Band gap determination in thick films from reflectance measurements

Optical Materials 12 (1999) 115±119

Band gap determination in thick ®lms from re¯ectance measurements Vipin Kumar a, Sachin Kr. Sharma a, T.P. Sharma a

a,*

, V. Singh

b

Department of Physics, C.C.S. University, Meerut 250 004, India b S.G. (P.G.) College, Saroorpur Khurd, Meerut, India Received 6 March 1998; accepted 6 July 1998

Abstract Spectroscopic techniques are very useful for characterising semiconducting materials. We demonstrate here a new formulation and method for measuring the energy band gap in thick ®lms from the re¯ectance data. Ó 1999 Elsevier Science B.V All rights reserved. PACS: 78.20.Ci; 78.50.Ge; 78.66.ÿw; 78.66.Hf Keywords: Band gap; Screen printing; Re¯ection spectra; Semiconductors

1. Introduction In recent years there has been considerable interest in thin ®lm semiconductors for use in various devices [1,2]. Studying semiconducting ®lms required the knowledge of the energy band gap of the material. We have established one of the simplest methods of determining the energy band gap [3±6]. In this communication we have given a method and formulation of the technique used for ®nding energy band gap of thick ®lms in which only the re¯ection spectra of the ®lm from the ®lm side is used. 2. Theory The re¯ectivity of an interface is de®ned as the ratio of the re¯ected energy to the incident energy. *

Corresponding author.

Let I0 be the incident intensity of the ®lm surface of thickness t; r1 ; r2 and r3 are the re¯ection coef®cients at front, inner and rear front faces of the ®lm. As shown in Fig. 1 light is made to fall at small angle of incidence on the ®lm (5 in our case). Some part is re¯ected back in the same medium (air), some part is re¯ected inside the ®lm. Let A0 be the amplitude of the incident beam. The amplitudes of re¯ected and refracted beams from the surface are given by  Arefl ˆ A0 r1 exp

 ÿjp ; 2

Arefr ˆ A0 …1 ÿ r1 † exp ‰ÿ…a ‡ jb†tŠ;

…1† …2†

where a is the attenuation constant, b is the phase factor and t is the ®lm thickness. This refracted beam is again re¯ected at the inner surface of the ®lm. The amplitude of the wave re¯ected at the inner face of the interface of medium I and III is given by

0925-3467/99/$ ± see front matter Ó 1999 Elsevier Science B.V All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 9 8 ) 0 0 0 5 2 - 4

116

V. Kumar et al. / Optical Materials 12 (1999) 115±119

so





A ˆ A0 exp

 jp ‡ ‰r1 ‡ …1 ÿ r1 †r2 …1 ÿ r3 † 2

 exp …ÿ2…a ÿ jb†t†Š; hence the intensity of re¯ected beam is I ˆ AA ; I ˆ A20 ‰A ‡ B exp …ÿ2…a ‡ jb†t†Š  ‰A ‡ B exp …ÿ2…a ÿ jb†t†Š; where A ˆ r1 and B ˆ …1 ÿ r1 †r2 …1 ÿ r3 †: So I ˆ A20 ‰A2 ‡ AB exp …ÿ2…a ‡ jb†t† Fig. 1. Representation of re¯ection and transmission phenomenon.

 AIrefl

ˆ A0 …1 ÿ r1 †r2 exp ‰ÿ…a ‡ jb†tŠ exp

 jp ÿ : 2

This beam is again re¯ected and refracted at the rear side of the incident face (interface of medium II & I). The amplitudes of the re¯ected and refracted beams are: AII refl

ˆ A0 …1 ÿ r1 †r2 r3 exp ‰ÿ2…a ‡ jb†tŠ   jp ;  exp ÿ 2

AII refr ˆ A0 …1 ÿ r1 †r2 …1 ÿ r3 † exp‰ÿ2…a ‡ jb†tŠ   jp :  exp ÿ 2 Neglecting further re¯ection, we have the resultant amplitude of re¯ected beam: A ˆ Arefl ‡ AII refr ;   jp ‡ A0 …1 ÿ r1 †r2 …1 ÿ r3 † A ˆ A0 r1 exp ÿ 2   jp ;  exp ‰ÿ2…a ‡ jb†tŠ  exp ÿ 2   jp A ˆ A0 exp ÿ ‰r1 ‡ …1 ÿ r1 †r2 …1 ÿ r3 † 2  exp …ÿ2…a ‡ jb†t†Š;

‡ AB exp …ÿ2…a ÿ jb†t† ‡ B2 exp …ÿ4at†Š i.e. I ˆ I0 ‰A2 ‡ 2AB cos…2bt† exp…ÿ2at† ‡ B2 exp …ÿ4at†Š; I ˆ I1 ‡ I2 exp …ÿ2at† ‡ I3 exp …ÿ4at†Š; 2

where I1 ˆ I0 A ˆ

…3†

I0 r12 :

I2 ˆ 2ABI0 ˆ 2I0 r1 …1 ÿ r1 †r2 …1 ÿ r3 † cos…2bt†; 2

2

I3 ˆ B2 ˆ I0 …1 ÿ r1 † r22 …1 ÿ r3 † : For the ®lms of thickness large compared with the wave length, the term cos 2bt can be treated as constant and equal to unity (for thick ®lms bt ˆ …2p=k†t is always an even multiple of p therefore cos 2bt ˆ 1). So we take I2 as constant term. Due to higher absorption the third term in Eq. (3) can be neglected in comparison of the second term, so I ˆ I1 ‡ I2 exp …ÿ2at†;

…4†

where I1 is the re¯ection from the upper surface of the ®lm and I2 exp …ÿ2at† is re¯ection from the inner surface. So, I1 ˆ Imin : From Eqs. (4) and (5) we can write I ˆ Imin ‡ I2 exp …ÿ2at†; I ÿ Imin ˆ I2 exp …ÿ2at†:

…5†

V. Kumar et al. / Optical Materials 12 (1999) 115±119

Taking the logarithm ln …I ÿ Imin † ˆ ln…I2 † ÿ 2at:

…6†

In order to ®nd out ln …I2 † consider the region where a ˆ 0, there I ˆ Imax or ln …Imax ÿ Imin † ˆ ln …I2 †;

…7†

which were conveniently used in a monochromatic system have all been replaced with mirrors. This is done to eliminate image deviation due to chromatic aberration. The PbS detector converges the light beam with a toroidal mirror located below the photomultiplier. This permits placing the PbS

using Eqs. (6) and (7) ln …I ÿ Imin † ˆ ln…Imax ÿ Imin † ÿ 2at;   …Imax ÿ Imin † 2at ˆ ln …I ÿ Imin †   …Rmax ÿ Rmin † ; ˆ ln …R ÿ Rmin †

…8†

where R is the re¯ectance and is given by R ˆ I=I0 . For a direct band gap material, the absorption coecient 1=2

ahm ˆ A…hm ÿ Eg †

;

…9†

where A is a constant which is di€erent for di€erent transitions, Imin is estimated by the fall in re¯ection spectra of the ®lm. From Eqs. (8) and (9) it is clear that there is a proportionality relation between ln ‰…Rmax ÿ Rmin †=…R ÿ Rmin †Š and a where Rmax and Rmin are the maximum and minimum re¯ectance in re¯ection spectra and R is the re¯ectance for any intermediate energy photons recorded by Hitachi spectrophotometer U-3400. As in case of absorption spectra, we plot a graph between …ahm†2 (as ordinate) and hm (as abscissa), a straight line is obtained. The extrapolation of 2 straight line to …ahm† ˆ 0 axis gives the value of the direct band gap. Similarly here we plot a graph between hm (abscissa) and the square of ln ‰hm…Rmax ÿ Rmin †=…R ÿ Rmin †Š. As the ordinate we can get the band gap of the semiconductor.

Fig. 2. Re¯ection spectra of CdS sintered ®lm.

3. Experimental details We are using here our theory for the measurement of the band gap of some of our sintered ®lms (like CdS, ZnS, CdZnS). Re¯ection spectra of these sintered ®lms were taken at room temperature with the help of Hitachi spectrophotometer model U-3400. In this model, the prism/grating double monochromatic system is used. The lenses

117

Fig. 3. Re¯ection spectra of ZnS sintered ®lm.

118

V. Kumar et al. / Optical Materials 12 (1999) 115±119

symmetrically against the sample and reference beams, whereby the two beams are completely balanced. A mechanical chopper is placed before the ®rst monochromator to chop the light beam emitted from the light source, with which a deviation in zero signal can be minimized. The visible wavelength light source is a long life tungsten (wl) lamp which can be attached easily. The instrument incorporates the Hitachi microcomputer exclusive for it to control the mechanisms data processing and so on. We have prepared our ®lms on glass substrate and the thicknesses of the ®lms are of the order of a micron. Figs. 2 and 3 show the re¯ection spectra for CdS and ZnS sintered ®lms recorded by the above described Hitachi spectrophotometer. As our materials CdS and ZnS are direct band gap

materials, therefore to measure the energy band gap of these ®lms we used Eqs. (8) and (9) and the Tauc relation [7] for direct band gap materials.

Fig. 5. Energy band gap of ZnS sintered ®lm.

Fig. 4. Energy band gap of CdS sintered ®lm.

Fig. 6. Re¯ection spectra of Cd0:6 Zn0:4 S ®lm sintered at 500°C for 10 min. in air atmosphere.

V. Kumar et al. / Optical Materials 12 (1999) 115±119

119

using the Tauc relation. The energy band gap comes out as 2:82 eV. The structure of CdZnS ®lms is con®rmed by X-ray analysis (using CuKa radiation (k ˆ 1:5404†). The X-ray di€raction pattern of sintered Cd0:6 Zn0:4 S ®lm is shown in Fig. 8 which indicates the formation of the desired composition. In this XRD all the peaks are identi®ed except one peak at 34 which may be attributed to the presence of CdCl2 as CdCl2 is used as adhesive in Cd0:6 Zn0:4 S ®lms. 4. Conclusion Fig. 7. Energy band gap of Cd0:6 Zn0:4 S sintered ®lm. 2

According to the above relation we plot …ahm† versus …hm† where a is absorption coecient and hm is the photon energy. The energy band gap determinations for CdS and ZnS sintered ®lms are shown in Figs. 4 and 5 respectively. The energy band gap of these ®lms are 2:37 eV for CdS and 3:51 eV for ZnS. Similarly Fig. 6 shows the re¯ection spectra for Cd0:6 Zn0:4 S ®lms prepared by sintering at 500 C for 10 min. Fig. 7 shows the energy band gap determination of Cd0:6 Zn0:4 S ®lm

We conclude that re¯ection spectra of a given semiconducting ®lm are just sucient to calculate its energy band gap value. We have veri®ed this method for several semiconducting materials such as Si, GaAs, CdSe etc. We found this method easiest, accurate and fastest for energy band gap determination. Acknowledgements The authors wish to acknowledge the ®nancial support from the Department of Science and Technology, Govt. of India, to carry out this work. References

Fig. 8. X-ray di€raction of Cd0:6 Zn0:4 S ®lm sintered at 500°C for 10 min. in air atmosphere.

[1] O.S. Heavens, Optical Properties of Thin Solid Films, Butterworth, London, 1955. [2] F. Tepehan, N. Ozer, Solar Energy Materials and Solar Cells 30 (1993) 353. [3] T.P. Sharma, S.K. Sharma, V. Singh, C.S.I.O. Communication 19 (3±4) (1992) 63. [4] V. Kumar, V. Singh, S.K. Sharma, T.P. Sharma, Optical Materials 11 (1998) 29. [5] V. Kumar, T.P. Sharma, Optical Materials 10 (1998) 253. [6] S.K. Sharma, V. Kumar, S. Sirohi, S.K. Kaushish, T.P. Sharma, C.S.I.O. Communications 4 (3) (1996) 189. [7] J. Tauc (Ed.), Amorphous and Liquid Semiconductor, Plenum Press, New York, 1974, p. 159.