Theoretical estimation of direct transitions of Fe2O3 thin film

ANNO LXVI NOVEMBRE-DICEMBRE 2011 N. 6

A T T I DELLA «FONDAZIONE GIORGIO RONCHI»

EDITORIAL BOARD

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Prof. Roberto BuonannoOsservatorio Astronomico di RomaMonteporzio Catorne, Roma, Italy

Prof. Ercole M. GloriaVia Giunta Pisano 2, Pisa, Italy

Prof. Franco GoriDip. di Fisica, Università Roma IIIRoma, Italy

Prof. Vishal GoyalDepartment of Computer SciencePunjabi University, Patiala, Punjab, India

Prof. Enrique Hita VillaverdeDepartamento de OpticaUniversidad de Granada, Spain

Prof. Irving KaufmanDepartment of Electrical EngineeringArizona State University, TucsonArizona, U.S.A.

Prof. Franco LottiI.F.A.C. del CNR, Via Panciatichi 64Firenze, Italy

Prof. Tommaso MaccacaroDirettore Osservatorio Astronomico di Brera,Via Brera 28, Milano

Prof. Manuel MelgosaDepartamento de OpticaUniversidad de Granada, Spain

Prof. Alberto MeschiariScuola Normale Superiore, Pisa, Italy

Prof. Riccardo PratesiDipartimento di FisicaUniversità di Firenze, Sesto Fiorentino, Italy

Prof. Adolfo PazzagliClinical PsychologyProf. Emerito Università di Firenze

Prof. Edoardo ProverbioIstituto di Astronomia e Fisica SuperioreCagliari, Italy

Prof. Andrea RomoliGalileo Avionica, Campi BisenzioFirenze, Italy

Prof. Ovidio SalvettiI.ST.I. del CNRArea della Ricerca CNR di Pisa, Pisa, Italy.

Prof. Mahipal SinghDeputy Director, CFSL, Sector 36 AChandigarh, India

Prof. Marija StrojnikCentro de Investigaciones en OpticaLeon, Gto Mexico

Prof. Jean-Luc TissotULIS, Veurey Voroize, France

Prof. Paolo VanniDipartimento di BiochimicaUniversità di Firenze

Prof. Sergio VillaniLatvia State University, Riga, Lettonia

Theoretical estimation of direct transitions of Fe2O3 thin fi lm

NADIR F. HABUBI (*)(+), KHUDHEIR A. MISHJIL (*), HAYFA G. RASHID (*)

SUMMARY. – Iron oxide Fe2O3 thin fi lms have been prepared using spray pyrolysis technique. Experimentally, the optical energy gap of Fe2O3 was found to be around 2.0 eV. To achieve and estimate an accurate energy gap value, there was a problem in the exact selection of tangent point. To overcome this problem, Newton-Raphson method and its modifi cation were used. It was found that Newton-Raphson method can predict correlated nearly satisfactory Eg value.

Key words: Thin fi lms, iron oxide, spray pyrolsis technique, optical energy gap, Newton-Raphson method.

1. Introduction

Hematite is the oldest known iron oxide and is widespread in rocks and soil, it is also known as ferric oxide, ion sesquioxide, red ochre, speularite, specular ion ore, kidney ore, or martite (1,2) Hematite is blood-red in color, the crystal structure can be described in terms of close-packed planes of oxygen anions with iron cations in octahedral interstitial sites the most stable iron oxide with n-type (3) semiconducting properties is of scientifi c and technological importance. Be-cause of its usage in photo-electro chemical hypo-gel production (4,5), sensing (6,7), microwave devices as well as high-density recording media (8). Negative electrode in rechargeable batteries (9), magnetic recording material (10), informa-tion storage, controllable drug delivery (11), as a photo degradation of polycyclic aromatic hydro carbon pyrene in solid phase (12).Several techniques have been used to produce Fe2O3 thin fi lms such as, metal organic chemical vapor deposi-tion (13), thermal oxidation method(14), pulsed laser deposition (15), fi ltered arc

(*) Al-Mustansiryah University, College of Education, Physics Dept., Bagdhad, Iraq(+) Corresponding author: Nadir F. Habubi, e-mail: [email protected]

ATTI DELLA “FONDAZIONE GIORGIO RONCHI” ANNO LXVI, 2011 - N. 6

THIN FILMS

N.F. Habubi - K.A. Mishjil - H.G. Rashid884

deposition(16). In this work Fe2O3 thin fi lms were prepared and their electric transitions were investigated using Newton-Raphson method and its modifi cation for optimizing the value of the optical energy gap.

2. Experimental work

Thin fi lms of iron oxide were prepared by chemical pyrolysis technique. The spray pyrolysis was done with a laboratory designed glass atomizer, which has an output nozzle about 1 mm. The fi lms were deposited on preheated glass sub-strates at temperature of 350°C, the starting solution was achieved by an aqueous solution of 0.1 M diluted with de-ionized water formed the fi nal spray solution.A total volume of 50 ml was used in each deposition, spray time was 10 s and the spray interval (2 min) were kept constant. The carrier gas (fi ltered compressed air) was maintained at a pressure of 105 Nm–2. Thickness of the sample was measured using the weighting method and was found to be around 0.5 µm. Optical trans-mittance and absorbance were recorded within wavelength range 300-900 nm using UV-visible spectrophotometer of Shimadzu type.

3. Results and discussion

The optical behavior of the material is generally utilized to determine its optical constants, for example the absorption coeffi cient α, which was estimated from the absorbance spectrum. In the high photon energy region, the energy dependence of the absorption coeffi cient (α > 104 cm–1) suggests the occurrence of direct electron transitions. Namely, the great value the absorption coeffi cient in the range 2.1-3.0 eV is due to optical excitation across the direct interband transitions. On the other side, in the range 1.4-2.0 eV, the absorption coeffi cient indicates indirect interband transitions, which is smaller compared with the ab-sorption coeffi cient, which is due to direct interband transitions, as shown in Fig. 1 which represents α vs. photon energy hν. Therefore, Fe2O3 exhibits direct in-terband transitions, with band gap determined from (16):

[1]

where A is a constant relatied to the effective mass of the electrons and holes and equals either 0.5 for allowed direct transitions, or 2 for allowed indirect transi-tions, hυ is the photon energy, Eg is the optical energy gap and can be obtained by extrapolating the straight line at α = 0, Fig. 1. The value deduced was ~ 1.9 eV, the known value of Fe2O3 band gap is ~ 2 eV (16).

This method was based mathematically on the fact (17) that the slope of the curve y = f(x) at the point P = f(x,y) is defi ned as the slope of the tangent line to the curve at P as shown in Fig. 2.

αhν = A hν −Eg( )n

Theoretical estimation of direct transitions … 885

FIG. 2

The slope of the curve y = f(x) at the point P = f(x,y)

With this method it’s diffi cult to select the exact tangent point (slope of the graph y = f(x) at this point) to achieve an accurate Eg value. This means that the in-stantaneous rate of change of y with respect to x was evaluated. As a result, an ap-proximated Eg value usually associated with error was obtained. To overcome this problem, Newton-Raphson method and its modifi cation (17) were introduced. The analysis of using these methods was summarized as follows:

Step 1

Using a graphing utility to generate the behavior of experimental data pairs (αhυ)2 vs. (hυ), shown in Fig. 3 A). This behavior may be represented in terms of xy-plane abbreviation, i.e. y = f(x). Then taking advantage of any general knowl-edge one has about the function to help in choosing the window.

FIG. 1

Absorption coeffi cient α vs. hv for Fe2O3 using Sigma-plot software.


Step 2

Finding the “best fi tting” or “good fi tting” equation. This may be down with the aid of Matlab (18) or Sigma-plot software package (19). It is found that the cubic equation yields “best” fi tting to experimental data pairs (αhυ)2 and (hυ):

Cubic equation: where

shown in Figs. 3 A) and 3 B).

Step 3

Finding the “roots”, if needed. This may be done using “Matlab” or “Ad-vantage plus TM” software (20).

Step 4

Calculating fi rst and second derivatives of y = f(x) for the cubic equation, this may be done analytically or numerically.

y = y0 + ax + bx 2 + cx 3

y0 = −3.678×1010, a = 9.2578×1010

b = −7.040×1010, c = 1.670×1010

AFIG. 3


Step 5

Finding the ‘intercepts’, using “Advantage plus TM” software, as shown in Fig. 4, the point of intercepts are summarized as follows:

Table 1Points of intercept of f (x) with f ́ (x) and f ̋ (x)

Functions Point of intercept Fig. No.Initial point Final point

f (x) and f ́ (x)(1.6977, 0.2316)

&(1.0248, 0.3088)

(4.3951, 46.2832) 4A)

f(x) and f ̋ (x) (1.3955, 0.2652) (3.8456, 25.2252) 4B)f ́ (x) and f ̋ (x) (1.3428, 0.0333) (3.40228, 20.7033) 4C)

Step 6

Applying Newton-Raphson method and it’s modifi cation. Usually this nu-merical method is used for solving nonlinear algebraic equations and looking for zeros of f(x). The basic idea of these methods is:

• when the real root x1 is known, then one may easily compute the func-tional f(x1).

BFIG. 3

Relation between (αhυ)2 vs.(hυ) of Fe2O3 before and after fi tting using: A) Sigma-plot software; B) Matlab software


Drawing a line tangent to the curve at point x1, then the tangent line inter-sects the x-axis at a point, say x2, which plays a signifi cant role in evaluating Eg values, as shown in Fig. 5.

• Evaluating intersection point of tangent line with x-axis:

[2]

A

B

xK+1 = xK −

f (xK )ʹ f (xK )

for K = 1, 2, 3,...


wherexK+1 = approximate root after K+1 iterations.xK = approximate root after K iterations.f(xK) = functional value at xK.f ́ (xK) = fi rst derivative value of the functional at xK.

• Modifi ed Newton-Raphson method:

[3]

As a result, intersection points gives a range of Eg values due to the cases were arranged in Table (2).

C

FIG. 4

Point of intercepts using “Graphing advantage TM plus” software:A - f(x) and f ́ (x); B - f(x) and f ̋ (x); C - f ́ (x) and f ̋ (x)

FIG. 5

Looking for zeros of f(x) using Newton-Raphson method.

xK+1 = xK −

ʹ f xK( )ʹ ́ f xK( )


Table 2Energy gap results using Newton-Raphson method.

Functions Energy gap (eV)

f(x) and f ́ (x)

4.3953.3952.7282.2771.9531.649

f ́ (x) and f ̋ (x)

3.4022.4021.9171.7001.6291.618

f (x) and f ̋ (x)

3.8463.0282.4822.1041.8091.389

The average Eg value is 2.009 eV. From this result, one may conclude that adapting Newton-Raphson theorem gives results correlate nearly satisfactorily with predicted Eg value (16).

4. Conclusion

From previous results one can conclude:The chemical spray pyrolysis method that has been used to perform the ex-

perimental measurements required for this investigation was found to work fairly successfully.

Eg value predicted using Newton-Raphson method and its modifi cation give good value and fi t the experimental results well.

The accuracy of Eg value depends on experimental conditions and deposi-tion technique Therefore, Eg deduced from optical measurements is less than the known value, this discrepancy may be related to the fi lms surface roughness and scattering.


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Theoretical estimation of direct transitions of Fe2O3 thin film

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