CZO films: Effect of different post-annealing temperatures on electron-phonon interactions, steepness parameters, coordination numbers, optical mathematical equations, and skin depth

Over the last few decades, dielectric-electric properties as a function of frequency and temperature have been comprehensively studied [1]. Zinc oxide belongs to group II-VI composite semiconductor and is known as an n-type semiconductor on account of the insufficiency of oxygen. ZnO has 60 meV excitation binding energy and a large direct band-gap of approximately 3.4 eV. ZnO is very suitable for gas sensor applications since its surface is so sensitive to oxygen and hydrogen and has a high thermal conductivity [2]. According to reported works, there could be several reasons for the development of RT ferromagnetism in TM-doped ZnO such as uncontrollable ferromagnetic clusters, secondary phases, and defects. It has been also reported that Co is highly soluble in ZnO, and Co-doped ZnO can have ferromagnetic or non-ferromagnetic properties at RT, depending on its growth technique and conditions. Moreover, using different additional doping elements, such as Al, Cu, and Ga in TM-doped ZnO can result in RT ferromagnetism and/or enhanced ferromagnetism, which is known as carrier-induced RT ferromagnetism [3]. Deposition of ZnO can be accomplished with techniques such as molecular beam epitaxy, metal-organic chemical vapor deposition, pulsed laser deposition, radio-frequency magnetron sputtering, and spray pyrolysis deposition that has been used to deposit ZnO on different substrates [4]. The aim of the present work is also to discuss the correlation between these parameters and temperature.

Details of the deposition processing of films are given in our previous report [5].

In our previous work, PL and EDX have been studied and the difference between photoluminescence emission peaks of ZnO films compared with copper-doped ZnO films is mainly due to a green emission at a position of about 530 nm which was related to the transition from donor levels to the valence band because of oxygen ions vacancies and the replacement of Cu atoms ions instead of Zn atoms in the ZnO structure. The percentages of copper and zinc atoms in as-deposited copper-doped ZnO films were 4.31 and 23.48%, respectively [5]. It is observed that the nano- and micro-roughness is created by fluctuations in the surface of short wavelengths, characterized by hills (local maxima) and valleys (local minima) of varying amplitudes and spacing. The root mean square (RMS) of CZO film for room temperature, 400, 500, and 600°C was 5.74, 4.70, 6.02, and 10.1 nm, respectively (these results are reported in table 1). According to AFM images and particle distribution (see fig. 1), surface morphology changes can be analyzed by changes in annealing temperature in the zinc oxide layers in such a way that in the unbacked and annealed layer, the particle shape is almost identical and with the same distributions. Also, the surface roughness of the layers changed in the range of 4 to 10 nm with annealing, and the lowest surface roughness is related to the annealed layer at 400°C. The CZO substrate at 600°C has a relatively large root mean square. In our previous work, XPS and XRD have been studied. In these patterns, only a well-defined diffraction peak is observed at around 34°, which is ascribed to the crystalline (002) planes of the ZnO phase. In the spectrum (XPS) of CZO film, broadbands at around 932.5 eV, 943 eV and 952 eV were observed, which correspond to the Cu $2p_{3/2}$ peak of Cu¹⁺, the Cu $2p_{3/2}$ of Cu²⁺ and the Cu $2p_{1/2}$, respectively [6,7].

Table 1:. The values of the direct band-gap energy (E_g ), the band tail width (EU), steepness parameter $(\sigma)$, and electron-phonon interaction ($E_{e-p}$) and RMS of the samples.

Sample	Wavelength of	En	Eg	The constant	Steepness	($E_{e-p}$)	RMS
	absorption	(eV)	(eV)	($\alpha_0$)	parameter $(\sigma)$	(eV)	(nm)
	edge (nm)			$({\text{nm}})^{-1}$	(eV)
CZO (as-deposited)	364.514	0.515	3.21	35.88	20E−10	3.33E10+8	7.238
CZO (400°C)	386.736	0.388	3.18	34.52	6.69E−10	9.96E10+8	7.999
CZO (500°C)	398.484	0.168	3.17	26.10	2.52E−10	26.43E10+8	7.056
CZO (600°C)	389.245	0.310	3.19	32.18	4.12E−10	16.16E10+8	7.817

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For each of these four samples, we measured the transmission and reflection spectrum at room temperature, and 400, 500, and 600°C to calculate the absorption coefficient from the ($1-R-T/T$) ratio, spectral dependence of the index of refraction. With the increase of energy from 3 to 3.2 eV, the optical absorption coefficient (see fig. 2) of the oxide layer has also increased sharply, because in this energy range the frequency of the electromagnetic wave is close to or equal to the frequency of the resonant electrons in the copper oxide molecules. Also, with the annealing of the layers and the increase of the annealing temperature, a shift of the absorption edge towards larger photon energy is observed. The shift in absorption edge might have resulted from the formation of the conjugated bond system caused by bond cleavage and reconstruction. The plots indicate that for all the samples, the optical density increased almost linearly with increasing photon energy up to the absorption edge after which a sharp rise in optical energy was noticed. The variation of the optical density with varying post-deposition annealing temperatures can also be linked in terms to the stoichiometric changes induced by copper vacancies and neutral spaces. The formation of these defects depends on the sticking coefficient, nucleation rates, and the movement of intruding zinc and copper ions during deposition which is a function of the concentration of the reacting species [8].

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The skin depth $(\delta)$ is a measure of the distance that an optical beam penetrates in glass samples before the beam is scattered. It is related to the absorption coefficient, ${\alpha}(\nu)$, according to the following simple relationship [9]:

$$\begin{equation} {\delta} = {\lambda} / 2{\pi}k = 2/{\alpha}, \qquad k = \alpha \lambda/4{\pi}, \end{equation} \tag{ 1 }$$

where k is the extinction coefficient of films. Figure 3 shows the variation of skin depth as a function of photon energy for all films. It is clear from the figure that the skin depth increases as the wavelength increases; this behavior could be seen in all samples. It is seen from fig. 3 that the skin depth changes from $8\times10^{2}$ to $25\times10^{2}\ \text{nm}$ with temperature. The value of the cut-off energy, $E_{\textit{cut-off}}$, of the films was about 3.20 eV and the value of the ${\lambda}_{\textit{cut-off}}$ was about 387 nm. The highest skin depth is found for the CZO films annealed at 500 and 600°C which indicates the effect of temperature. The complex dielectric constant, $\overline{\varepsilon}(h{\nu})$, of semiconducting thin films is based on the real, ${\varepsilon}_{r}=(n^{2}-k^{2})$, and imaginary, ${\varepsilon}_{Im}(h{\nu}) =(2nk)$, parts of the optical dielectric constant and can be characterized by the following relation [10]:

$$\begin{align} \overline{\varepsilon }(h{\nu})&={\varepsilon}_{r} (h{\nu}) +i{\varepsilon}_{Im}(h{\nu}), \end{align} \tag{ 2 }$$

$$\begin{align} {\varepsilon}_{r}(h{\nu})&={n}^{2}-k^{2}=\varepsilon_{\infty }-(e^{2}N/4\pi^{2}c^{2}\varepsilon_{0}m^{*})\lambda^{2}, \end{align} \tag{ 3 }$$

$$\begin{align} {\varepsilon}_{Im}(h{\nu})&=2nk = (\varepsilon_{0}w_{p}/8\pi^{2}c^{3}\tau)\lambda^{3}. \end{align} \tag{ 4 }$$

The spectra of real and imaginary parts of the optical dielectric constant are shown in fig. 3. The variation of ${\varepsilon}_{r}$ behaves with the same trend as n, while following the behavior of k with photon energy [11]. The optical band-gap is obtained from the intersection of the real and imaginary parts vs. photon energy. This observed trend may also be correlated with the distribution defects of atoms. The magnitude of the optical energy band-gap values differs within the range of 3.17 eV and 3.21 eV.

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The tail of the absorption edge is exponential, indicating the presence of localized states in the energy band-gap. The amount of tailing can be predicted to a first approximation by plotting the absorption edge data in terms of an equation originally given by Urbach [12]. The absorption edge gives a measure of the energy band-gap and the exponential dependence of the absorption coefficient, in the exponential edge region the Urbach rule is expressed as [13,14]

$$\begin{equation} {\alpha} ={\alpha}_o \text{exp} (h{\nu}/E_{u}), \end{equation} \tag{ 5 }$$

where ${\alpha}_{o}$ is a material constant of films. The parameter hν corresponds to the energy of the lowest free excited state at zero lattice temperature, while E_u is the Urbach energy, which characterizes the slope of the exponential edge. This parameter determines the steepness of the Urbach tail. Urbach energy values change inversely with the optical band-gap (see fig. 4). The Urbach energy values of the films increase with a decrease in doping temperature. The decrease of Urbach energy suggests that the atomic structural disorder of doped films is decreased by increasing the doping temperature. This behavior can result from increasing the grain size. The size of the grains varies with the doping and influences the value of the optical energy gap; this increased crystallinity leads to a redistribution of states from band to tail and may be attributed to the improvement. As a result, both a decrease in the optical gap and expansion of the Urbach tail take place. The optical gap is reduced with increasing temperature. It is worth bearing in mind that the Urbach tail energy is closely related to the disorder in the film network. Since the ion radius of Cu is larger than ZnO, the Cu introduction into the films is then followed by the lattice distortion and consequently disorder creation which causes the optical reduction. Therefore, as the band-gap decreases, the magnitude of defect energy increases. This supports our argument that sub-band states formed in between the valence and conduction bands result in the narrowing of the band-gap. With increasing temperature, the number of defect levels below the conduction band increases to such an extent that the band edge is shifted deep into the forbidden gap, thereby reducing the effective band-gap of CZO. The empirical equation from this linear fitting is given as [15]

$$\begin{equation} E_{g} = 0.115-0.557 E_{u}. \end{equation} \tag{ 6 }$$

The increase in Urbach energy confirms the increase in the width of localized states in the band-gap and hence is responsible for band-gap decay of CZO thin films with increasing temperature. The steepness parameter, which demonstrates the broadening of the absorption band, can be determined empirically using the following equation [16]:

$$\begin{equation} E_{U} = k_{B}T/{\sigma}(T), \end{equation} \tag{ 7 }$$

where the steepness parameter is the Boltzmann constant and T is the absolute temperature. Furthermore, the steepness parameter also determines the strength of electron-phonon interaction ($E_{e-p}$) and both are related to each other via the following relationship [17]:

$$\begin{equation} E_{e-p} =2/ 3{\sigma}. \end{equation} \tag{ 8 }$$

It is demonstrated that the values of the steepness parameter are decreasing while electron-phonon interaction is increasing with the substitution of CZO thin films. This may be ascribed to the change in iconicity and caution valence of CZO thin films.

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The optical conductivity ${\sigma}$ of films was calculated using the relation [18]

$$\begin{equation} {\sigma} = {\alpha}nc / 4{\pi}, \end{equation} \tag{ 9 }$$

where c is the velocity of the incident light. It can be seen that the optical conductivity for thin films increases with the increase of photon energy; this increase is due to the high absorbance of the films in that region (see fig. 5). This suggests that the increase in optical conductivity is due to electrons excited by photon energy. The origin of this increase may be attributed to some changes in the structure of the film due to the doping and the value of temperature.

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Decrease in extinction coefficient implies that these sudden intense peaks indicate the region of deeper penetration for electromagnetic waves and these show the high optical conductivity.

The electrical conductivity $(\gamma)$ in terms of wavelength $({\lambda})$ and optical conductivity $(\sigma)$ of films can be calculated using the formula [19]

$$\begin{equation} \gamma = 2\lambda\sigma/ \alpha. \end{equation} \tag{ 10 }$$

The electrical conductivity of the crystal is decreased gradually as the photon energy increases. The decrease in electrical conductivity is due to the high absorbance of thin films and may be due to electron excitation by the photon [20].

Plotting $(n^{2}-1)^{-1}$ vs. $(h\upsilon)^{2}$ allows us to determine the oscillator parameters. The refractive index $n_\infty$ of films in infinite wavelength can be determined by the following relation [21]:

$$\begin{equation} n^{2}-1= [E_{d}E_{o} / E_{o}^{2}- (h\upsilon)^{2}], \end{equation} \tag{ 11 }$$

where E_d and E_o are single-oscillator constants, E_o is the energy of the effective dispersion oscillator, and E_d the so-called dispersion energy, which measures the intensity of the inter-band optical transitions. Experimental verification of relation (11) can be obtained by plotting $(n^{2}-1)^{-1}$ vs. $(h\upsilon)^{2}$, as illustrated in fig. 6, which yields a straight line for normal behavior having the slope $(E_{o}E_{d})^{-1}$, and the intercept with the vertical axis equal to $E_{o}/E_{d}$. According to the single-oscillator model, the single-oscillator parameters E_o and E_d are related to the imaginary part of the complex dielectric constant. The dispersion energy E_d obeys the following empirical relationship [22]:

$$\begin{equation} E_{d} = \beta N_{c}Z_{a}N_{e}, \end{equation} \tag{ 12 }$$

where N_c is the coordination number of the caution nearest neighbor to the anion, Z_a is the formal chemical valiancy of the anion, N_e is the effective number of valence electrons per anion and β is a constant (0.37±0.04) for covalently bonded crystalline and amorphous chalcogenides and $0.26\,\pm\,0.04$ eV for halides and most oxides that have ionic structure). Taking $N_{c} = 4$, $Z_{a} = 2$, $N_{e} = 8$ for ZnO [22], the β values for the different ZnO nanocrystalline films were determined and are given in table 2. As observed, the obtained values of β for films are in agreement with that published for CZO [22]. The crude refractive index (n_crud ) of the as-prepared and annealed films under investigation can be computed based on the envelope method via transmission spectrum proposed using the Swanepoel method [23]. The values of n can be computed at any wavelength via the relation

$$\begin{align*} 1/T_{m} -1/T_{M} &=4 (n^{2}-1)(n^{2}-s^{2})/16n^{2}s;\nonumber\\ n_{crude} &= [(N+(N^{2} - S^{2})^{1/2}]^{1/2}, \end{align*}$$

where

$$\begin{align} N = 2S[(T_{M}-T_{m})/T_{M}T_{m}]+[(S^{2}+1)/2]. \end{align} \tag{ 13 }$$

This states that refractive index n can be determined only from the values of T_m and T_M obtained from experimental envelope generation.

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Table 2:. The empirical equation from linear fitting $\ln(\alpha)$ of samples upon the ${\lambda}^{-1}\,({\text{nm}})^{-1}\,\beta$ and the single-oscillator energy and dispersion energy parameters, constant.

Sample	Line equation	Eo	Ed	β
		(eV)	(eV)
CZO	$y = 1.94 x+ 14.31$	8.59	18.00	0.28
(as-deposited)
CZO (400°C)	$y = 2.58 x+ 14.20$	5.79	13.17	0.21
CZO (500°C)	$y = 5.97 x+13.72$	6.08	12.42	0.19
CZO (600°C)	$y = 3.23 x+13.02$	5.89	11.33	0.18

T_M and T_m are the transmission maximum and the corresponding minimum at a certain wavelength. The transmittance of the substrate in the absence of film T_s gives an interference-free spectrum [23],

$$\begin{equation} T_{s} = 2s/s^{2}+1\quad \text{or}\quad {s}=1/T_{s}^{-1}+1/(T_s^{2}-1)^{1/2}. \end{equation} \tag{ 14 }$$

The transmittance and reflectance spectrum of thin films at different temperatures together with the transmittance of the uncoated glass substrate is shown in fig. 1. Values of n_crud at any adjacent maximal (or minimal), that have been computed by eq. (13), are used to deduce crude thickness thin films, $d_{crude}= d_{1}$. If n_e1 and n_e2 are the refractive indices of two adjacent maxima or minima at wavelengths $\lambda_{e1}$ and $\lambda_{e2}$, then the crude thickness of the films is expressed as [24]

$$\begin{align} &2n_{m+1}d/ \lambda_{m+1 }- 2n_{m} d/ \lambda_{m} = (m+1) -(m)=1;\nonumber\\ &d=\lambda_{m}\lambda_{m+1} / 2(\lambda_{m}n_{m+1} - \lambda_{m+1}n_{m}). \end{align} \tag{ 15 }$$

The values of film thickness are listed in table 3 (d₁), where n_m and $n_{m+1}$ are the refractive indices at two adjacent maxima (or minima) and $\lambda_{m}$ and $\lambda_{m+1}$ are the corresponding wavelengths. To improve the accuracy of thickness films there is a set of the order number m_o for the interference fringes which are deduced via the relation [24]

$$\begin{equation} m_{crud} = 2n_{crud}\bar{d}_{crud}/ \lambda_{ext}. \end{equation} \tag{ 16 }$$

The order number m can increase precision, by rounding it to an integer, if n and λ were taken at maxima, or half-integer if n and λ were taken at minima. We would denote an improved order number after rounding as m_new . The thickness d_crude can be improved with improved m_new at each extreme $\lambda_{ext}$, by rearranging eq. (16) in terms of d,

$$\begin{equation} d_{new} = m_{new}\lambda_{ext} /2n_{crude}. \end{equation} \tag{ 17 }$$

We would denote an improved thickness as d_new . After obtaining the thicknesses of all the films at all extremes, the new thickness should be averaged, $\bar{d}_{new}$. In previous reports [24], the transmittance and reflectance spectrum for samples were shown. Next, by taking the approximate value of m_o , a new order number m is produced, where $m = 1, 2, 3,\ldots$, at the maximum points in the optical transmission spectrum and $m =1/2, 3/2, 5/2$, at minimum points in the optical transmission spectrum. The final values of the new refractive index n₂ and other mentioned values are listed in table 3. Then, to compute the standard deviation values $\sigma_{i}$ and deviation ratio p_i about the actual value for each of d₁ and d₂, we use the following relation [24]:

$$\begin{align} \sigma_{i}&=\bigg[1/n\sum _{1}^{n}d_{i}^2-\overline{d}_{i}^{2}\bigg]^{1/2};\nonumber\\ \overline{d}_{i} &= 1/n\sum_{1}^{n}d_{i};\\ p_{i}[\%]&= \sigma_{i}/d_{i},\nonumber \end{align} \tag{ 18 }$$

where i is a number equal to 1 or 2 and n refers to the number of thicknesses.

Table 3:. Two values of envelopes T_M and T_m of the samples and the computed values of refractive index and film thicknesses are based on the envelope method.

Sample (no)	λ (nm)	TM	Tm	Ts	S	n1	n2	d1 (nm)	m0	m	d2 (nm)	n
CZO (as-deposited)
$\overline{d_1}=695.75~\sigma_1= 1.83\%$	496.03	0.67	0.57	0.90	1.44	2.60	2.57	398.19	4.17	6.94	662.32	2.34
$\overline{d_2}=475.34~\sigma_2 =0.88\%$	580.37	0.69	0.59	0.88	1.50	2.01	2.01	595.44	4.12	3.49	504.32	2.06
	714.23	0.70	0.61	0.85	1.59	2.07	2.06	740.96	4.29	3.14	543.45	1.94
	942.50	0.73	0.63	0.85	1.59	2.10	2.10	890.96	3.97	1.42	319.06	1.87
	1420.25	0.75	0.63	0.83	1.66	2.52	2.51	853.21	3.03	1.23	347.55	1.93
CZO (400°C)
$\overline{d_1}=429.26~\sigma_1=0.79\%$	460.19	0.65	0.57	0.90	1.44	1.72	1.71	450.36	3.37	3.78	507.87	2.28
$\overline{d_2}=517.88~\sigma_2 =0.95\%$	536.92	0.67	0.59	0.90	1.44	2.09	2.08	663.09	5.16	3.38	434.33	2.03
	654.46	0.69	0.60	0.87	1.54	1.97	1.95	875.11	5.27	3.68	613.08	1.96
	846.48	0.70	0.61	0.86	1.57	2.01	1.83	803.08	3.81	1.83	423.46	1.91
	1268.30	0.73	0.62	0.85	1.59	2.25	2.24	545.10	1.93	1.37	387.69	2.11
CZO (500°C)
$\overline{d_1}=667.35~\sigma_1=5.33\%$	450.23	0.59	0.55	0.91	1.41	1.27	1.26	460.35	2.60	3.27	581.37	2.18
$\overline{d_2}=575.87~\sigma_2=4.05\%$	522.36	0.65	0.56	0.88	1.50	2.02	4.03	698.77	5.40	4.65	602.41	1.97
	636.92	0.71	0.65	0.86	1.57	1.60	3.19	944.42	4.74	3.51	700.18	1.90
	812.91	0.85	0.74	0.87	1.54	1.76	3.50	790.27	3.42	2.38	552.53	1.85
	1228.04	0.90	0.78	0.86	1.57	1.81	3.60	550.43	1.62	1.30	442.89	2.09
CZO (600°C)
$\overline{d_1}=592.81~\sigma_1=1.96\%$	462.16	0.65	0.59	0.91	1.41	1.41	2.81	413.92	2.53	6.11	1004.51	1.98
$\overline{d_2}=646.33~\sigma_2 =1.84\%$	492.28	0.71	0.62	0.90	1.44	1.67	3.88	556.68	3.78	3.90	495.21	1.88
	600.98	0.74	0.65	0.89	1.48	1.68	2.04	699.64	3.91	3.37	992.37	1.84
	778.12	0.77	0.63	0.87	1.54	1.74	3.47	644.68	2.88	1.52	341.03	1.91
	1152.56	0.80	0.69	0.86	1.57	1.95	1.95	649.15	2.20	1.35	398.54	1.89

The accuracy of d can now be significantly increased by taking the corresponding exact integer or half-integer values of m associated with each extreme of the optical transmittance and reflectance spectrum and deriving a new thickness, d₂, from relation (18). By using the values of n₁, the values of d₂ found in this way have a smaller dispersion $(\sigma_{1} > {\sigma}_{2})$. It should be highlighted that the accuracy of the final thickness is approximately better than 1% (listed in table 3). As shown in table 3, using the values of n₁, the values of d₂ found in this way have a smaller dispersion $(\sigma_{1} > \sigma_{2})$. The accurate value of n in eq. (18) is obtained as listed in table 3 and can be solved by the exact value of m at each λ and, thus, the final values of the refractive index n₂. The variation in the refractive index and extinction coefficients of samples can be attributed to the optical absorption spectra and various impurities and perfection in the sample's materials. So, it is reasonable to expect that the structural disorders and defects in the samples are caused by changing the annealing temperature [25].

In this work, we found that the value of annealing temperature has an important effect on the morphology of CZO films. The CZO films annealed at 500°C have a minimum value of RMS roughness of about 7.056 nm. We found that the CZO films annealed at 500°C have the maximum value of optical density and the skin depth, especially in the low-energy range. It can be seen that the CZO films annealed at 500°C have a minimum value of the steepness parameter about of $2.52 \times 10^{-10}$ but have a maximum value of electron-phonon interaction at about $26.43 \times 10^{+8}$. The variations of $\ln(\alpha)$ of films vs. the $\lambda^{-1}\,({\text{nm}})^{-1 }$ were studied by the empirical equations linear fitting as $y=Ax+B$, where 1.9 < A < 6 and 13 < B < 14.35. We observed that as-deposited CZO films have maximum values of dispersion energy (E_d ) at about 18 eV.

Data availability statement: No new data were created or analysed in this study.