1 Introduction

Aluminium nitride (AlN) exhibits excellent properties such as a high acoustic wave velocity (Tsubouchi and Mikoshiba 1985), high chemical inertness (Kar et al. 2005), a wide bandgap, and good electrical isolation (Engelmark et al. 2003), in addition to its well-known piezoelectric properties with remarkable values for the piezoelectric coefficients d31, d33 (Ababneh et al. 2010) and to its CMOS compatible fabrication process. These outstanding properties of AlN emphasize it as a very attractive material for the fabrication of micromachined devices in various applications, such as surface acoustic-wave devices (Ingrosso et al. 2007), energy harvesters (Marzencki et al. 2008), pressure sensors (Akiyama et al. 2006), resonators (Piazza and Pisano 2007) and gyroscopes (Gunthner et al. 2006). Furthermore, the development of new photonic devices requires precise information about optical properties, such as refractive index, optical losses and electro-optic constants (Lu et al. 2018; Xiong et al. 2014; Sun et al. 2017). Single-crystalline AlN is a very promising material for the application in optoelectronic devices, such as UV light-emitting diodes and laser diodes (Nishida et al. 2004; Kipshidze et al. 2003). Also, sputtered c-axis oriented AlN films were used to demonstrate photonic microdisk resonators (Ghosh and Piazza 2013).

AlN thin-films can be deposited using various deposition techniques, e.g. chemical vapor deposition (CVD) (Rodriguez-Clemente et al. 1993), molecular beam epitaxy (Mackenzie et al. 1995), or sputtering. The advantage of using a reactive sputtering process is that AlN films can be deposited at or near room temperature while maintaining a high degree of c-axis orientation. This property is considered to be a precondition for obtaining high piezoelectric coefficients (Ababneh et al. 2010). AlN represents the highest value of the piezoelectric constants, compared to other common III-V nitrides like GaN or InN with similar processing properties (Ambacher 1998). Additionally, in contrast to ferroelectric materials with permanent electrical dipole like lead zirconate titanate (PZT), no high-temperature polarization step is necessary after deposition.

In this work, AlN thin-films with different degrees of c-axis orientation were deposited and optically analyzed. The purpose of the present work is to investigate the influence of different sputtering pressures with different degrees of c-axis orientation on the structural and optical properties of AlN thin-films, such as lattice parameters, crystalline size, microstrain, dislocation density, transmittance, refractive index, and dielectric constant. Such a study may facilitate the implementation of the sputtered AlN in optical and optoelectronic devices.

2 Experimental details

2.1 Fabrication process

Different samples of AlN thin-films with thicknesses of 550 nm were deposited on (100) silicon and glass substrates in a production-type DC-magnetron sputtering machine [Ardenne (LS 730 S)]. Prior to deposition, silicon and glass substrates were cleaned with acetone and deionized water. The aluminium target had a purity of 99.999%, a diameter of 200 mm, and was positioned with a substrate-to-target separation of 65 mm. After the evacuation of the deposition chamber to a pressure below 1 × 10−7 mbar, Argon gas was used to pre-sputter the target for 10 min prior to each film deposition for purification purposes. The substrates were nominally unheated and the sputtering was performed at room temperature.

Sputtering conditions, such as plasma power, sputtering pressure, and Ar/N2 flow can substantially influence the degrees of c-axis orientation of AlN thin-films. Therefore, during the sputtering process, the plasma power was fixed at 1000 W and the deposition chamber was filled with a 100% N2 atmosphere. The sputtering pressure was varied between 2 × 10–3 and 6 × 10–3 mbar. Table 1 summarizes the sputtering conditions of AlN thin-films.

Table 1 Sputtering conditions for AlN thin-films
Full size table

2.2 Film characterization

The thickness of the AlN films was measured with a surface profilometer (from Tencor Instr.) after a step was locally generated using phosphoric-based acid. The crystal structure of the AlN thin-films was revealed using the X-ray diffraction (XRD) technique (from Philips X-Pert) by performing the phase analysis using the θ–2θ scans in the Bragg–Brentano mode. The XRD was performed at monochromatized CuKα-line with a wavelength of 1.5418 Å. Double-Beam UV–Vis Spectrophotometer (U-3900H) in the wavelength range of 250–800 nm was used to investigate the optical properties of the AlN thin-films.

3 Results and discussion

3.1 X-ray diffraction (XRD)

Powder X-ray data analysis was used to investigate the crystalline nature of AlN thin-films deposited at different parameters using an X-ray diffractometer. Figure 1 shows the XRD pattern of AlN thin-films sputtered at a plasma power of 1000 W in a pure nitrogen atmosphere and different sputtering pressures. The results showed that all the samples have a wurtzite hexagonal structure with a single peak for the (002) orientation, which is attributed to the grain growth associated with the preferred orientation. For the samples with the same thickness (about 550 nm), increasing the sputtering pressure from 2 × 10–3 to 6 × 10–3 mbar showed a decrease in the (002) peak intensity and a widening of its corresponding full-width at half-maximum (FWHM) value. These results indicate a degradation in the AlN crystal quality and, consequently, a reduction in the piezoelectric coefficients. In a previously published study, more details related to the influence of the sputtering conditions on the c-axis orientation of AlN thin-films were presented (Ababneh et al. 2010). Additionally, from Fig. 1 a slight shift has been noticed in all diffraction lines towards the lower diffraction angles (2θ) when the sputtering pressure increases. The (002) peak position for AlN thin-film sputtered at 2 × 10–3 mbar on (001) silicon substrates is 36.058° and shifted to a lower angle of 35.834° at 6 × 10–3 mbar. This shift in the peak position to the lower angles can be attributed to the internal film stress and the increase in the oxygen contamination within the film as the sputtering pressure increased.

Fig. 1
figure 1

The X-ray diffraction patterns of AlN thin-films with thicknesses of 550 nm at a plasma power of 1000 W in 100% N2 atmosphere and different sputtering pressures

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The lattice constants ‘\(a\)’ and ‘\(c\)’ for the hexagonal crystal structure can be determined using the relation (Cullity 1978):

$$\frac{1}{{d}_{(hkl)}^{2}}=\frac{4}{3}\left(\frac{{h}^{2}+hk+{k}^{2}}{{a}^{2}}\right)+\frac{{l}^{2}}{{c}^{2}}$$
(1)

where \({d}_{(hkl)}\) is the interplanar spacing, which can be calculated using the Bragg's law (Lu et al. 2015):

$$\lambda =2{d}_{hkl}\mathrm{sin}{\theta }_{hkl}$$
(2)

with λ is the wavelength of X-ray (0.1540598 nm) and \(\theta \) is the angle of incidence.

The lattice constants ‘a’ and ‘c’ of the hexagonal wurtzite structure of AlN thin-films can thus be determined using the following equations:

$$a=\frac{\lambda }{\sqrt{3}\mathrm{sin}{\theta }_{(100)}}$$
(3)
$$c=\frac{\lambda }{\mathrm{sin}{\theta }_{(002)}}$$
(4)

From Fig. 1, the AlN samples have a single peak of (002) and thus, the lattice constant ‘a’ cannot be determined. The lattice constant ‘c’ was then determined using Eq. (4) and the results were correlated with the bulk value of wurtzite AlN (i.e. 4.9795 Å for 2 × 10–3 mbar sputtering pressure). The lattice constant ‘c’ has increased slightly to 5.0207 Å at 6 × 10–3 mbar sputtering pressure (see Fig. 2). A lattice constant ‘a’ with 3.104 Å was reported by a previous study (Alsaad et al. 2020) for AlN thin-film deposited with other sputtering parameters with different X-ray diffraction peaks, e.g. (100), (002), and (101).

Fig. 2
figure 2

The Lattice constant ‘c’ of AlN thin-films determined using the X-ray diffraction pattern as a function of sputtering pressure

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3.1.1 Strain

Crystallite size and lattice strain are counted among the fundamental properties of thin-films. They can be extracted using the peak-width analysis from the XRD measurements. The crystallite size and the lattice strain both affect the peak width, peak intensity, and 2θ peak-shift position. The crystallite size represents the size of coherently diffracting domains (Solliard and Flueli. 1985; Fu et al. 2011). The average crystallite size (\(D\)) is calculated using Debye Scherrer's formula (Fu et al. 2011) as:

$$D=\frac{k\lambda }{\beta \mathrm{cos}\theta }$$
(5)

where \(\lambda \) is the wavelength of X-ray (λ = 0.154184 nm), β is the full-width at half-maximum (FWHM) after correcting for instrumental peak broadening (β expressed in radians), θ is the Bragg angle and k is the Scherrer constant.

The k-value depends on many factors such as the crystal shape, the diffraction line indices, and the dispersion in the crystallite sizes of the powder (Shull 1946; Pielaszek 2003). Typically, \(k\) is between 0.8 and 1.39 and for the spherical particles k is nearly 0.94. Lattice strain or micro-strain is a measure of the distribution of lattice constants arising from crystal imperfections, such as lattice dislocations, the grain boundary triple junction, contact or sinter stresses, stacking faults and coherency stresses (Zhang et al. 2006). The broadening due to the micro-strain is caused by the non-uniform displacements of the atoms with respect to their reference-lattice positions. The micro-strain within the domain is considered as lattice defects and is equivalent to the variations in the \(d\)-spacing within the domains by an amount depending on the elastic constants of the material and the nature of internal stresses. The micro-strain \(\langle \varepsilon \rangle \) of thin-films was determined according to the following relation (Fu et al. 2011):

$$\langle \varepsilon \rangle =\frac{\beta \mathrm{cot}\theta }{4}$$
(6)

The associated thermal and kinetic energies of the molecules during the deposition process above the substrate causes a random crystal growth. This leads to internal or residual stress (extended or compressed). Due to this internal stress, the deformations of crystallization of the thin-films will be created. The amount of deformations depends on the elastic constants of the material and the nature of internal stresses. The relative deformation is known as micro-strain, and it is equivalent to variations in the d-spacing within the domains. The origin of the micro-strain \(\langle \varepsilon \rangle \) is related to the lattice misfit, which depends on the growth conditions (Jona and Shirane 1962).

Figure 3 shows the crystallite size \(D\) and the average micro-strain \(\langle \varepsilon \rangle \) as a function of sputtering pressure. The crystallite size for AlN thin-film at 2 × 10–3 mbar was 31.5 nm. Increasing the sputtering pressure to 4 × 10–3 and 6 × 10–3 mbar leads to a decrease in the crystallite size to 26.2 nm and 18.6 nm, respectively. It is obvious that increasing the sputtering pressure leads to a decrease in the crystallite size and to a decrease in the degree of crystallinity in the thin-film. It is also evident that increasing the sputtering pressure leads to an increase in the micro-strain. The inverse relation between the micro-strain and the crystallite size is also shown in Fig. 3. This can be due to the decrease in the volume occupied by the arranged atoms inside of the agglomerate crystallite; and thereby the total surface area is increased. This change in the surface will decrease the shift of plane position and reduce the micro-strain (Horiuchi et al. 2010).

Fig. 3
figure 3

The crystallite size \(D\) and micro-strain \(\langle \varepsilon \rangle \) of AlN thin-films as a function of sputtering pressure

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3.1.2 Stress

Stress in thin-films is caused by various factors, for example; temperature changes, phase transitions, and defects, such as dislocations in the film. The total internal stresses (\(\sigma )\) were determined according to:

$$\sigma =Y*\langle \varepsilon \rangle $$
(7)

with Y is the Young's Modulus of the material.

In this study, the results showed that the total internal stress of AlN thin-film deposited at 2 × 10–3 mbar was 1.12 × 1012 N/m2 and increases to 1.88 × 1012 N/m2 when the sputtering pressure increases to 6 × 10–3 mbar (see Fig. 4a).

Fig. 4
figure 4

Effect of the sputtering pressure for AlN thin-films on the a Total internal stress, b dislocation density, and c crystalline density

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Dislocation is considered as one of the main defects categories in the crystal, which can be related to the impact of internal stress. Dislocation density (δ) is defined as the length of dislocation lines per unit volume of the crystal (Akl and Hassanien 2015). The value of the dislocation density represents the vacant spaces between the crystalline clusters or the crystallite agglomerations (Fu et al. 2011; Akl and Hassanien 2015; Williamson and Smallman 1956). The strength and ductility of materials are controlled by dislocations. The dislocation density can be determined using the line profile analysis of the X-ray diffraction (LPA-XRD), with the simple Williamson–Smallman equation (Fu et al. 2011; Akl and Hassanien 2015; Williamson and Smallman 1956):

$$\delta =1/{D}^{2}$$
(8)

where: \(D\) is the value of the crystallite size. Figure 4b shows the dislocation density of AlN thin-films as a function of sputtering pressure.

Here, the dislocation density for AlN thin-films at 2 × 10–3 mbar was found to be 1.01 × 1012 lines/cm2, and increases when increasing the sputtering pressure.

The crystalline density (N) is the crystallite number per unit volume which depends on various parameters such as crystallite size, crystallite shape, equi-dimensions crystallites, and the degree of agglomeration of thin-films. The crystalline density of AlN thin-films was calculated from the estimated values of crystallite size, using the following formula (Dawber et al. 2005; Scott 2007):

$$N=t/{D}^{3}$$
(9)

where: t is the film thickness which in this work is fixed to 550 nm.

Figure 4c shows the crystalline density for AlN thin-films as a function of sputtering pressure. The crystalline density of AlN thin-film deposited at 2 × 10–3 mbar was found to be 1.756 × 1012 crystal/cm2 and increases with the increase in sputtering pressure.

3.2 UV–Vis spectroscopy

In this work, a UV–Vis spectrophotometer with the total internal integrating sphere was employed to investigate the transmittance spectra \(T\% \left(\lambda \right)\) for sputtered AlN thin-films in the spectral range 250–800 nm. The AlN thin-films were deposited on glass substrates at different sputtering pressure (2 × 10–3 to 6 × 10–3 mbar). The transmittance \(T\% \left(\lambda \right)\) spectra used to investigate all-optical properties to obtain the relevant parameters for optoelectronic fields.

Figure 5 shows the optical transmittance spectra of AlN thin-films at different sputtering pressures. The results showed that the transmittance rapidly increases with increasing the wavelength, which is again due to the absorption edge. It can be observed that distinct interference fringes were present in the transparent region with a relatively large intensity of up to 91%. In the short wavelength range, the absorption increases, and the intensity of the interference fringes decrease towards the zero at the fundamental absorption edge of the film. The number of the observed interference fringes is limited to five fringes due to the relatively small thickness of samples (i.e. 550 nm). Emam-Ismail et al. reported that increasing the film thickness leads to an increase in the number of interference fringes (Emam-Ismail et al. 2012). The existence of interference patterns in the measured transmittance spectra indicates a good quality of the thin-film with a uniform thickness.

Fig. 5
figure 5

The transmittance spectra of AlN thin-films sputtered on glass substrates with a film thickness of about 550 nm at different sputtering pressures

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Figure 5 reveals also that, increasing the sputtering pressure leads to a shift of the threshold transmittance towards the lower wavelength range (i.e. the blue shift in the optical bandgap). This shift of the bandgap towards higher energies can be attributed to the decrease in the crystalline nature of the sputtered films with increasing the deposition pressure. Figure 5 illustrates the transmittance spectrum of AlN at 2 × 10–3 mbar, which shows the lowest bottom and thereby the highest refractive index (n). On the contrary, the transmittance spectrum of AlN at 6 × 10–3 mbar has the highest bottom, and thereby it has the lowest refractive index.

Figure 6 shows a typical transmittance spectrum of AlN thin-film. The transmittance curve is enveloped between two values; TM and Tm. From Fig. 6, the refractive index calculations started with the peak at 675 nm wavelength. This means that the first order will behave like an integer.

Fig. 6
figure 6

Typical transmission spectrum for AlN thin-film sputtered at 2 × 10–3 mbar on a glass substrate. The TM and Tm are the experimental curves according to the text

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The spectral dependence of the refractive index for AlN thin-films was evaluated based on Swanepoel's envelope method (SWEM) (1983) and Manifacier et al. (1976), the first approximation values of the refractive index of the film can be determined according to the following relation (Dahshan et al. 2008; Aly et al. 2009):

$${n}^{2}=\left[N+{({N}^{2}-{S}^{2})}^{1/2}\right]$$
(10)

where:

$$N=2S\left(\frac{{T}_{M}-{T}_{m}}{{T}_{M}.{T}_{m}}\right)+\left(\frac{{S}^{2}+1}{2}\right)$$
(11)
$$S=\frac{1}{{T}_{s}}+{\left(\frac{1}{{T}_{S}^{2}}-1\right)}^{1/2}$$
(12)

with \({T}_{M}\) and \({T}_{m}\) denoted the maximum and minimum transmittance values at a given wavelength, respectively, \(S\) is the refractive index of the substrate and \({T}_{s}\) is the transmission spectrum of the glass substrate.

The first approximation value of the film thickness (d) can be found in the form of Aly et al. (2009):

$$d=\frac{{\lambda }_{1}{\lambda }_{2}}{2({n}_{c2}{\lambda }_{1}-{n}_{c1}{\lambda }_{2})}$$
(13)

where nc1 and nc2 are the refractive indices at two adjacent maxima (or minima) at λ1 and λ2.

The film thicknesses were estimated and found to be about 550 nm for all samples, which agrees with the actual thickness of the deposited samples. Using the calculated values of the refractive index determined by SWEM and performing the least-square fitting according to Cauchy's dispersion equation, a reasonable function of two-terms can be obtained, such as:

$$n\left(\lambda \right)=a+\frac{b}{{\lambda }^{2}}$$
(14)

where a and b are Cauchy's constants.

By substituting a and b values in Eq. (14), the refractive index over the entire spectral range of measurement 250–800 nm can be then determined (Swanepoel 1983). Figure 7 depicts the relation between the refractive index and the wavelength of incident photons. It is obvious that the refractive index gradually decreases while increasing the wavelength of the incident photon. Additionally, increasing the sputtering pressure reduces the refractive index for the AlN thin-films. In a previous work (Ababneh et al. 2020), the relationship between the refractive index and the degree of c-axis orientation was presented and discussed.

Fig. 7
figure 7

Dispersion of the refractive index of AlN thin-films as a function of sputtering pressure deposited on glass substrates with a 550 nm film thickness

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The amount of light energy loss by absorption and scattering in the thin-films is represented by the extinction coefficient (\(k\)). The extinction coefficient depends on the density of free electrons and the structural defects in the thin-films. The extinction coefficient can be determined using the following relation with Ahmad et al. (2018):

$$k=\alpha \lambda /4\pi $$
(15)

where: \(\alpha \) is the absorption coefficient defined by \(\alpha =\left(1/d\right)\mathrm{ln}(1/T)\), with d is the film thickness which is found to be 550 nm.

Figure 8 shows the extinction coefficient of AlN thin-films as a function of wavelength at different sputtering pressure. It is clear that the extinction coefficient values are dramatically decreasing in the wavelength range of 250–300 nm. This results in increasing the transmittance of the incident photons within the visible wavelength range. Furthermore, the extinction coefficient of AlN thin-films in the visible wavelength range decreases as the sputtering pressure is increased. This indicates that the AlN samples deposited at 2 × 10–3 mbar have a higher amount of light energy loss by scattering and absorption compared with the AlN samples deposited at 4 × 10–3 and 6 × 10–3 mbar.

Fig. 8
figure 8

Extinction coefficient spectra of AlN thin-films sputtered on a glass substrate with a film thickness of about 550 nm at different sputtering pressures

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3.3 Optoelectronic parameters

3.3.1 Optical bandgap energy

The Tauc plot (Tauc 2012) is commonly used in the determination of the bandgap energy of semiconductor materials. However, the existence of a glass substrate blocks UV light in the spectral range between 250 and 300 nm. Therefore, the energy values obtained with this method for the bandgap might be inaccurate. Therefore, we applied the mathematical model described by Al Bataineh et al. for the determination of the bandgap energy values of AlN thin-films (Al-Bataineh et al. 2020). In this model, the transmission values are related to the incident photon wavelength λ by the relation:

$$T\left(\lambda \right)={e}^{- \frac{4\pi d}{\lambda }\frac{A{\left(hc/\lambda -{E}_{g}\right)}^{2}}{{\left(hc/\lambda \right)}^{2}-B\left(hc/\lambda \right)+C}}$$
(16)

where: \(hc = 1240 eV\), \(d\) is the thickness of the film, \({E}_{g}\) is the bandgap energy, \(A, B, C\) are fitting constants.

Figure 9 shows the measured transmittance spectra of AlN thin films sputtered on glass substrates at different sputtering pressures, together with the fitted data according to Al-Bataineh et al. (2020) in the spectral range 300–800 nm. It is clear that the fitted function is passed approximately at FWHM of the interference fringes, which means that the model fitted well the transmission spectra. The bandgap of AlN thin-film sputtered at 2 × 103 mbar was found to be (6.1254 ± 0.0247) eV, while the bandgap of the film sputtered at 4 × 10–3 mbar was found to be (6.1457 ± 0.0953) eV. Finally, the bandgap of AlN thin-film sputtered at 6 × 10–3 mbar was found to be (6.1578 ± 0.0247) eV. The reason for the slight increase in the bandgap energy can be attributed to the decrease in the agglomerations of atoms and the poor crystallinity at high deposition pressures.

Fig. 9
figure 9

Transmission spectra of AlN thin films sputtered on a glass substrate with a film thickness of about 550 nm at different sputtering pressures. The experimental (circles) and the model (solid lines)

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3.3.2 Optical dispersions and Optoelectronic parameters

The optical dispersion has a great influence on the design of optical materials. Based on Wemple–DiDomenico (WDD) single effective-oscillator model, the dispersion of refractive index has been used to determine the dispersion parameters such as the Sutcliffe and Wilson (2003), Wemple and DiDomenico (1971):

  • effective single oscillator (\({E}_{0}\)), which gives quantitative information on the overall band structure of the material;

  • dispersion energy (\({E}_{d}\)), which measures the average strength of inter-band optical transitions and associated with the changes in the structural order of the material. It is related to the ionicity, anion valency and coordination number of the material, and the incident photon energy, \(h\upsilon \).

  • zero-frequency dielectric constant (\({\varepsilon }_{0}\)).

  • optical moments.

The advantage of using the single effective oscillator model for fitting the experimental data is that it provides an intuitive physical interpretation of the measured quantities (Fu et al. 2011). The WDD single effective-oscillator model presented in Eq. (17) provides a physical interpretation of the measured quantities:

$${n}^{2}-1=\frac{{E}_{d}{E}_{0}}{{E}_{0}^{2}-{E}^{2}}$$
(17)

By plotting \({({n}^{2}-1)}^{-1}\) on y-axis vs. \({(hv)}^{2}\) on the x-axis, as shown in Fig. 10, the dispersion parameters can be evaluated by fitting a straight line. The values of \({E}_{0}\) and \({E}_{d}\) can be calculated utilizing the slope \({({E}_{0}{E}_{d})}^{-1}\), which is the interception with the vertical axis \(({E}_{0}/{E}_{d})\). The estimated values of \({E}_{d}\) and \({E}_{0}\) for AlN thin-films at different sputtering pressure are shown in Table 2. These parameters have a crucial role in identifying the characteristics of optical materials, which allow calculating the required factors for the design of the spectral dispersion devices and optical communication. It is clear that the \({E}_{0}\) values for AlN thin-films increases as the sputtering pressure is increased. This increase in the \({E}_{0}\) values can be related to the increase in the energy of the bonds formed between the film elements as well as the change in iconicity values of these bonds (Hassanien and Akl 2015). The results of \({E}_{0}\) and \({E}_{d}\) can be employed to determine the zero-frequency dielectric constant \({\varepsilon }_{0}\) and zero-frequency refractive index \({n}_{0}\) by rewriting Eq. (18) and assuming \(hv=0\) as follows:

Fig. 10
figure 10

The dispersion parameters determination using the \({({n}^{2}-1)}^{-1}\) vs.\({(hv)}^{2}\) plot for AlN thin films sputtered on a glass substrate at different sputtering pressures with a film thickness of about 550 nm

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Table 2 The estimation results of the essential optical parameters of AlN thin films sputtered on a glass substrate with a film thickness of about 550 nm at different sputtering pressures and 1000 W plasma power in 100% N2 atmosphere
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$${\varepsilon }_{0}={n}_{0}^{2}=1+\frac{{E}_{d}}{{E}_{0}}$$
(18)

Table 2 shows the estimated values of the zero-frequency dielectric constant \({\varepsilon }_{0}\) and zero-frequency refractive index \({n}_{0}\) for all AlN samples. The results show that the values of the zero-frequency refractive index coincide with the theoretical and experimental values of the normal refractive index. The zero-frequency dielectric constant \({\varepsilon }_{0}\) and zero-frequency refractive index \({n}_{0}\) for AlN thin-film at 2 × 10–3 mbar were found to be 4.726 and 2.174, respectively. Increasing the sputtering pressure to 4 × 10–3 and 6 × 10–3 mbar leads to a decrease in the \({\varepsilon }_{0}\) to 4.581 and 4.445, respectively.

The moments \({M}_{-1}\) and \({M}_{-3}\) of the optical spectra can be determined for the present thin-films by Badran (2012):

$${{\mathrm{E}}_{0}}^{2}={\mathrm{M}}_{-1}/{\mathrm{M}}_{-3}$$
(19)

and

$${{\mathrm{E}}_{\mathrm{d}}}^{2}={{\mathrm{M}}^{3}}_{-1}/{\mathrm{M}}_{-3}$$
(20)

The resulted values of the moments \({M}_{-1}\) and \({M}_{-3}\) are shown in Table 2. It can be observed that the optical moments decrease as the sputtering pressure is increased. The low values of \({M}_{-1}\) and \({M}_{-3}\) reveals a low polarization of the AlN thin-films under consideration (Okutan et al. 2005).

The average oscillator wavelength \(({\lambda }_{0})\) and oscillator length strength \(({S}_{0})\) parameters of the AlN samples can be determined using the modified WDD single effective-oscillator model given by Eq. (21) Jundale et al. (2011):

$${n}^{2}-1=\frac{{S}_{0}{\lambda }_{0}^{2}}{(1-{\lambda }_{0}^{2})/{\lambda }^{2}}$$
(21)

The oscillator parameters can be determined by plotting \({({n}^{2}-1)}^{-1}\) vs.\({\lambda }^{-2}\) and applying a straight-line fit, as shown in Fig. 11. The values of \({S}_{0}\) and \({\lambda }_{0}\) can be determined using the slope \({1/S}_{0}\), which represents the intercept on the vertical axis \((1/{S}_{0}{\lambda }_{0}^{2})\). Table 2 summarises the estimated oscillator parameters.

Fig. 11
figure 11

The determination of the oscillator parameters using the \({({n}^{2}-1)}^{-1}\) vs.\({\lambda }^{-2}\) plot for AlN thin-films sputtered on a glass substrate at different sputtering pressures with a film thickness of about 550 nm

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The dielectric constant of the material is determined by electronic, ionic, dipolar, and space charge polarization. The space charge contribution depends on the purity and the perfection of the glass substrates. Generally, the influence of the space charge contribution is strongly obvious in the low-frequency region. The refractive index accurate data can be analyzed to obtain the density of states (ratio of free carrier to the effective mass) \(N/{m}^{*}\) and the high-frequency dielectric constant \({\varepsilon }_{\infty }\) from the following expression (Fasasi et al. 2018; Spitzer and Fan 1957):

$${\varepsilon }^{^{\prime}}={n}^{2}={\varepsilon }_{\infty }-\frac{1}{4{\pi }^{2}{\varepsilon }_{0}}\left(\frac{{e}^{2}}{{c}^{2}}\right)\left(\frac{{N}_{c}}{{m}^{*}}\right){\lambda }^{2}$$
(22)

where: \(e\) is the electronic charge, \({N}_{c}\) is the density of states or charge carrier density, \({m}^{*}\) is the effective mass of the carrier and \({\varepsilon }_{\infty }\) is the high-frequency dielectric constant.

By plotting \({\varepsilon }^{^{\prime}}\) on the y-axis vs.\({\lambda }^{2}\) on the x-axis (see Fig. 12) generates a straight line in the long-wavelength region. The \(N/{m}^{*}\) can be determined by the slope of the linear line, while \({\varepsilon }_{\infty }\) is determined by the extrapolation of the linear part of this curve at \({\lambda }^{2}=0\) (see Table 2).

Fig. 12
figure 12

The \(N/{m}^{*}\) determination using the variation of the real part of the dielectric constant (\({\varepsilon }^{^{\prime}}\)) vs. \({\lambda }^{2}\) plot for AlN thin films sputtered on a glass substrate at different sputtering pressures with a film thickness of about 550 nm

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The \({\varepsilon }_{\infty }\) for AlN thin-film at 2 × 10–3 mbar was found to be 4.934 and decreases with the increase in the sputtering pressure. This behavior is consistent with the behavior of the refractive index. The obtained values of the dielectric constant at zero wavelength, \({\varepsilon }_{\infty }\) are also found to be greater than the index of refraction, \(n\). This means that the available free charge carriers in AlN thin-films are strongly contributing to the polarization process (Hassanien 2016; Farag et al. 2012). Hence, the concentration of the charge carriers also decreases when sputtering pressure is increased.

4 Conclusions

In this work, AlN thin-films with different degrees of the c-axis orientation were sputtered on nominally unheated silicon and glass substrates while varying the deposition pressure. X-ray diffraction measurements confirmed the wurtzite structure of the AlN thin-films with the dependence of the c-axis orientation on the deposition pressure. They were also used for determining the mechanical properties of the AlN thin films (such as lattice constant, crystallite size, dislocation density and film stress). Additionally, the optical properties of AlN thin-films have been investigated. The optical bandgap, optical dispersion and optoelectronic parameters (such as refractive index, extinction coefficient and dielectric constants) were calculated using optical techniques. The deposited AlN thin-films exhibited a slight increase in the bandgap from 6.125 to 6.158 eV with increasing the sputtering pressure. Contrarily, the refractive index of the films exhibits a decrease when increasing the sputtering pressure and was in the range from 2.31 to 2.21 at 250 nm wavelength. The optical moments showed a decrease as the sputtering pressure is increased similar to the dielectric constants trend. This indicates a strong dependence of the optical parameters of AlN thin-films with the sputtering pressure.