ISRN Condensed Matter PhysicsVolume 2012 (2012), Article ID 738023, 10 pagesdoi:10.5402/2012/738023

Research Article

Quantum Effects of Indium/Ytterbium Doping on ZnO-Like Nano-Condensed Matter in terms of Urbach-Martienssen and Wemple-DiDomenico Single-Oscillator Models Parameters

A. Boukhachem,¹ B. Ouni,¹ A. Bouzidi,¹ A. Amlouk,¹ K. Boubaker,² M. Bouhafs,³ and M. Amlouk¹

¹Unité de Physique des Dispositifs à Semi-Conducteurs, Faculté des Sciences Mathematiques, Physiques et Naturelles de Tunis, Université de Tunis El Manar, 2092 Tunis, Tunisia²Département de Physique et Chimie, École Supérieure des Science et Techniques de Tunis, Université de Tunis, 1007 Tunis, Tunisia³Unité de Recherche MA2I, Ecole Nationale d'Ingénieurs de Tunis (ENIT), BP 37, Le Belvédère, 1022 Tunis, Tunisia

Received 22 November 2011; Accepted 2 January 2012

Academic Editors: A. Krimmel and S. Krukowski

Copyright © 2012 A. Boukhachem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Conducting and transparent optical ZnO thin films were deposited on glass substrates by a simple mini spray technique. Alternatively, some of the obtained films were doped with indium and ytterbium at the molar rates of: 1, 2, and 3% (In) and 100, 200, and 300 ppm (Yb). In addition to the classical structural investigations including XRD, microhardness vickers (Hv), and optothermal techniques, thorough optical measurements have been carried out for comparison purposes. The refractive indices and the extinction coefficients of the differently doped layers have been deduced from their transmission-reflection spectra over an extended wavelength range. Analysis of the refractive index data through Wemple-DiDomenico single oscillator model yielded quantum characteristics along with the values of long-wavelength dielectric constant, average oscillator wavelength, average oscillator strength, average oscillator energy and dispersion energy. Real and imaginary parts of dielectric constant have also been used to calculate free carrier plasma resonance frequency, optical relaxation time, and free carriers concentration-to-effective mass ratio. Finally, analysis of Urbach-Martienssen model parameters allowed proposing nanoscale explanations to the divergence about doping-related evolution of Urbach tails, this intriguing item having been intensively discussed in the literature in the last decades.

1. Introduction

Transparent conducting oxides (TCOs) such as tin oxides and indium-doped oxide systems have been used in several optoelectronic devices such as gas sensors, panel displays [1, 2], and photovoltaic solar cells (PVCs). Among these oxides, zinc oxide has attracted considerable attention from those interested in the application to devices working in ultraviolet regions, with the interest specially lying in its wide bandgap, quantum confinement effects in accessible size ranges, and large exciton binding energy (≈60 meV) [1, 2].

It has been recorded that zinc oxide is a hexagonal wurtzite structured semiconductor with high piezoelectric and gas-detecting properties [1–6]. Its deposition on glass-like substrates has been widely experimented and applied [7–11].

On the other hand, many doping elements for ZnO have been tried [12–15]. In some studies, the merits of indium and aluminium as effective doping agents have been pointed out [16–19].

In this work, explanations to the paradoxical effects of indium and ytterbium doping on ZnO crystalline structure are proposed. The elaborating techniques and doping protocols have been detailed along with common characterizing techniques. The last part of this paper is dedicated to the nanoscale analyses and discussions in reference to Urbach-Martienssen and Wemple-DiDomenico single-oscillator models.

2. Experimental Details

2.1. Undoped ZnO Films Preparation

ZnO thin films have been first prepared at a glass substrate temperature of 460°C, using propanol and zinc acetate Zn(CH₃CO₂)₂ : 10⁻² M according to the chemical protocol summarised elsewhere [20, 21].

The precursor mixture was acidified using acetic acid (pH = 5) and the carrier gas was nitrogen (pressure ≈ 0.35 bar) through a 0.5 mm diameter nozzle. The nozzle-to-substrate plane distance was fixed at the optimal value of 27 cm as demonstrated earlier, for the same disposal, by K. Boubaker et al. [22]. During the whole deposition process, precursor mixture flow rate was approximately 4 mL/min.

Thicknesses of the obtained layers have been deduced according to the method established by Belgacem and Bennaceur [23].

2.2. Indium and Ytterbium Doping

Under similar experimental conditions, indium-doped ZnO, In thin films solutions have been fabricated by adding hydrated indium chloride (InCl₃, xH₂O, 99.9% purity) to the precursor solution while maintaining acidity level. In the three elaborated samples, the indium-to-zinc molar ratios were 0.01, 0.02, and 0.03 for the samples designated as (), (), and (), respectively.

In ytterbium-doped ZnO, Yb thin films solutions were fabricated similarly by adding small amounts of ytterbium chloride (YbCl₃) to the precursor solution in which ytterbium-to-zinc molar ratios were 100, 200, and 300 ppm for the respective samples (), (), and ().

Undoped ZnO samples are designated by .

2.3. Common Characterization Disposals

The prepared films , , and were investigated using X-ray diffractometry by means of a Philips (PW1429) system. The optical transmittance and reflectance ) of the films were recorded using a Shimadzu UV 3100 double-beam spectrophotometer, within a (300–1800 nm) wavelength range. Finally, a mechanical characterization has been carried out using a diamond-pyramidal-indenter Vickers test disposal whose details are presented in precedent studies [20–23].

3. Already Achieved XRD, Mechanical, and Optothermal Characterization

XRD, mechanical, and optothermal characterization of the as-grown films has been already carried out and discussed in precedent studies [20, 21]. XRD patterns of the deposited In-doped ZnO films along with the Yb-doped ones are shown in Figures 1(a) and 1(b), respectively.

(a)

(b)

Figure 1: (a) In-doped ZnO films XRD diagrams and (b) Yb-doped ZnO films XRD diagrams.

The mechanical microhardness of the ytterbium- and indium-doped ZnO layers has been evaluated in terms of Vickers hardness (Hv). The characterization test has been carried out using a common diamond-pyramidal-indenter Vickers test disposal whose main parameters are detailed in precedent studies [20, 21].

Optothermal investigation has been based on variation of the Amlouk-Boubaker Optothermal expansivity , which is a synthetic physical parameter defined and used in precedent studies [24–39]. This parameter has been defined by where is the thermal diffusivity and is the effective absorptivity, defined [30–33] as the mean normalized absorbance weighted by , the solar standard irradiance, with being the normalised wavelength: where is the reference solar spectral irradiance, fitted using the Boubaker polynomials expansion scheme (BPES).

The calculated values of the mechanical Vickers microhardness (Hv), along with Amlouk-Boubaker Optothermal expansivity , for the doped layers are gathered in Table 1.

Table 1: Microhardness (Hv) and Amlouk-Boubaker Optothermal expansivity .

Sample
Microhard. (Hv)
(×10⁻¹² m³ s⁻¹)

Material
Name
Doping rate

ZnO

—
341
18.2

In-doped ZnO

1%
575
14.5

2%
649
12.9

3%
739
10.1

Yb-doped ZnO

100 ppm
301
9.21

200 ppm
399
8.24

300 ppm
496
11.2

4. Wemple-DiDomenico and Urbach-Martienssen Models Investigations

4.1. Theoretical Fundaments

The Wemple-DiDomenico single-oscillator model is based on a particular dependence [40] of the dispersion of refractive index below the interband absorption edge: where is the single-oscillator energy, is the frequency, h is Plank constant, and is the dispersion energy, which is also a measure of the interband optical transition intensity.

In the Urbach-Martienssen model [40, 41], Urbach energy represents the main parameter that determines the photon capture efficiency of a semiconductor layer. It is defined through the system: Mott and Davis [42] noted that oppositely to crystalline structures, where the fundamental edge is mainly determined by conduction band minimum and valence band maximum levels, ion-doped binary semiconductor compounds present a particular optical absorption edge profile. In these materials, the absorption coefficient increases exponentially with the photon energy near the energy gap. This variation results in “blurring” of the valence-conduction bands and narrows slightly the band gap by appearance of the so-called Urbach tails (Figure 2).

Figure 2: Urbach tails scheme.

4.2. Experiment

Figures 3 and 4 present, respectively, the reflectance and transmission spectra of the In- and Yb-doped samples.

(a)

(b)

Figure 3: Transmission spectra of the (a) In-doped and (b) Yb-doped ZnO films.

(a)

(b)

Figure 4: Reflectance spectra of the (a) In-doped and (b) Yb-doped ZnO films.

For the six elaborated samples and , it can be noticed that the transmission coefficient increases with doping (from 40% to 85%) while the reflectance remains inside a narrow interval (5%–20%). These observations confirm the XRD-related consequences of doping on crystal c-axis alignment, which are favorable to light transmission.

Moreover, it is obvious that doping alters transmitting performance of the initial oxide ZnO. The synthetic absorption coefficient, which is calculated according to [43–46], is involved [47] in the relation: where is a constant and is the bandgap energy.

The optical bandgap energy is deduced from Figure 5, which presents plots of versus , by extrapolating the straight line portion of each graph to the energy axis.

(a)

(b)

Figure 5: Plots of versus .

Similarly, Urbach energy is deduced from local straight line portions slope in the plots of versus (Figure 6).

(a)

(b)

Figure 6: Plots of versus .

On the other hand, the optical characteristics of dispersion n(λ) and k(λ) (refractive index and extinction coefficient, resp.), for values of the wavelength λ between 300 and 1800 nm, have been calculated using optical experimental measurements and the methods of Bathe and Patil [48] and Belgacem and Bennaceur [23]. The plots of n(λ) and k(λ) are presented in Figures 7 and 8.

(a)

(b)

Figure 7: Plots of n(λ).

(a)

(b)

Figure 8: Plots of k(λ).

The evolution of the index of refraction has been fitted to Cauchy [49] law: The calculated values of A and B, along with and Urbach parameters , , and , defined in Section 4.1, are gathered in Table 2.

Table 2: Relevant Cauchy and Urbach parameters.

Sample
Cauchy constants
Bandgap
Urbach parameters

(nm²)
(eV)
(eV)

(eV)
(meV)

1.80
31232.59
3.25
6.10
1.87
13.91
119.68

1.82
28810.59
3.26
6.39
1.95
14.97
93.11

1.923
29376.15
3.35
6.77
2.02
18.44
88.49

1.62
35511.20
3.28
5.25
1.59
8.74
67.19

1.69
18848.93
3.28
6.68
2.03
12.59
51.14

1.73
27190.68
3.28
6.49
1.98
13.34
57.31

Finally, interaction of the doped layers with electromagnetic radiation has been modeled through complex dielectric constant [50, 51] defined by The calculated values of and are presented in Figures 9 and 10.

(a)

(b)

Figure 9: Plots of .

(a)

(b)

Figure 10: Plots of .

The obtained values were used [51] to evaluate , , and , high frequency dielectric constant, plasma frequency and relaxation time, respectively, through the relations where represents free carriers concentration-to-effective mass ratio.

Calculated values of high frequency dielectric constant, plasma pulsation, relaxation time, and free carriers concentration-to-effective mass ratio are gathered in Table 3.

Table 3: Calculated parameters.

Sample

(10¹⁴ rad·S⁻¹)
(10⁻¹⁵ s)
(10⁴⁶ g⁻¹ cm⁻³)

3.42
2.40
8.09
6.82

3.50
2.73
2.40
9.04

3.86
2.25
0.82
6.80

2.82
2.44
1.87
5.83

3.01
2.83
2.74
8.38

3.20
2.94
10.50
9.57

5. Discussion and Analysis

Precedent analyses (Figures 1(a) and 1(b)) showed that the doped layers and developed an enhanced preferred orientation of the crystallites with respect to the (002) reflection. Since ZnO layers are generally identified by XRD peaks, (101), (100), and (002) in hexagonal wurtzite system (JCPDS card file no. , c = 5.20661A>), the domination of the doped films c-axis-oriented (002) peaks perpendicularly to the glass substrate plane represents a drastic optical enhancement. This feature has been confirmed by mechanical and Optothermal measurements, which recorded an increase of the microhardness of the doped layers along with an obvious decrease of .

The actual analyses have been carried out on this basis: the ZnO wurtzite matrix is not meaningfully altered by In or Yb doping. Tiburcio-Silver et al. [52] excluded, in the same context, the existence of any indium or ytterbium chemically established compound. In other words, In³⁺ and Yb³⁺ ions seem to be incorporated inside ZnO würtzite unaltered matrices (Figure 11).

Figure 11: Doping ions intercalation inside ZnO wurtzite matrices.

Since In³⁺, Al³⁺, and Yb³⁺ have the same valences and perform close oxygen-related electronegativity, the explanation of concentration-related stark differences of evolution of Urbach energy (Figure 12) and free carriers lifetime (Figure 13) should be investigated from the side of the doping element distinctive physical properties.

Figure 12: Doping-related Urbach energy evolution.

Figure 13: Doping-related free carriers lifetime evolution.

In fact, the regular decrease of Urbach energy and free carriers lifetime (Figures 12 and 13) for In-doped layers, as recoded also by Cody et al. [53] for incrementally aluminum-doped binary compounds, contrasts with the Yb-doped layers behavior. A phenomenon of saturation seems to occur for increasing doping amounts inside these latter layers (Figure 11).

In this context, nanoscale comparative estimation of atomic and ionic radii revealed an intriguing element: among the prospected entities, ytterbium represents the highest radius, which is moreover comparable to upper ZnO–O bond dimension inside ZnO würtzite matrix (Figure 11).

As Dao [54], Wasim et al. [55], Narayanan et al. [56], Tauc et al. [57], and Grus and Sikorska [58] evoked a strong dependence of the Urbach energy versus electron-phonon and exciton-phonon interactions as well as crystalline structural static disorder, radiation inside the crystal and deviation from the ideal stoichiometry, it can consequently be suggested that indium incremental incorporation does not induce any meaningful additional static disorder, due to size adequacy and ion-matrix compatibility, while high-sized atoms (like ytterbium) cease to be incorporated at a given amount level and hence disturb the global layer crystalline order and widen Urbach tails.

6. Conclusion

Zinc oxide layers have been prepared and doped with indium and ytterbium using a low cost spray pyrolysis process. During elaboration and characterisation phases, it has been verified that in absence of any conventionally verified chemical reaction of In³⁺ and Yb³⁺ ions, the würtzite structure of ZnO crystals was globally unaltered. Comparison between In³⁺ and Yb³⁺ doping in terms of Urbach energy and free carriers lifetime led to interesting observations. The intriguing problem of the doping-related Urbach tails distortion has been discussed relatively to atomistic nanoscale considerations and some explanations have been proposed. Further studies on vanadium- (V-) and molybdenum- (Mo-) doped samples are actually in progress in order to confirm the suspected effects of ionic size on Urbach tails width and carriers lifetime.

References

P. C. Liao, A. Korotcov, C. W. Huang, Y. S. Huang, D. S. Tsai, and K. K. Tiong, “Synthesis of IrO₂ nanocrystals on sapphire via metal-organic chemical vapor deposition,” Journal of Alloys and Compounds, vol. 442, no. 1-2, pp. 313–315, 2007.

M. H. Brodsky, Ed., Amorphous Semiconductors, Springer, Berlin, Germany, 1979.

W. Lin, J. Pak, D. C. Ingram, and A. R. Smith, “Molecular beam epitaxial growth of zinc-blende FeN(111) on wurtzite GaN(0001),” Journal of Alloys and Compounds, vol. 463, no. 1-2, pp. 257–262, 2008.

D. Y. Wang, J. Zhou, and G. Z. Liu, “Effect of Li-doped concentration on the structure, optical and electrical properties of p-type ZnO thin films prepared by sol-gel method,” Journal of Alloys and Compounds, vol. 481, no. 1-2, pp. 802–805, 2009.

M. R. Barati, “Influence of zinc substitution on magnetic and electrical properties of MgCuZn ferrite nanocrystalline powders prepared by sol-gel, auto-combustion method,” Journal of Alloys and Compounds, vol. 478, no. 1-2, pp. 375–380, 2009.

D. Beena, K. J. Lethy, R. Vinodkumar, A. P. Detty, V. P. Mahadevan Pillai, and V. Ganesan, “Photoluminescence in laser ablated nanostructured indium oxide thin films,” Journal of Alloys and Compounds, vol. 489, no. 1, pp. 215–223, 2010.

I. Saita, T. Toshima, S. Tanda, and T. Akiyama, “Hydrogen storage property of MgH₂ synthesized by hydriding chemical vapor deposition,” Journal of Alloys and Compounds, vol. 446-447, pp. 80–83, 2007.

Y. M. Kim, M. Yoon, I. W. Park, Y. J. Park, and J. H. Lyou, “Synthesis and magnetic properties of O films prepared by the sol-gel method,” Solid State Communications, vol. 129, no. 3, pp. 175–178, 2004.

S. K. Hong, H. Y. Koo, D. S. Jung, I. S. Suh, and Y. C. Kang, “Preparation of Bi₂O₃–B₂O₃–ZnO–BaO–SiO₂ glass powders with spherical shape by spray pyrolysis,” Journal of Alloys and Compounds, vol. 437, no. 1-2, pp. 215–219, 2007.

G. Hu, X. Deng, Z. Peng, Y. Cao, and K. Du, “Morphology and luminescence of (Y,Gd)BO₃ : Eu phosphor particles prepared by urea-assisted spray pyrolysis,” Journal of Alloys and Compounds, vol. 452, no. 2, pp. 462–466, 2008.

P. M. Devshette, N. G. Deshpande, and G. K. Bichile, “Growth and physical properties of O thin films prepared by spray pyrolysis technique,” Journal of Alloys and Compounds, vol. 463, no. 1-2, pp. 576–580, 2008.

M. Joseph, H. Tabata, and T. Kawai, “p-Type electrical conduction in ZnO thin films by Ga and N codoping,” Japanese Journal of Applied Physics, vol. 38, no. 11 A, pp. L1205–L1207, 1999.

C. S. Hsi, B. Houng, B. Y. Hou, G. J. Chen, and S. L. Fu, “Effect of Ru addition on the properties of Al-doped ZnO thin films prepared by radio frequency magnetron sputtering on polyethylene terephthalate substrate,” Journal of Alloys and Compounds, vol. 464, no. 1-2, pp. 89–94, 2008.

E. Bacaksiz, M. Parlak, M. Tomakin, A. Özçelik, M. Karakiz, and M. Altunbaş, “The effects of zinc nitrate, zinc acetate and zinc chloride precursors on investigation of structural and optical properties of ZnO thin films,” Journal of Alloys and Compounds, vol. 466, no. 1-2, pp. 447–450, 2008.

K.-S. Kim, H. W. Kim, and C. M. Lee, “Effect of growth temperature on ZnO thin film deposited on SiO₂ substrate,” Materials Science and Engineering B, vol. 98, no. 2, pp. 135–139, 2003.

T. Moriga, Y. Nishimura, H. Suketa et al., “Effects of Al, Ga-doping on transparent conducting properties of amorphous ZnO–SnO₂ films,” International Journal of Modern Physics B, vol. 20, no. 25-27, pp. 3902–3907, 2006.

A. Takagi, K. Nomura, H. Ohta et al., “Carrier transport and electronic structure in amorphous oxide semiconductor, a-InGaZnO₄,” Thin Solid Films, vol. 486, no. 1-2, pp. 38–41, 2005.

K. Nomura, T. Kamiya, H. Ohta, K. Ueda, M. Hirano, and H. Hosono, “Carrier transport in transparent oxide semiconductor with intrinsic structural randomness probed using single-crystalline InGaO₃(ZnO)₅ films,” Applied Physics Letters, vol. 85, no. 11, pp. 1993–1995, 2004.

S. W. Kim, S. Matsuishi, T. Nomura et al., “Metallic state in a lime-alumina compound with nanoporous structure,” Nano Letters, vol. 7, no. 5, pp. 1138–1143, 2007.

A. Boukhachem, S. Fridjine, A. Amlouk, K. Boubaker, M. Bouhafs, and M. Amlouk, “Comparative effects of indium/ytterbium doping on, mechanical and gas-sensitivity-related morphological, properties of sprayed ZnO compounds,” Journal of Alloys and Compounds, vol. 501, no. 2, pp. 339–344, 2010.

B. Ouni, J. Ouerfelli, A. Amlouk, K. Boubaker, and M. Amlouk, “Structural, mechanical and opto-thermal properties of non-crystalline thin films,” Journal of Non-Crystalline Solids, vol. 356, no. 25-27, pp. 1294–1299, 2010.

K. Boubaker, A. Chaouachi, M. Amlouk, and H. Bouzouita, “Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition,” The European Physical Journal, vol. 37, no. 1, pp. 105–109, 2007.

S. Belgacem and R. Bennaceur, “Propriétés optiques des couches minces de SnO₂ et CuInS₂ airless spray,” Revue de Physique Appliquée, vol. 25, no. 12, pp. 1245–1258, 1990.

S. Fridjine and M. Amlouk, “A new parameter: an abacus for optimizing PVT hybrid solar device functional materials using the boubaker polynomials expansion scheme,” Modern Physics Letters B, vol. 23, no. 17, pp. 2179–2191, 2009.

O. D. Oyodum, O. B. Awojoyogbe, M. Dada, and J. Magnuson, “Comment on “enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition” by K. Boubaker, A. Chaouachi, M. Amlouk and H. Bouzouita: on the earliest definition of the boubaker polynomials,” The European Physical Journal, vol. 46, no. 2, Article ID 21201, 2 pages, 2009.

A. Belhadj, O. F. Onyango, and N. Rozibaeva, “Boubaker polynomials expansion scheme-related heat transfer investigation inside keyhole model,” Journal of Thermophysics and Heat Transfer, vol. 23, no. 3, pp. 639–640, 2009.

A. Belhadj, J. Bessrour, M. Bouhafs, and L. Barrallier, “Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and boubaker polynomials expansion,” Journal of Thermal Analysis and Calorimetry, vol. 97, no. 3, pp. 911–915, 2009.

K. B. Ben Mahmoud, “Solution to heat equation inside cryogenic vessels using boubaker polynomials,” Journal of Thermophysics and Heat Transfer, vol. 23, no. 2, pp. 409–411, 2009.

T. G. Zhao, Y. X. Wang, and K. B. Ben Mahmoud, “Limit and uniqueness of the Boubaker-Zhao polynomials single imaginary root sequence,” International Journal of Mathematics & Computation, vol. 1, no. 8, pp. 13–16, 2008.

K. Boubaker, “On modified boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation,” Trends in Applied Sciences Research, vol. 2, no. 6, pp. 540–544, 2007.

K. Boubaker, “A new polynomial sequence as a guide to numerical solutions for applied-physics-related partial differential equations under dirichlet-newman-type exogenous boundary conditions,” Numerical Methods for Partial Differential Equations, vol. 25, no. 4, pp. 802–809, 2009.

S. Slama, J. Bessrour, B. Karem, and M. Bouhafs, “investigation of A3 point maximal front spatial evolution during resistance spot welding using 4Q-boubaker polynomal sequence,” in Proceedings of the COTUME, pp. 79–80, 2008.

H. Labiadh, M. Dada, O. B. Awojoyogbe, K. B. Ben Mahmoud, and A. Bannour, “Establishment of an ordinary generating function and a christoffel-darboux type first-order differential equation for the heat equation related boubaker-turki polynomials,” Differential Equations and Control Processes, vol. 1, pp. 51–66, 2008.

O. B. Awojoyogbe and K. Boubaker, “A solution to Bloch NMR flow equations for the analysis of hemodynamic functions of blood flow system using m-boubaker polynomials,” Current Applied Physics, vol. 9, no. 1, pp. 278–283, 2009.

J. Ghanouchi, H. Labiadh, and K. Boubaker, “An attempt to solve the heat transfer equation in a model of pyrolysis spray using 4q-order m-boubaker polynomials,” International Journal of Heat and Technology, vol. 26, no. 1, pp. 49–53, 2008.

S. Slama, M. Bouhafs, and K. B. Ben Mahmoud, “A boubaker polynomials solution to heat equation for monitoring A3 point evolution during resistance spot welding,” International Journal of Heat and Technology, vol. 26, no. 2, pp. 141–145, 2008.

K. Boubaker, “The boubaker polynomials, a new function class for solving bi-varied second order differential equations,” Far East Journal of Applied Mathematics, vol. 31, no. 3, pp. 299–320, 2008.

D. H. Zhang and F. W. Li, “A Boubaker Polynomials Expansion Scheme BPES-Related Analytical Solution to Williams-Brinkmann Stagnation Point Flow Equation at a Blunt Body,” International Journal of Applied Physics Letters, vol. 2, p. 25, 2009.

S. Tabatabaei, T. Zhao, O. B. Awojoyogbe, and F. O. Moses, “Cut-off cooling velocity profiling inside a keyhole model using the boubaker polynomials expansion scheme,” Heat and Mass Transfer, vol. 45, no. 10, pp. 1247–1251, 2009.

F. Urbach, “The long-wavelength edge of photographic sensitivity and of the electronic absorption of solids,” Physical Review, vol. 92, no. 5, p. 1324, 1953.

W. Martienssen, “Über die excitonenbanden der alkalihalogenidkristalle,” Journal of Physics and Chemistry of Solids, vol. 2, no. 4, pp. 257–267, 1957.

N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials, Clarendon Press, Oxford, UK, 1979.

A. Amlouk, K. Boubaker, M. Amlouk, and M. Bouhafs, “Study of ytterbium doping effects on structural, mechanical and opto-thermal properties of sprayed ZnO thin films using the boubaker Polynomials Expansion Scheme (BPES),” Journal of Alloys and Compounds, vol. 485, no. 1-2, pp. 887–891, 2009.

C. Khelia, M. Amlouk, and K. Boubaker, “Boubaker Polynomials Expansion Scheme (BPES)-related optical properties of b-SnS₂ sprayed layers,” Fizika A, vol. 18, no. 2, pp. 81–88, 2009.

A. Amlouk, K. Boubaker, and M. Amlouk, “Effects of substrate temperature on sprayed ZnO thin films optical and morphological properties in terms of Amlouk-Boubaker opto-thermal expansivity ψ AB,” Journal of Alloys and Compounds, vol. 482, no. 1-2, pp. 164–167, 2009.

K. B. Ben Mahmoud and M. Amlouk, “The 3D Amlouk-Boubaker expansivity-energy gap-vickers hardness abacus: a new tool for optimizing semiconductor thin film materials,” Materials Letters, vol. 63, no. 12, pp. 991–994, 2009.

S. Dabbous, T. Ben Nasrallah, J. Ouerfelli, K. Boubaker, M. Amlouk, and S. Belgacem, “Study of structural and optical properties of sprayed WO₃ thin films using enhanced characterization techniques along with the boubaker Polynomials Expansion Scheme (BPES),” Journal of Alloys and Compounds, vol. 487, no. 1-2, pp. 286–292, 2009.

S. R. Bathe and P. S. Patil, “Electrochromic characteristics of fibrous reticulated WO₃ thin films prepared by pulsed spray pyrolysis technique,” Solar Energy Materials and Solar Cells, vol. 91, no. 12, pp. 1097–1101, 2007.

H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry, John Wiley & Sons, New York, NY, USA, 1999.

M. Sesha Reddy, K. T. Ramakrishna Reddy, B. S. Naidu, and P. J. Reddy, “Optical constants of polycrystalline CuGaTe₂ films,” Optical Materials, vol. 4, no. 6, pp. 787–790, 1995.

J. I. Pankove, Optical Processes in Semiconductors, Prentice-Hall, New Jersey, NJ, USA, 1971.

A. Tiburcio-Silver, J. C. Joubert, and M. Labeau, “Propriétés électriques et optiques de couches minces de ZnO et ZnO dopé à l'indium, obtenues par le procédé Pyrosol,” Journal of Physics, vol. 2, no. 7, pp. 1287–1303, 1992.

G. D. Cody, T. Tiedje, B. Abeles, B. Brooks, and Y. Goldstein, “Disorder and the optical-absorption edge of hydrogenated amorphous silicon,” Physical Review Letters, vol. 47, no. 20, pp. 1480–1483, 1981.

T. H. Dao, Dépôt de couches minces de silicium à grande vitesse par plasma MDECR, Doctoral Thesis, Ecole Polytechnique de France, May 2007.

S. M. Wasim, C. Rincon, G. Marin, et al., “Effect of structural disorder on the Urbach energy in Cu ternaries,” Pysical Reviews B, vol. 64, no. 19, Article ID 195101, 8 pages, 2001.

K. L. Narayanan, K. P. Vijayakumar, K. G. M. Nair, and N. S. Thampi, “Effect of irradiation-induced disorder on the optical absorption spectra of CdS thin films,” Physica B, vol. 240, no. 1-2, pp. 8–12, 1997.

J. Tauc, R. Grigorovic, and A. Vancu, “Optical properties and electric structure of amorphous germanium,” Physica Status Solidi, vol. 15, no. 2, pp. 627–637, 1966.

M. Grus and A. Sikorska, “Characterization of the absorption edge in crystalline CdS : Cu powder by use of photoacoustic and reflection spectroscopy,” Physica B, vol. 266, no. 3, pp. 139–145, 1999.