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

Authors: A. Boukhachem; B. Ouni; A. Bouzidi; A. Amlouk; K. Boubaker; M. Bouhafs; M. Amlouk
Journal: ISRN Condensed Matter Physics
 (2012)

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## 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.


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## Body

## 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 [In3+]/[Zn2+] were 0.01, 0.02, and 0.03 for the samples designated as (AInd,1), (AInd,2), and (AInd,3), 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 [Yb3+]/[Zn2+] were 100, 200, and 300 ppm for the respective samples (BYb,1), (BYb,2), and (BYb,3).Undoped ZnO samples are designated byAZnO,0.


### 2.3. Common Characterization Disposals

The prepared filmsAZnO,0, AInd,i|i=1··3, and BYb,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 [20–23].


## 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 [In3+]/[Zn2+] were 0.01, 0.02, and 0.03 for the samples designated as (AInd,1), (AInd,2), and (AInd,3), 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 [Yb3+]/[Zn2+] were 100, 200, and 300 ppm for the respective samples (BYb,1), (BYb,2), and (BYb,3).Undoped ZnO samples are designated byAZnO,0.


## 2.3. Common Characterization Disposals

The prepared filmsAZnO,0, AInd,i|i=1··3, and BYb,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 [20–23].


## 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) In-doped ZnO filmsAInd,i|i=1..3  XRD diagrams and (b) Yb-doped ZnO films BYb,i|i=1..3  XRD diagrams.
(a)(b)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 expansivityψAB, which is a synthetic physical parameter defined and used in precedent studies [24–39]. This parameter has been defined by(1)ψAB=Dα̂,
where D is the thermal diffusivity and α̂ is the effective absorptivity, defined [30–33] as the mean normalized absorbance weighted by I(λ̃)AM1.5, the solar standard irradiance,  with λ̃ being the normalised wavelength:(2)α̂=∫01I(λ̃)AM1.5×α(λ̃)dλ̃∫01I(λ̃)AM1.5dλ̃,
where I(λ̃)AM1.5 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 expansivityψAB, for the doped layers are gathered in Table 1.Table 1
Microhardness (Hv) and Amlouk-Boubaker Optothermal expansivityψAB.
SampleMicrohard. (Hv)ψAB (×10−12 m3 s−1)MaterialNameDoping rateZnOAZnO,0—34118.2In-doped ZnOAInd,11%57514.5AInd,22%64912.9AInd,33%73910.1Yb-doped ZnOBYb,1100 ppm3019.21BYb,2200 ppm3998.24BYb,3300 ppm49611.2


## 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:(3)n=1+EdE0E02-(hν2),
where E0 is the single-oscillator energy, ν is the frequency, h is Plank constant, and Ed is the dispersion energy, which is also a measure of the interband optical transition intensity.In the Urbach-Martienssen model [40, 41], Urbach energy Eu represents the main parameter that determines the photon capture efficiency of a semiconductor layer. It is defined through the system:(4)α(hν)=α0e(hν/Eu),Eu=α(hν)(d[α(hν)]d[hν])-1=h[ddν(Lnα(ν))]-1. 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

Figures3 and 4 present, respectively, the reflectance and transmission spectra of the In- and Yb-doped samples.Transmission spectra of  the (a) In-doped and (b) Yb-doped ZnO films.
(a)(b)Reflectance spectra of  the (a) In-doped and (b) Yb-doped ZnO films.
(a)(b)For the six elaborated samplesAInd,i|i=1··3 and BYb,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,(5)α=1d2(Ln(1-R2T))+(Ln((1-R)2T))2
which is calculated according to [43–46], is involved [47] in the relation:(6)(αhν)=Ahν-Eg,
where A is a constant and Eg is the bandgap energy.The optical bandgap energyEg 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.Plots ofαhν versus hν.
(a)(b)Similarly, Urbach energyEu is deduced from local straight line portions slope in the plots of Ln(α) versus hν (Figure 6).Plots ofLn(α) versus hν.
(a)(b)On the other hand, the optical characteristics of dispersionn(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.Plots ofn(λ).
(a)(b)Plots ofk(λ).
(a)(b)The evolution of the index of refractionn(λ) has been fitted to Cauchy [49] law:(7)n(λ)=A+Bλ2.
The calculated values of A and B, along with Eg and Urbach parameters E0, Ed, and Eu, defined in Section 4.1, are gathered in Table 2.Table 2
Relevant Cauchy and Urbach parameters.
SampleCauchy constantsBandgapUrbach parametersAB (nm2)Eg (eV)E0 (eV)E0/EgEd (eV)Eu
(meV)AInd,11.8031232.593.256.101.8713.91119.68AInd,21.8228810.593.266.391.9514.9793.11AInd,31.92329376.153.356.772.0218.4488.49BYb,11.6235511.203.285.251.598.7467.19BYb,21.6918848.933.286.682.0312.5951.14BYb,31.7327190.683.286.491.9813.3457.31Finally, interaction of the doped layers with electromagnetic radiation has been modeled through complex dielectric constantε(λ) [50, 51] defined by(8)ε(λ)=(n(λ)+ik(λ))2=ε1(λ)+iε2(λ),ε1(λ)=n(λ)2-k(λ)2,ε2(λ)=2n(λ)k(λ).
The calculated values of ε1(λ) and ε2(λ) are presented in Figures 9 and 10.Plots ofε1(λ).
(a)(b)Plots ofε2(λ).
(a)(b)The obtained values were used [51] to evaluate ε∞, ωp, and τ,  high frequency dielectric constant, plasma frequency and relaxation time, respectively, through the relations(9)ε1=ε∞-ε∞ωp24π2c2λ2,ε2=2nk≈ε∞ωp28π3c3τλ3,ωp=4πNe2ε∞me*,
where N/me* 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 Table3.Table 3
Calculated parameters.
Sampleε∞ωp(1014 rad·S−1)τ   (10−15 s)N/m* (1046 g−1 cm−3)AInd,13.422.408.096.82AInd,23.502.732.409.04AInd,33.862.250.826.80BYb,12.822.441.875.83BYb,23.012.832.748.38BYb,33.202.9410.509.57


## 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:(3)n=1+EdE0E02-(hν2),
where E0 is the single-oscillator energy, ν is the frequency, h is Plank constant, and Ed is the dispersion energy, which is also a measure of the interband optical transition intensity.In the Urbach-Martienssen model [40, 41], Urbach energy Eu represents the main parameter that determines the photon capture efficiency of a semiconductor layer. It is defined through the system:(4)α(hν)=α0e(hν/Eu),Eu=α(hν)(d[α(hν)]d[hν])-1=h[ddν(Lnα(ν))]-1. 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

Figures3 and 4 present, respectively, the reflectance and transmission spectra of the In- and Yb-doped samples.Transmission spectra of  the (a) In-doped and (b) Yb-doped ZnO films.
(a)(b)Reflectance spectra of  the (a) In-doped and (b) Yb-doped ZnO films.
(a)(b)For the six elaborated samplesAInd,i|i=1··3 and BYb,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,(5)α=1d2(Ln(1-R2T))+(Ln((1-R)2T))2
which is calculated according to [43–46], is involved [47] in the relation:(6)(αhν)=Ahν-Eg,
where A is a constant and Eg is the bandgap energy.The optical bandgap energyEg 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.Plots ofαhν versus hν.
(a)(b)Similarly, Urbach energyEu is deduced from local straight line portions slope in the plots of Ln(α) versus hν (Figure 6).Plots ofLn(α) versus hν.
(a)(b)On the other hand, the optical characteristics of dispersionn(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.Plots ofn(λ).
(a)(b)Plots ofk(λ).
(a)(b)The evolution of the index of refractionn(λ) has been fitted to Cauchy [49] law:(7)n(λ)=A+Bλ2.
The calculated values of A and B, along with Eg and Urbach parameters E0, Ed, and Eu, defined in Section 4.1, are gathered in Table 2.Table 2
Relevant Cauchy and Urbach parameters.
SampleCauchy constantsBandgapUrbach parametersAB (nm2)Eg (eV)E0 (eV)E0/EgEd (eV)Eu
(meV)AInd,11.8031232.593.256.101.8713.91119.68AInd,21.8228810.593.266.391.9514.9793.11AInd,31.92329376.153.356.772.0218.4488.49BYb,11.6235511.203.285.251.598.7467.19BYb,21.6918848.933.286.682.0312.5951.14BYb,31.7327190.683.286.491.9813.3457.31Finally, interaction of the doped layers with electromagnetic radiation has been modeled through complex dielectric constantε(λ) [50, 51] defined by(8)ε(λ)=(n(λ)+ik(λ))2=ε1(λ)+iε2(λ),ε1(λ)=n(λ)2-k(λ)2,ε2(λ)=2n(λ)k(λ).
The calculated values of ε1(λ) and ε2(λ) are presented in Figures 9 and 10.Plots ofε1(λ).
(a)(b)Plots ofε2(λ).
(a)(b)The obtained values were used [51] to evaluate ε∞, ωp, and τ,  high frequency dielectric constant, plasma frequency and relaxation time, respectively, through the relations(9)ε1=ε∞-ε∞ωp24π2c2λ2,ε2=2nk≈ε∞ωp28π3c3τλ3,ωp=4πNe2ε∞me*,
where N/me* 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 Table3.Table 3
Calculated parameters.
Sampleε∞ωp(1014 rad·S−1)τ   (10−15 s)N/m* (1046 g−1 cm−3)AInd,13.422.408.096.82AInd,23.502.732.409.04AInd,33.862.250.826.80BYb,12.822.441.875.83BYb,23.012.832.748.38BYb,33.202.9410.509.57


## 5. Discussion and Analysis

Precedent analyses (Figures 1(a) and1(b)) showed that the doped layers AInd,i|i=1··3 and BYb,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-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 ψAB.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 (Figures12 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 (Figure11).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 Eu 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.



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