extinction coefficient constant VS wavelength for UV, visible and infrared of electromagnetic wave region; calcium sulphide (CaS) thin film.

However, it was generally observed that the behavior of the duo appeared to be irregular in their pattern.

Also, the energy band gap as deduced from the figure for various wavelengths have been shown in Table 1 based on the considered regions of the EM wave.

	Band gaps
Thin film	Computed band gaps (eV)	Average computed band gap (eV)	Experimental values of band gap (eV)
CaS; (i); (ii); (iii); (iv); (v)	250 nm – 3.06; 500 nm – 2.53; 750 nm – 2.42; 1000 nm – 2.28; 1200 nm – 2.24	2.51	3.00–3.39

Table 1.

Computed band gap as indicated by the extrapolated graphs.

The electromagnetic wave spectra propagated through the thin film material. As indicated, each wavelength has a unique band gap associated with it as shown in Table 1 just as deduced from Figure 2. It is observed that the band gap for near-infrared is seen to be narrower while that of UV (i) is wider. However, the average of the duo is considered to be the actual band gap of the thin film that is 2.51 eV. At the initial stage, it was inferred from the literature based on the experimental result that the band gap of the thin film is within the range of 3.00–3.39 eV. Thus, appropriately it is reasonably considered that the computed band gap of CaS is narrower than the result from the experimental value though with just little margin of about ∼0.697 eV.

4. Conclusion

The analysis of the optical and solid state properties of CaS thin film has been carried out successfully using a general scalar wave equation that was made solvable by the use of Green’s function technique and which finally led to the deduction of the propagation wave through the CaS thin film material. However, from the computed result and analysis in terms of the energy band gaps, it was discovered that there is a discrepancy between the values of the energy band gaps as observed in the table. This is the fact that the computed band gap is narrower than the one obtained experimentally However, the reason may not be farfetched from the assumptions and approximations that were involved during the mathematical deductions in formulating the governing equations that were used in computation which might, of course, affected the computed results of the band gaps, and perhaps the results of other graphs that showcased the optical properties as shown in graphs and also as recorded in the table some of which often appeared to be negative contrary to the experimental results in so many cases. From the computation and analysis, it was discovered that the computed band gap is narrower than the one obtained experimentally.

However, the unique feature of this work is that it has indicated that the energy band gap of any material can be studied in terms of the wavelength of the radiation propagating through the material since the wavelength of every region of the electromagnetic wave spectrum has a band gap associated with it.