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Journal of Alloys and Compounds 696 (2017) 1244e1254

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Structural, optical properties and characterization of (C2H5NH3)2CdCl4, (C2H5NH3)2CuCl4 and (C2H5NH3)2Cd0.5Cu0.5Cl4 compounds Ramzi Lefi*, Ferid Ben Naser, Hajer Guermazi Unit e de recherche: Physique des Mat eriaux Isolants et Semi Isolants, Faculty of Sciences of Sfax, University of Sfax, Tunisia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 April 2016 Received in revised form 24 November 2016 Accepted 6 December 2016 Available online 9 December 2016

In this work, we are interested in the synthesis of hybrid perovskite crystals (C2H5NH3)2CdCl4, (C2H5NH3)2CuCl4 and (C2H5NH3)2Cd0.5Cu0.5Cl4, by the conventional method. X-ray diffraction indicates that these compounds crystallize at room temperature in the orthorhombic system. The values of lattice parameters and volume were determined. The values of the average crystallite size and the strain were calculated and compared by two models: Scherrer and Williamson-Hall. We note that the (C2H5NH3)2CdCl4 compound has the best crystal quality. Infrared spectroscopy (FTIR) analysis and UV evisible spectrophotometer were used. FTIR technique confirms the presence of the organic chain of the perovskite compounds. On the other hand, UVevisible absorption measurements show that the optical absorbance is higher in the visible-NIR domain for these compounds. The optical band gap (Eg) is deduced to be 3.11; 2.4 and 3.04 eV for (C2H5NH3)2CdCl4, (C2H5NH3)2CuCl4 and (C2H5NH3)2Cd0.5Cu0.5Cl4 respectively. The dispersion parameters (E0 and Ed) of these compounds were estimated using the WempleeDiDomenico model. The different optical parameters such as refractive index, dielectric permittivity and conductivity were also determined. It is noted that the conductivity increases with the introduction of copper in (C2H5NH3)2CdCl4 compound. © 2016 Elsevier B.V. All rights reserved.

Keywords: Perovskites Strain Average crystallite size FTIR Gap energy

1. Introduction Recently, hybrid compounds have been under extensive investigations for the past two decades due to their interesting structure, optical and electric properties [1e4]. The crystalline structure of compounds with the type [(CnH2nþ1)NH3]2MX4, where n ¼ 2,3…, M represents divalent metals (M: Cu, Cd, …) and X halogens (X: Cl and I, …), may be described as sequences of alternating organic-inorganic layers [5]. For these hybrid materials, the ammonium ion of the organic group forms hydrogen bonds with the halide ion of the inorganic layer, which makes these organicinorganic compounds good candidates for proton conduction with high potential for technological applications [6,7]. The electronic structure of [(CnH2nþ1)NH3]2MX4 is regarded as a selforganized multiple-quantum-well structure consisting of inorganic wells and organic barriers. For compounds with short chain length n  4 and M ¼ Mn2þ, Fe2þ, Cu2þ, Cd2þ or Pb2þ, the metal ion extends in layers of associated [MX6] octahedra [8e11] and for

M ¼ Co2þ and Zn2þ unassociated tetrahedra [MCl4] arrangement is observed [12,13]. For these hybrid compounds, the physical properties depend on the characteristics of the organic cation and the inorganic anion, which allows the properties of the material to be tailored [14,15]. Organiceinorganic hybrid materials have recently drawn massive attention because of their photovoltaic applications and excellent performances as light absorbers and carrier transporters [16,17]. In this work, we seek to improve the performances of these materials by preparation of a new perovskite compounds based on cadmium and copper. We aim to study the effect of the introduction of copper on structural, optical and electrical properties of (C2H5NH3)2CdCl4 compound. The results of the hybrid (C2H5NH3)2CuyCd1yCl4 compounds (y ¼ 0; 0.5 and 1) were reported and discussed. 2. Experimental 2.1. Synthesis of material

* Corresponding author. E-mail address: ramzilefi@gmail.com (R. Lefi). http://dx.doi.org/10.1016/j.jallcom.2016.12.078 0925-8388/© 2016 Elsevier B.V. All rights reserved.

The crystals of (C2H5NH3)2CuyCd1yCl4 compound were synthesized starting from C2H5NH3Cl, CuCl2 and CdCl2 precursors of

R. Lefi et al. / Journal of Alloys and Compounds 696 (2017) 1244e1254

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Table 1 Lattice parameters and volume of A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 compounds.

A2Cd Cl4 A2CuCl4 A2Cu0.5Cd0.5Cl4

a (Å)

b (Å)

c (Å)

b( )

V (Å3)

7.352 7.375 7.35

7.503 7.480 7.47

19.05 21.254 21.18

90 90 90

1076.56 1172.47 1162.48

Fig. 1. (a), (b) and (c): X-ray diffractogram of the compounds A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4, respectively.

Fig. 2. (a), (b) and (c): variation of cos bhkl vs 4sinq of the compounds A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4, respectively.

high purity, which were weighed in the stoichiometric proportion conforming to the equation:

2C2H5NH3Cl þ y CuCl2 þ (1y) CdCl2þH2O / (C2H5NH3)2CuyCd1y Cl4 (y ¼ 0; 0.5 and 1) The mixture was slowly evaporated at room temperature. After

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Table 2 Parameters of the crystallite size and the dislocation density estimated from Scherrer's and W-H. Compounds

Sherrer's

A2CdCl4 A2CuCl4 A2Cu0.5Cd0.5Cl4

Table 3 Different peaks attribution by FTIR of the three samples A2CdCl4, A2CuCl4 and A2Cu0.5Cd0.5Cl4. Wavenumber (cm1)

Williamson-Hall

Attribution

D (nm)

4 (nm2)

D (nm)

4 (nm2)

A2CdCl4

A2CuCl4

A2Cu0.5Cd0.5

113.9 99.61 70.81

7.7 105 1.007 104 1.99 104

87.8 83.33 65.67

1.3 104 1.44 104 2.31 104

405 794 873 970 1045 1191 1340 1405 1492 1587 1631 1920 2956 3122 3197 3409

414.6 794.5 e 972 1051 1197 1336 1406 1498 e 1637 1968 e e 3183 3415

524 792 869 968 1045 1190 1339 1403 1488 1583 e 2065 2950 e e 3415

bold : This material has the best structural and optical properties

a few days, the product compound was obtained. The prepared samples are (C2H5NH3)2CdCl4 (y ¼ 0), (C2H5NH3)2Cd0,5Cu0,5Cl4 (y ¼ 0.5) and (C2H5NH3)2CdCl4 (y ¼ 1) which are named in the following sections as A2CdCl4, A2Cu0.5Cd0.5Cl4 and A2CuCl4 respectively, where (A ¼ C2H5NH3). For the optical measurements we used pellets of about 1.2 mm of thickness. 2.2. Characterization techniques The structural results were investigated at room temperature by X-ray diffraction using a «Bruker D8» apparatus having CuKa (l¼ 1.5418 Å) radiation. The lattice parameters and the peak indexation of the synthesized compounds have been determined by X Pert PROMPD software. The infra-red spectra were recorded in the 4004000 cm1 range using a Bruker spectrometer reference Vertex 80v. Finally, the optical measurements, in the UVevisible range were carried out using a spectrophotometer, JENWAY 6400, within a (320e1000 nm) wavelength range. A2CdCl4

d(CeCeN) r(CH2) ʋ(CeC)

r(NH3)

ʋ(CeN)

r(CH3) g(CH2) ds(CH3) ds(NH3) das(NH3) d(NeH) das(NH3) ʋas(CH3) ʋs(NH3) ʋas(NH3) ʋas(NH3)

ʋ: stretching; d: bending or in-plane deformation; r: Rocking; s: symmetric; as: asymmetric; g: out-plane deformation.

3. Results and discussion 3.1. Structural study 3.1.1. X-ray diffraction The room temperature powder X-ray diffractogram of the 110

(a)

(b)

A2CuCl4

100

100

Transmittance (%)

80

60

80 70 60 50 40

40

30 4000

3500

3000

2500

2000

-1

1500

1000

500

0

4000

3500

Wavelength(cm )

105

3000

2500

2000

1500 -1

Wavelength (cm )

A2Cd0.5Cu0.5Cl4

(c)

98

Transmittance (%)

Transmittance (%)

90

91

4000

3000

2000

1000

0

-1

Wavelength (cm ) Fig. 3. (a), (b) and (c): FTIR spectras of the different samples A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4, respectively.

1000

500

0

R. Lefi et al. / Journal of Alloys and Compounds 696 (2017) 1244e1254

Fig. 4. Variation of absorption coefficient of the samples as a function of wavelength.

A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 compounds are shown in Fig. 1(a)e(c), respectively. The reflection peaks of the X-ray profile were indexed. These compounds were found to have orthorhombic structure at room temperature with a space group Pnma for A2CdCl4 and Pbca for A2CuCl4 and A2Cd0.5Cu0.5Cl4. The lattice parameters a, b and c, and volumes are given in Table 1. The obtained results are in good agreement with literature [18,19]. The diffraction peak broadening evidences grain refinement

1247

along with the large strain associated with the crystal imperfections. The instrumental broadening (bhkl) was corrected, corresponding to every different peak of the material using Eq. (1). The average crystallite size was calculated using Scherrer's formula Eq. (2) [20]. The strain induced in powders due to crystal imperfection and distortion can be calculated using Eq. (3) [21]. From equations (2) and (3), it is confirmed that the XRD peak broadening from crystallite size varies as cos1 q, and from strain varies as tan1 q. Assuming that the particle size and strain contributions to line broadening are independent to each other and both have a Cauchylike profile, the observed line breadth is simply the sum of equations (2) and (3), this leads to Eq. (4). By rearranging Eq. (4) of Williamson-Hall (W-H) model [21], we get Eq. (5). W-H plots for the prepared nanocrystals are show in Fig. 2(a)e(c). From the linear fit to the data, the crystallite size D was estimated from the y-intercept, and the strain ε , from the slope of the fit. The strain values estimated from the fit are about (3.5  104), 1.34  103 and (4.36  104) for the A2CdCl4, A2CuCl4 and A2Cu0.5Cd0.5Cl4, respectively. The negative sign of ε for the A2CdCl4 and A2Cu0.5Cd0.5Cl4 shows that these compounds undergo a compressive strain. The values of the crystallite size and the dislocation density estimated from the two models Scherrer's and W-H are presented in Table 2. Where the dislocation density is calculated using Eq. (6) [22]. It is noted that the hybrid A2CdCl4 compound has the largest crystallite size and the lowest value of the dislocation density

Fig. 5. (a), (b) and (c): plots (ahn)2 versus hn of A2CdCl4 and A2CuCl4 and A2Cd0.5Cu0.5Cl4, respectively.

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Table 4 Values of gap-energy, refractive index and Urbach energy for different compounds.

A2CdCl4 A2Cd0.5Cu0.5Cl4 A2CuCl4

Eg (eV)

n

EU (eV)

3.11 3.04 2.4

2.37 2.38 2.58

0.89 1.05 0.97

bold : This material has the best structural and optical properties

Fig. 6. Variation of gap-energy versus refractive index.

(Table 2). This explains that this material has the best crystal quality. We noted that the substitution of cadmium by copper in the A2CdCl4 compound reduces the size of crystallites D. This can be related to the difference in the ionic radius (r2þ Cd ¼ 155pm and r2þ Cu ¼ 135pm). It has been reported by B. Amaya Moral et al. [23], that the M-Cl bond length is different for M ¼ Cd and M ¼ Cu. They reported that the equatorial and axial M-Cl distances are Req ¼ 2.67 and Rax ¼ 2.52 for A2CdCl4 while for A2CuCl4 are Req ¼ 2.28 and Rax ¼ 2.98 [23]. The value of crystallite size D determined by WilliamsoneHall method is lower than that obtained by Scherrer's one. This difference is attributed to the strain value. Thus, using the Scherrer's method without considering the strain may yield to inaccurate results.

3.1.2. FTIR analysis At room temperature, Fig. 3(a)e(c) show infrared (IR) spectra of the studied compounds A2CdCl4, A2CuCl4 and A2Cu0.5Cd0.5Cl4, respectively. The infrared spectra are recorded for the wavenumber region between 4000 and 400 cm1. A detailed assignment of all bands is difficult, but we can attribute some of them by comparison with similar compounds [24e26]. The assignments are listed in Table 3. The bands observed at about 970, 1492, 1587, 3193 cm1 in IR spectra, for the three samples A2Cd Cl4, A2CuCl4 and A2Cu0.5Cd0.5Cl4, are attributed to rocking, symmetric and asymmetric deformation, and stretching of NH3, respectively. The band observed at about 405 and 1045 cm1, for these

Fig. 7. (a), (b) and (c): (Ln a(hn)) versus energy hn plots of the compounds A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4, respectively.

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Fig. 8. (a), (b) and (c): Plots of the refractive index versus wavelength of the compounds A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4, respectively.

Table 5 Cauchy's parameters of the A2CdCl4, A2Cd0.5Cu0.5Cl4 and A2CuCl4 compounds.

A2CdCl4 A2Cd0.5Cu0.5Cl4 A2CuCl4

A

B(mm)2

C (mm)4

D (mm)6

E(mm)8

1.42 1.1 0.92

0.242 0.166 0.160

0.747 0.219 0.308

0.428 0.358 0.337

0.067 0.023 0.031

samples, are attributed to the deformation and stretching vibrations of (CeN), respectively. The bands observed at about 1191, 1405 and 2956 cm1 are attributed to rocking, symmetric deformation and asymmetric stretching of CH3, respectively. The bands observed at about 970 and 1336 cm1, for different samples, are attributed to rocking and deformation out of plan of CH2, respectively. The peaks appear at about 873 cm1 are attributed to the stretching of (CeC) for A2CdCl4 and A2Cu0.5Cd0.5Cl4. Finally, we note that these results confirm the presence of the organic chains of hybrid compounds A2Cd Cl4, A2CuCl4 and A2Cu0.5Cd0.5Cl4. 3.2. Optical properties 3.2.1. The absorption coefficient The absorption coefficient a was calculated using Eq. (7) [27,28]. The variation of absorption coefficient a of our samples as a function of wavelength is shown in Fig. 4. It is clearly seen from the

figure that the absorption coefficient tends to decrease as the wavelength increases particularly for A2CuCl4 and A2Cd0.5Cu0.5Cl4. This behavior was observed for many semiconductors and can occur for many reasons, such as internal electric fields within the crystal, deformation of lattice due to strain caused by imperfection and inelastic scattering of charge carriers by phonons [29]. The absorption coefficient a of the sample A2Cd0.5Cu0.5Cl4 decreases faster than that of A2CdCl4 and A2CuCl4, this indicates a high dislocation density. This behavior of the absorbance coefficient is in agreement with the analysis of the dislocation density using the WeH and Scherrer models. 3.2.2. Optical band gap In the case of a direct transition, the optical band gap energy is given by the Tauc's relation Eq. (8) [30]. Fig. 5(a)e(c) show the plot of ½a$ðhnÞ2 versus the photon energy hn which leads to the Eg value by a linear fit. The calculated value of optical band gap energy Eg of A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 are given in Table 4. These values are comparable to that found in the literature [31]. The substitution of cadmium by copper modifies interatomic distances in the crystal lattice. This creates a change in the positions of the valence band and the conduction band, hence the change in the gap energy from 3.11 to 3.04 eV. We noted that the introduction of copper in A2CdCl4 compound actually decreased the values of gapenergies. This can be due to the creation of impurities such Cu interstitial or Cd vacancies. This leads to creation of defect levels in the band-gap of the material. The values of the gap energy show

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Fig. 9. (a), (b) and (c): Plots of (n21)1 versus (hy)2 of the A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 hybrid compounds, respectively.

Table 6 Calculated values of dispersion parameters of different compounds.

A2CdCl4 A2Cd0.5Cu0.5Cl4 A2CuCl4

S0(mm)2

l0 (mm)

E0(eV)

Ed (eV)

2.52 1.586 0.267

0.375 0.432 0.533

2.52 1.586 2.19

0.19 0.74 1.28

that our materials can be classified as semiconductor. The refractive index n of the samples is calculated from the gap energy by using the following equation Eq. (9) [32]. The values of the refractive index n are also presented in Table 4. Fig. 6 shows the variation of gap-energy versus refractive index in all compounds. We can clearly note that the increase of copper content leads to an increase in refractive index and significant decrease of the gap energy. 3.2.3. Urbach energies The incorporation of impurity into the semiconductor often reveals the formation of band tailing in the band gap due to the interaction of phonons. The presence of tail absorption profile follows the Urbach law Eq. (10) [33,34]. The width of the located states (band tail energy or Urbach energy EU) was estimated from the slopes of Ln aðhyÞ versus energy plots (Fig. 7(a)e(c)). The estimated values are presented in Table 4. The values prove the presence of disorder in these hybrid compounds. In fact, the Urbach energy is associated with the microstructural lattice disorder. It can be originated from the existence of defects and impurities in both organic and inorganic phases. It is noted that the hybrid A2CdCl4 compound has the small value of EU. This result explains that this material has the best crystal quality. This confirms the results found by XRD.

Fig. 10. Extinction coefficient (k) as a function of wavelength.

3.2.4. Refractive index and coefficient extinction The dispersive behavior of refractive index n(l) was deduced from optical experimental measurements using the following equation Eq. (12) [35]:

R. Lefi et al. / Journal of Alloys and Compounds 696 (2017) 1244e1254

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Fig. 11. (a), (b) and (c): Plots of ε1 versus l2 of the A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 hybrid compounds, respectively.

The variations of refractive index n of these compounds with the wavelength are shown in Fig. 8(a)e(c). The refractive index of the sample decreases with increasing the wavelength. The experimental data were fitted in the range (500e1000) using Cauchy formula Eq. (13) [36]. The best fit of experimental data was added in Fig. 8. It demonstrates that these samples have a normal dispersion for the entire wavelength range. Values of Cauchy's parameters are gathered in Table 5. In addition to the Cauchy extrapolation described above, which was done to provide reasonable values for refractive index in the spectral range of low absorption, we have also carried out some other optical constants based on WempleeDiDomenico, single oscillator model using Eq. (14) [37,38]. The WempleeDiDomenico model is used to fit the experimental data of Fig. 9(a)e(c). E0 and Ed can be determined from the intercept, (EE0d ) and the slope (-E01Ed ). Their values are listed in Table 6. The usual situation in insulators and semiconductors is to find an inverse correlation between the refractive index and the characteristic interband energy E0. In semiconductors, this correlation works very well, and even extends to an analogous correlation with the band gap energy [39]. Under the same model, the refractive index can also be analyzed to determine the average oscillator wavelength l0 and oscillator length strength S0 of the studied samples. These values can be obtained by using the following relations Eq. (15) [40,41]: From Fig. 9, we obtain S0 and l0 values from the linear fit of ðn2 ðlÞ  1Þ1 versus l2 . The obtained values are given in Table 6.

Fig. 10 shows the variation of extinction coefficient k of different compounds as function of wavelength. This latter shows similar dependence on wavelength to the refractive index. 3.2.5. Optical dielectric constants The complex dielectric constant (ε(u) ¼ ε1(u)  iε2(u)) characterizes the optical properties of the solid material. The real and imaginary parts of the dielectric constant for these hybrid compounds are also determined by the following relations Eq. (16), Eq. (17) and Eq. (18) [42]: For these samples, in infrared domain, it is found that ε1 is a linear function of the square of the wavelength (Fig. 11). While ε2 is 3 linear with l (Fig. 12). These results can be used to determine the optical constants: ε∞ , up and which are the dielectric constant at high frequencies, the plasma pulse and relaxation time, respectively. We can also calculate the values of the conductivity given by Eq. (19) [43]. c is the velocity of light in vacuum. The calculated values of these constants are given in Table 7. It is noted that the copper introduction in the perovskite structure improves the conduction properties and increases the free carrier concentration-to-effective mass ratio N/m*. 4. Conclusion In this work, we have studied the structure and optical properties of A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 perovskites, prepared by the conventional method: evaporation at room condition. The

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Fig. 12. (a), (b) and (c): Plots of ε2 versus l3 of the A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 hybrid compounds, respectively.

Table 7 Calculated values of ε∞ , up and other constants.

A2CdCl4 A2Cd0.5Cu0.5Cl4 A2CuCl4

ε∞

up(1014 rad s1)

(1013s)

N/m*(1046 g1cm3)

(S.cm1)

2.33 1.12 1.18

6.07 13.1 9.73

1.53 0.52 1.21

2.36 5.31 3.45

924.36 956.11 1013.6

atomic arrangement of these hybrid compounds can be described by the alternation of organic-inorganic layers. Moreover, the material cohesion of these compounds are assured by hydrogen bonds (NeHeCl) established between anions and cations. These hybrid compounds A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 are crystallized in the orthorhombic system with Pnma, Pcab and Pbca space group, respectively. XRD analysis indicates that the sample A2CdCl4 has a high crystallite size and a low strain. This result explains that this material has the best crystal quality. FTIR technique confirms the presence of the organic chain of different hybrid compounds A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4. The optical band gap and Urbach energies of these samples were estimated from theoretical fits. The A2CdCl4 compound has a low value of Urbach energy; this confirms that this sample has the best crystal quality. The dispersion of these hybrid compounds were studied using the WempleeDiDomenico method. The oscillator energy E0 and the dispersion energy Ed were deduced. Moreover, the real and imaginary parts of dielectric permittivity as well as the optical parameters of different compounds A2CdCl4, A2CuCl4 and A2Cd0.5Cu0.5Cl4 were determined. It is noted that the synthesized

(C2H5NH3)2CuyCd1yCl4 compounds (y ¼ 0; 0.5 and 1) have strong absorption in visible and near infrared regions, and the substitution of copper actually improve the conductivity of the obtained material. These properties make these compounds good candidate to photovoltaic applications.

Acknowledgments The financial support of the Ministry of high education and scientific research of Tunisia is acknowledged.

Appendix

bhkl ¼

h

b2hkl

 Measured

  2  bhkl

i1 2

Instrumental

(1)

R. Lefi et al. / Journal of Alloys and Compounds 696 (2017) 1244e1254



kl bhkl cosq

(2)

Where D is the crystallite size, bhkl is the full width at half maximum of the diffraction peak, q the Bragg's angle, l is the wavelength of Cuka radiation and k is the shape factor ¼ 0.9.

ε¼

bhkl

bhkl ¼

kl þ 4εtanq Dcosq

bhkl cosq ¼ 4¼

(3)

4tanq

kl þ 4εsinq D

(5)

(6)

D2

(8)

Where A is a constant, hn is the photon energy and Eg is the optical band gap energy.

3 n2 ¼ qffiffiffiffiffi  2

(9)

Eg 20

Ln aðhnÞ ¼ Lnða0 Þ þ

hn  Eg EU

(10)

where a0 is a constant and EU is the Urbach energy determined through the following equations Eq. (11):

  d½LnðaÞ dðhnÞ

ð1 þ RÞ þ n¼ ð1  RÞ

(11)

"

ð1 þ RÞ ð1  RÞ

2



2

 1þk

#1  2

(12)

Where R is the reflectance (R(%) ¼ 100-A(%)-T(%)) and k is the extinction coefficient given by the following equation:

al

4p

2

l

þ

ε1 ðlÞ ¼ nðlÞ2  kðlÞ2

(17)

ε2 ðlÞ ¼ 2nðlÞkðlÞ

(18)

Ne2 t m*

(19)

Where e is the electron charge and N/m* is the free carriers' concentration-to-effective mass ratio, which can be calculated from the following equations [43]:

ε∞ u2p 4p2 c2

l2

(20)

(7)

ðahn Þ2 ¼ Aðhn  EgÞ

B

(16)

ε1 zε∞ 

Where d is the thickness of sample and T is the transmittance.

n¼Aþ

(15)

εðlÞ ¼ ½ðnÞ  ik2 ¼ ε1 ðlÞ  iε2 ðlÞ

sdc ¼

1 100 a ¼ ln d Tð%Þ



S0 l20  2  1  ll0

(4)

1

EU ¼

n2 ðlÞ ¼ 1 þ 

1253

C 4

l

þ

D 6

l

þ

E

l8

(13)

Where A, B, C, D and E are the Cauchy's parameters and l is the wavelength.

E E n2 ðlÞ ¼ 1 þ h 0 d i E20  E2

(14)

Where E ¼ hy is the incident photon energy, E0 is the energy of the effective dispersion oscillator, which is expected to correspond to the photon-energy position of the ultraviolet band gravity center. Ed is the dispersion energy.

ε2 ¼ 2nkz

u2p ¼

ε∞ u2p

l3

8p3 c3 t

4pNe2 ε∞ m*

(21)

(22)

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