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

1 Unité de Physique des Dispositifs à Semi-Conducteurs, Faculté des Sciences Mathematiques, Physiques et Naturelles de Tunis, Université de Tunis El Manar, 2092 Tunis, Tunisia 2 Département de Physique et Chimie, École Supérieure des Science et Techniques de Tunis, Université de Tunis, 1007 Tunis, Tunisia 3 Unité de Recherche MA2I, Ecole Nationale d’Ingénieursde Tunis (ENIT), BP 37, Le Belvédère, 1022 Tunis, Tunisia


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].
In this work, explanations to the paradoxical effects of indium and ytterbium doping on ZnO crystalline structure 2 ISRN Condensed Matter Physics 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 singleoscillator models.

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(CH 3 CO 2 ) 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 nozzleto-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].

Common Characterization Disposals. The prepared films
A ZnO,0 , A Ind,i | i=1··3 , and B Yb,i | i=1··3 were investigated using X-ray diffractometry by means of a Philips (PW1429) system. The optical transmittance T(λ) and reflectance R(λ) 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][21][22][23].

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 Indoped ZnO films along with the Yb-doped ones are shown in Figures 1(a) and 1(b), respectively. 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].
The calculated values of the mechanical Vickers microhardness (Hv), along with Amlouk-Boubaker Optothermal expansivity ψ AB , for the doped layers are gathered in Table 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 E 0 is the single-oscillator energy, ν is the frequency, h is Plank constant, and E d is the dispersion energy, which is also a measure of the interband optical transition intensity.
In the Urbach-Martienssen model [40,41], Urbach energy E u 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). Figures 3 and 4 present, respectively, the reflectance and transmission spectra of the In-and Yb-doped samples.

Experiment.
For the six elaborated samples A Ind,i | i=1··3 and B Yb,i | i=1··3 , 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, Ln which is calculated according to [43][44][45][46], is involved [47] in the relation: where A is a constant and E g is the bandgap energy. The optical bandgap energy E g is deduced from Figure 5, which presents plots of αhν versus hν, by extrapolating the straight line portion of each graph to the energy axis.
Similarly, Urbach energy E u is deduced from local straight line portions slope in the plots of Ln(α) versus hν ( Figure 6).
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.
The evolution of the index of refraction n(λ) has been fitted to Cauchy [49] law: The calculated values of A and B, along with E g and Urbach parameters E 0 , E d , and E u , defined in Section 4.1, are gathered in Table 2. Finally, interaction of the doped layers with electromagnetic radiation has been modeled through complex dielectric constant ε(λ) [50,51] defined by ε(λ) = (n(λ) + ik(λ)) 2 = ε 1 (λ) + iε 2 (λ), The calculated values of ε 1 (λ) and ε 2 (λ) are presented in Figures 9 and 10.    The obtained values were used [51] to evaluate ε ∞ , ω p , and τ, high frequency dielectric constant, plasma frequency and relaxation time, respectively, through the relations where N/m * e 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.

Discussion and Analysis
Precedent analyses (Figures 1(a) and 1(b)) showed that the doped layers A Ind,i | i=1··3 and B Yb,i | i=1··3 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. 361451 < a = 3.24982, c = 5.20661A>), the domination of the doped films c-axisoriented (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 ψ AB . 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, In 3+ and Yb 3+ ions seem to be incorporated inside ZnO würtzite unaltered matrices ( Figure 11).
Since In 3+ , Al 3+ , and Yb 3+ 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.
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 aluminumdoped 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 E u 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.

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 In 3+ and Yb 3+ ions, the würtzite structure of ZnO crystals was globally unaltered. Comparison between In 3+ and Yb 3+ 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.