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Thickness dependent optical properties of CdI2 films
Physica B 304 (2001) 166}174
Thickness dependent optical properties of CdI "lms Pankaj Tyagi , A.G. Vedeshwar *, N.C. Mehra
Thin Film Laboratory, Department of Physics & Astrophysics, University of Delhi, Delhi } 110 007, India University of Science & Instrumentation Center, University of Delhi, Delhi } 110 007, India Received 14 September 2000; received in revised form 22 November 2000; accepted 8 January 2001
Abstract Structural, compositional, morphological and optical studies were carried out on thermally evaporated CdI "lms as a function of "lm thickness. The stoichiometric "lms of hexagonal structure show a wavy thickness dependence of lattice parameter c and a thickness independent a. The optical absorption data show a best "t for a direct type of inter-band transition near the absorption edge yielding a direct optical energy gap of 3.6 eV. Part of the optical data was "tted to an indirect type of transition to determine the indirect optical energy gap of about 3 eV. Both energy gaps show thickness dependences, which can be explained qualitatively by a thickness dependence of the grain size through decreasing grain boundary barrier height with grain size. Similarly, the variation of the refractive index with thickness can be explained by the variation of c with thickness. 2001 Elsevier Science B.V. All rights reserved. PACS: 68.55.Jk; 78.20.Ci; 81.15.Ef; 78.66.Nk Keywords: Optical properties; Polycrystalline "lms; Halides
1. Introduction Cadmium iodide is an important compound having a layered structure with a hexagonal unit cell, which is a common structural type in many dihalides and MX type dichalcogenides [1]. The basic structure consists of an in"nite hexagonal sheet of Cd atoms sandwiched between two similar sheets of I atoms, the Cd atoms being octahedrally coordinated. These three sheet sandwiches are then stacked to form the three-dimensional compound. CdI is a well-known material having a number of
* Corresponding author. Tel.: #91-11-725-7793; fax: #9111-725-7061. E-mail address: [email protected] (A.G. Vedeshwar).
polytypes as high as 200 out of which only very few are commonly occurring [2]. The possibility of di!erent stacking sequences of three layer sandwiches along the third direction due to the weak bonding between layers leads to various polytype structures. Eventhough there have been some studies on CdI regarding structural and optical prop erties, the reports are quite diverging. The optical absorption measurements carried out on singlecrystal samples were "tted to an indirect energy gap of 3.2 eV while the re#ectivity spectra reveal a direct transition at 3.8 eV [3]. Some later reports [4,5] indicate an indirect energy gap of similar value. However, the band structure calculations show the presence of a direct band gap and a slightly smaller indirect band gap, the di!erence between the two is only about 0.3}0.6 eV [6}10]. The band structure
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determined by electron energy loss spectroscopy on CdI "lms shows a direct interband transition at about 4.1 eV [11] and the electronic structure determined by optical re#ectivity on CdI crystals shows forbidden direct (3.8 eV at 300 K) and allowed direct (4.3 eV at 30 K) transitions [10]. However, excitonic features have also been observed in CdI "lms at low temperatures [12}14]. The op tical absorption spectra of CdI crystals have also been analyzed in terms of indirect transitions up to 40 K and in terms of the Urbach rule (exponential tail) above 80 K [15]. Studies on CdI "lms re garding structure [11,16] and optical properties [12}15] are quite limited. Therefore, we thought it worthwhile to carry out detailed optical studies near the absorption edge on well characterized (by structural, compositional, morphological studies) CdI "lms in order to understand the material better, as it is a basic isostructural compound of many halides and chalcogenides. We report here the results of such analyses.
2. Experimental details CdI "lms were grown on glass substrates at room temperature by thermal evaporation at a vacuum of about 10\ Torr using a molybdenum boat. The starting material was analar grade powder which was pelletized for evaporation. Film thickness was monitored by a quartz crystal thickness monitor during evaporation and con"rmed subsequently by DEKTAK IIA surface pro"ler measurements. The deposition rate was kept slow (2}5 nm/s) because the higher deposition rates led to non-uniform "lm growth and low sticking. Films up to 100 nm were completely transparent and become translucent for higher thickness. The "lms of thickness below 50 nm were non-uniform, discontinuous and thicker "lms than 600 nm peel o! the substrate. Small pieces of 5 mm;5 mm were cut to carry out various analyses on the same "lm. The structural study of the "lms was carried out by X-ray di!raction analysis (PHILIPS X-Pert model PW-1830 generator di!ractometer). The "lm composition was found to be stoichiometric as determined by EDAX (JEOL-840). The morphology of the "lm was studied by SEM (JEOL-840). The
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optical absorption measurements were carried out in UV/VIS region using a (Shimadzu UV-260) spectrophotometer.
3. Results and discussions All the "lms grown at room temperature were polycrystalline and stoichiometric (from X-ray diffraction and EDAX analysis) irrespective of the "lm thickness without any exception. However, the earlier report [16] on structure of CdI "lms shows that the "lms thinner than 100 nm were amorphous and thicker "lms tend to crystallize. It is also known that CdI "lms grown at substrate temper ature less than 3003 K are amorphous and those either grown or annealed above 3003 K are polycrystalline [17]. We have shown the results of Xray di!raction analysis in Table 1. The high degree of orientation with the basal plane parallel to substrate and c-axis normal to the substrate plane can be seen as indicated by the total absence of (0 0 1), (1 0 1) or (1 1 1) re#ections. Earlier studies [11,16] have also observed this kind of c-axis alignment in CdI "lms. Our X-ray di!raction data agree very well with the earlier report [11] and ASTM No. (12}574). However, we have observed a small amount of (1 1 1) re#ection only for 160 and 500 nm thick "lms as can be seen from Table 1. In Table 1 we have shown only those peaks having relative intensity greater than one or little less than one. There were few smaller peaks from non-zero lre#ections of smaller intensities. Interesting result is the constancy of lattice parameter a and variation of c with "lm thickness. The variation of c with "lm thickness is shown in Fig. 1 which is quite reproducible as con"rmed by the X-ray analysis carried out on number of "lms of same thickness grown at same experimental conditions. No such thickness dependence on structure was studied earlier in the range 50}500 nm. Earlier report [16] on structural studies of "lms in the range 5}100 nm (based on electron di!raction) gives the di!raction pattern only for 5 and 20 nm thick "lms. Most of the descriptions were qualitative, like "lms below 20 nm are discontinuous (island structure) and "lms above 100 nm show translucent spots of increasing size with thickness etc. These qualitative
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Table 1 The X-ray di!raction data for various thicknesses of CdI "lms showing observed lattice spacing d with relative intensity and assigned (h k l) for each thickness. Lattice parameter c is shown at the bottom for each thickness d (nm)
Thickness (nm) P
55
160
250
360
500
1.529
Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) c (nm)
(%)
*
*
*
*
(%)
*
*
0.4 (0 0 1) *
*
(%)
100.0 (1 1 0) *
100.0 (1 1 0) 5.0 (1 1 1) *
100.0 (1 1 0) *
0.8 (1 0 2) 100.0 (1 1 0) *
*
*
100.0 (1 1 0) 7.9 (1 1 1) *
14.9 (2 2 0) *
17.6 (2 2 0) *
21.2 (2 2 0) *
20.6 (2 2 0) *
1.1 (3 3 0) *
2.4 (3 3 0) 0.6 (4 0 5) 4.2 (4 4 0) 1.529
1.8 (3 3 0) *
1.8 (3 3 0, 3 1 6) 0.9 (5 0 4) 5.8 (4 4 0) 1.913
0.7557 0.6846 0.6446 0.3447 0.3428 0.3332 0.2287 0.2121 0.1714
(%) (%) (%) (%) (%) (%) (%)
5.3 (1 1 4) 19.0 (2 2 0) 1.9 (2 0 4) 1.6 (3 3 0, 2 1 6) * 7.0 (4 4 0, 4 2 6) 1.599
Fig. 1. Thickness dependence of lattice parameter c of hexagonal structured CdI "lms.
descriptions are well observed in our studies also. However, we are not able to compare the dependence of c on "lm thickness, as it is carried out for the "rst time. This kind of variation with "lm thickness is hard to understand, in general, for a normal
2.9 (4 4 0) 1.891
5.1 (4 4 0, 1 1 11) 1.961
compound. However, CdI is known to be an im portant layer structured material having more than 200 polytype structures [2]. Polytypes di!er only in the sequence of stacking of I}Cd}I layers along c-axis. In our case, the very slow deposition rate might have led to the formation of many polytypes and as a result of admixture of these polytypes a variation of c can be expected. Therefore, it is quite hard to specify the single polytype of the "lms. This may be the reason why none of the earlier reports [11,16,12}15] on "lms speci"ed the structural polytype of the "lms. Usually, the variation of c with various polytypes takes place with the multiplicity of minimum c of 2H polytype of CdI crystal. In the present case, it is neither the transformation of one polytype to the other nor the preferred growth of a single polytype at a particular "lm thickness. The other reason for c-variation is due to the reduction of Van der Waal's gap due to the localization in octahedral voids of overstoichiometric Cd atoms from di!erent layers [18]. However, in our case no appreciable trace of
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Fig. 2. Variation of grain size with thickness of CdI "lms determined from the full width at half maximum of most intense X-ray di!raction peak using Scherrer formula and scanning electron micrograph (SEM).
overstoichiometry was detected in the "lm of any thickness. Therefore, we feel that there is some other mechanism concerned with the polytypism of CdI responsible for the c-variation with "lm thick
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ness. At present, we have no clue about the exact mechanism responsible for the variation of c with "lm thickness. We have also determined the average grain size from the full width at half maximum (FWHM) of the most intense peak using the Scherrer formula [19] as a function of "lm thickness which is compared with the grain size dependence on thickness determined by SEM in Fig. 2. Some SEM pictures are displayed in Fig. 3 for few "lms of di!erent thickness. We have carried out optical absorption/transmission measurements on a large number of samples of di!erent thickness. Some representative optical absorbance curves as a function of wavelength are shown for "lms of di!erent thickness in Fig. 4. The wavy nature of the absorbance away from the fundamental absorption edge is due to the interference fringes arising from the substrate-"lm and "lm-air interfaces. It can be seen from the "gure that these fringes smoothen out as "lm thickness increases. The fringe pattern is much more pronounced in transmission spectra and will
Fig. 3. SEM photographs of CdI "lms for "lms of di!erent thickness: (a) 55 nm (b) 160 nm (c) 250 nm and (d) 360 nm.
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calculated by (2.303A) " , t
Fig. 4. The optical absorption spectra of CdI "lms of di!erent thickness.
be utilized to determine refractive index (n) as a function of wavelength. The steep rise in the absorbance near the absorption edge hints a direct type of transition. The absorption coe$cient was
(1)
where A is the absorbance and t is the "lm thickness, neglecting the re#ection coe$cient, which is insigni"cant near the absorption edge. Dependence of on h near the absorption edge is shown in Fig. 5. In a crystalline or polycrystalline material the nature of optical transitions (direct or indirect) near the absorption edge can be determined by the relation between and the optical energy gap E . Assuming parabolic bands, the relation between and E for a direct type of transition is given by [20] h"Constant(h!E )L and for an indirect type of transition by [20] A(h!E #E )L B(h!E !E )L # , h" exp( /¹)!1 1!exp(! /¹) " "
(2)
(3)
Fig. 5. The absorption coe$cient as a function of h. The inset A shows (h) versus h plot for the determination of E (direct) while inset B shows (h) versus h plot for the determination of E (indirect).
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where E is the phonon energy assisting the transition, is the Debye temperature and A, " B are constants. For a direct type of transition n" or depending on whether the transition is allowed or forbidden in quantum mechanical sense. Similarly, n"2 or 3 for an indirect type of allowed and forbidden transitions respectively. The usual method of determining energy gap is to plot a graph between (h) L and h and look for that value of n which gives best linear graph in the absorption edge region. Obviously, there will be a single linear region in case of a direct type of transition and two linear portions for an indirect type of transition as can be seen from Eqs. (2) and (3). We have plotted (h) L versus h for CdI "lms of various thickness and the best "t was obtained for n" indicating a direct type of allowed transition as shown in the inset A of Fig. 5. In earlier reports [3,4] the absorption data on single-crystals were "tted to n"2 to obtain an indirect energy gap of 3.2 eV by the plot of () versus h. However, in [4] authors clearly indicate that the data "t equally well with n"3 and the exact nature of the transition is not certain. Using the data of versus h in [3,4] we have found that the best "t is obtained for n", that is a direct type of transition. However, of CdI single-crystal was shown to obey the Urbach rule (exponential dependence on h) for temperatures above 80 K [15]. In our case does not show exponential dependence on h near the absorption edge. The excitonic peaks at 5.62 and 6.12 eV have been observed in re#ectivity spectra (at room temperature) of singlecrystal [13] and in absorption spectra of "lms at 80 K [12]. The two groups of excitonic peaks in re#ectivity spectra of single-crystal, one near the absorption edge (at 3.68, 3.58 and 4.8 eV) and the other at 5.7, 6.2 eV have been observed at 30 K [10]. It is also shown that super thin crystalline "lm (2.8 nm) shows excitonic peak at 3.9 eV while comparatively thicker "lm (38 nm) shows excitonic peaks at 5.7 and 6.2 eV at 77 K [14]. At room temperature, however, excitonic peaks are not seen in the absorption spectra. A peak corresponding to a direct band to band transition has been identi"ed in the re#ectivity spectra at 3.8 eV (at room temperature) [3] and at 4.3 eV (at 30 K) [10]. We can see
171
that at room temperature both these reports agree very well when we apply the linear temperature coe$cient of energy gap (!1.2;10\ eV [3]) [10]. A direct transition from valence band maxima to conduction band minima at 4.1 eV has been determined from electron energy loss spectroscopy [11]. Band structure calculations [6}11] clearly show the existence of both direct and indirect band gaps of similar magnitudes, the di!erence between the two is about 0.3}0.6 eV. However, the band structure [9] and band gaps [21] of CdI were shown to be insensitive to its polytypes. Therefore, we can realize that both indirect and direct band gaps exist in CdI and they are separated by just 0.3}0.6 eV. Since indirect gap is just less than the direct gap, it lies near the onset of direct gap and can hardly be noticed in (h) versus h plot of much dominated direct transition. Therefore, part of the optical absorption data near the knee or tail of the direct absorption edge have to be re-plotted as (h) versus h to determine indirect gap as shown in the inset B of Fig. 5. We have plotted (h) versus h for the sake of comparison with the reported values of E (indirect) in the literature. However, we still believe that the optical absorption data reveal clearly a direct energy gap showing the best "t to n". Thus determined value of E (indirect) agrees well with earlier experimental results as well as band structure calculations. Our value of E (direct) of 3.6 eV determined for "lms of thickness )250 nm agrees well with the predicted value of 3.8 eV from band structure calculations. However, both energy gaps show thickness dependence as shown in Fig. 6. In general, thickness dependence of energy gap can arise due to one or combined e!ect of the following causes: (a) The change in barrier height due to change in grain size in polycrystalline "lms. (b) A large density of dislocations and (c) Quantum size e!ect. However, "rst one looks reasonable cause in the present case with small contributions from dislocation density as well. Since the "lms in the present study were quite thick, the quantum size e!ect can completely be ruled out. The decreasing energy gap with grain size as shown in Fig. 7 is similar to its thickness dependence and indicates the decreasing barrier height with grain size. The variation of grain
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Fig. 6. Thickness dependence of E (direct) and E (indirect).
Fig. 7. The grain size dependence of direct optical energy gap along with the estimated grain boundary barrier height variation factor (X!fd) as a function of grain size.
boundary barrier height with grain size is given by [21] E "E #C(X}fd), (4) where E is the original barrier height, C is a con stant depending on the density of charge carriers and dielectric constant of the material, X is the barrier width (20}30 nm), d is the grain size and f is a fraction of the order of to depending on the charge accumulation and carrier concentration. We can make the estimation of the change in barrier height as a function of grain size in CdI . We take X"20 nm as the average barrier width, f"
considering the low carrier concentration of CdI and the grain size as a function of "lm thickness from Fig. 2. Thus, the calculated variation factor (X!fd) as a function of grain size is compared with the experimentally observed variation of energy gap with grain size in Fig. 7. The striking qualitative agreement indicates the dominance of barrier height contribution to the observed optical energy gap variation with thickness. As mentioned earlier, the interference fringes observed in the transmission spectra can be used to determine optical constants of the material like refractive index (n), absorption index (k) etc by drawing an envelope on maxima and minima of the fringe pattern as shown in Fig. 8. Detailed description of the procedure is given in [22]. The fringe pattern (number of maxima and minima) show thickness dependence. Inset of Fig. 8 shows the refractive index (n) as a function of wavelength determined for 250 nm thick "lm from the transmission spectra shown with envelope in the "gure. Similarly, n has been determined for all the "lms of di!erent thickness. Accuracy of n decreases near the band edge due to large absorption. The refractive index also shows thickness dependence as shown in Fig. 9. We have compared the thickness dependence of refractive index for three di!erent wavelengths well below the absorption edge. The behavior is similar. The physical interpretation of the variation of refractive index can be ascribed to the variation of both density and electronic structure. The density e!ect on the variation of n can be estimated as (n!1) is proportional to 1/< [23], where < is the unit cell volume. For the present case of hexagonal unit cell < "0.866ac. Since lattice constant a is thick ness independent, we see that (n!1) is proportional to 1/c. Therefore, the thickness dependence of n must show an inverse behavior to that of c, which is indeed true as can be seen by comparing Figs. 1 and 9. This indicates the dominance of density e!ect in the thickness dependence of refractive index. However, small discrepancies for higher thickness side can be understood to be compensated by large electronic structure di!erence observed in thickness dependence of energy gap.
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Fig. 8. The interference fringes in transmission spectra for two di!erent thicknesses of CdI "lms showing the construction of envelopes on maxima and minima of the fringe pattern for the determination of refractive index n. The inset shows thus determined n as a function of wavelength.
Fig. 9. Variation of refractive index n with "lm thickness for three di!erent wavelengths well below the absorption edge.
4. Conclusions The lattice parameter c of hexagonal structured CdI "lms shows a wavy dependence on "lm thick
ness in the range 50}500 nm. The exact mechanism responsible for this dependence is not known at present. However, the polytypism of CdI may be believed to be the main cause in some way. Further experimental investigations are being pursued on similar materials to understand this behavior. The variation of c is very much re#ected in variation of refractive index (n) with "lm thickness. Therefore, we believe that the thickness dependence of n is mainly due to the thickness dependence of c. The optical absorption data clearly show a direct type of interband transition which has already been identi"ed both experimentally and theoretically as discussed. An indirect type of transition also exists along a di!erent direction in the Brillouin zone as revealed by the band structure calculations. The indirect gap can also be determined near the absorption edge by a careful analysis of part of absorption data. The thickness dependences of both optical energy gaps correlate quite well qualitatively with thickness dependence of grain size via the decreasing grain boundary barrier height with
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grain size. Absence of correlation between thickness dependence of energy gap and c parameter could be due to the insensitivity of band structure to polytypes. This is quite possible because only I}Cd}I sandwich contributes mainly to the band structure and their stacking along c-axis has very little e!ect.
Acknowledgements We would like to thank Mr. P.C. Padmakshan, Department of Geology, University of Delhi, for carrying out X-ray di!raction measurements. Also, we would like to thank Dr. P. Arun, Vinod Kumar Paliwal and Hina for their helpful discussions.
References [1] F. Hulliger, in: F. Levy, D. Reidel (Eds.), Structural Chemistry of Layer*Type Phases, publishing company, Dordrecht-Holland, 1976, pp. 31}34, 275. [2] G.C. Trigunayat, Sol. Stat. Ion. 48 (1991) 3}70. [3] D.L. Greenaway, R. Nitsche, J. Phys. Chem. Solids 26 (1965) 1445. [4] P.A. Lee, G. Said, R. Davis, T.H. Lim, J. Phys. Chem. Solids 30 (1969) 2719.
[5] Y. Takemura, T. Komatsu, Y. Kaifu, Phys. Stat. Sol. (B) 72 (1975) K87. [6] J.V. McCanny, R.H. Williams, R.B. Murray, P.C. Kemeny, J. Phys. C 10 (1977) 4255. [7] J. Bordas, J. Robertson, A. Jakobsson, J. Phys. C 11 (1978) 2607. [8] J. Robertson, J. Phys. C 12 (1979) 4753. [9] R. Coehoorn, G.A. Sawatzky, C. Haas, R.A. deGroot, Phys. Rev. B 31 (1985) 6739. [10] I. Pollini, J. Thomas, R. Coehoorn, C. Hass, Phys. Rev. B 33 (1986) 5747. [11] R.D. Bringans, W.Y. Liang, J. Phys. C 14 (1981) 1065. [12] S. Brahms, Phys. Lett. 19 (1965) 272. [13] S. Kondo, S. Matasuoka, T. Saito, Phys. Stat. Sol. (B) 186 (1994) K77. [14] S. Kondo, S. Matasuoka, T. Saito, Phys. Stat. Sol. (A) 165 (1998) 271. [15] H. Nakagawa, T. Yamada, H. Matsumoto, T. Hayashi, J. Phys. Soc. Japan 56 (1987) 1185. [16] R.M. Yu, Philos. Mag. 16 (1967) 1167. [17] S. Kondo, T. Suzuki, T. Saito, J. Phys. D 31 (1998) 2733. [18] I.M. Bolesta, I.V. Kityk, V.I. Kovalisko, R.M. Turchak, Ferroelectrics V192 (1997) 107. [19] B.D. Cullity, Elements of X-ray Di!raction, 2nd Edition, Addison-Wessley, California, USA, 1978, p. 102. [20] N.F. Mott, E.A. Davis, Electronic Processes in Non-Crystalline Materials, Clarendon Press, Oxford, 1979, pp. 273}274. [21] J.C. Slater, Phys. Rev. 103 (1956) 1631. [22] R. Swanepoel, J. Phys. E 16 (1983) 1214. [23] M. Wohlecke, V. Marrello, A. Onton, J. Appl. Phys. 48 (1977) 1748.
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Thickness dependent optical properties of CdI "lms Pankaj Tyagi , A.G. Vedeshwar *, N.C. Mehra
Thin Film Laboratory, Department of Physics & Astrophysics, University of Delhi, Delhi } 110 007, India University of Science & Instrumentation Center, University of Delhi, Delhi } 110 007, India Received 14 September 2000; received in revised form 22 November 2000; accepted 8 January 2001
Abstract Structural, compositional, morphological and optical studies were carried out on thermally evaporated CdI "lms as a function of "lm thickness. The stoichiometric "lms of hexagonal structure show a wavy thickness dependence of lattice parameter c and a thickness independent a. The optical absorption data show a best "t for a direct type of inter-band transition near the absorption edge yielding a direct optical energy gap of 3.6 eV. Part of the optical data was "tted to an indirect type of transition to determine the indirect optical energy gap of about 3 eV. Both energy gaps show thickness dependences, which can be explained qualitatively by a thickness dependence of the grain size through decreasing grain boundary barrier height with grain size. Similarly, the variation of the refractive index with thickness can be explained by the variation of c with thickness. 2001 Elsevier Science B.V. All rights reserved. PACS: 68.55.Jk; 78.20.Ci; 81.15.Ef; 78.66.Nk Keywords: Optical properties; Polycrystalline "lms; Halides
1. Introduction Cadmium iodide is an important compound having a layered structure with a hexagonal unit cell, which is a common structural type in many dihalides and MX type dichalcogenides [1]. The basic structure consists of an in"nite hexagonal sheet of Cd atoms sandwiched between two similar sheets of I atoms, the Cd atoms being octahedrally coordinated. These three sheet sandwiches are then stacked to form the three-dimensional compound. CdI is a well-known material having a number of
* Corresponding author. Tel.: #91-11-725-7793; fax: #9111-725-7061. E-mail address: [email protected] (A.G. Vedeshwar).
polytypes as high as 200 out of which only very few are commonly occurring [2]. The possibility of di!erent stacking sequences of three layer sandwiches along the third direction due to the weak bonding between layers leads to various polytype structures. Eventhough there have been some studies on CdI regarding structural and optical prop erties, the reports are quite diverging. The optical absorption measurements carried out on singlecrystal samples were "tted to an indirect energy gap of 3.2 eV while the re#ectivity spectra reveal a direct transition at 3.8 eV [3]. Some later reports [4,5] indicate an indirect energy gap of similar value. However, the band structure calculations show the presence of a direct band gap and a slightly smaller indirect band gap, the di!erence between the two is only about 0.3}0.6 eV [6}10]. The band structure
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determined by electron energy loss spectroscopy on CdI "lms shows a direct interband transition at about 4.1 eV [11] and the electronic structure determined by optical re#ectivity on CdI crystals shows forbidden direct (3.8 eV at 300 K) and allowed direct (4.3 eV at 30 K) transitions [10]. However, excitonic features have also been observed in CdI "lms at low temperatures [12}14]. The op tical absorption spectra of CdI crystals have also been analyzed in terms of indirect transitions up to 40 K and in terms of the Urbach rule (exponential tail) above 80 K [15]. Studies on CdI "lms re garding structure [11,16] and optical properties [12}15] are quite limited. Therefore, we thought it worthwhile to carry out detailed optical studies near the absorption edge on well characterized (by structural, compositional, morphological studies) CdI "lms in order to understand the material better, as it is a basic isostructural compound of many halides and chalcogenides. We report here the results of such analyses.
2. Experimental details CdI "lms were grown on glass substrates at room temperature by thermal evaporation at a vacuum of about 10\ Torr using a molybdenum boat. The starting material was analar grade powder which was pelletized for evaporation. Film thickness was monitored by a quartz crystal thickness monitor during evaporation and con"rmed subsequently by DEKTAK IIA surface pro"ler measurements. The deposition rate was kept slow (2}5 nm/s) because the higher deposition rates led to non-uniform "lm growth and low sticking. Films up to 100 nm were completely transparent and become translucent for higher thickness. The "lms of thickness below 50 nm were non-uniform, discontinuous and thicker "lms than 600 nm peel o! the substrate. Small pieces of 5 mm;5 mm were cut to carry out various analyses on the same "lm. The structural study of the "lms was carried out by X-ray di!raction analysis (PHILIPS X-Pert model PW-1830 generator di!ractometer). The "lm composition was found to be stoichiometric as determined by EDAX (JEOL-840). The morphology of the "lm was studied by SEM (JEOL-840). The
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optical absorption measurements were carried out in UV/VIS region using a (Shimadzu UV-260) spectrophotometer.
3. Results and discussions All the "lms grown at room temperature were polycrystalline and stoichiometric (from X-ray diffraction and EDAX analysis) irrespective of the "lm thickness without any exception. However, the earlier report [16] on structure of CdI "lms shows that the "lms thinner than 100 nm were amorphous and thicker "lms tend to crystallize. It is also known that CdI "lms grown at substrate temper ature less than 3003 K are amorphous and those either grown or annealed above 3003 K are polycrystalline [17]. We have shown the results of Xray di!raction analysis in Table 1. The high degree of orientation with the basal plane parallel to substrate and c-axis normal to the substrate plane can be seen as indicated by the total absence of (0 0 1), (1 0 1) or (1 1 1) re#ections. Earlier studies [11,16] have also observed this kind of c-axis alignment in CdI "lms. Our X-ray di!raction data agree very well with the earlier report [11] and ASTM No. (12}574). However, we have observed a small amount of (1 1 1) re#ection only for 160 and 500 nm thick "lms as can be seen from Table 1. In Table 1 we have shown only those peaks having relative intensity greater than one or little less than one. There were few smaller peaks from non-zero lre#ections of smaller intensities. Interesting result is the constancy of lattice parameter a and variation of c with "lm thickness. The variation of c with "lm thickness is shown in Fig. 1 which is quite reproducible as con"rmed by the X-ray analysis carried out on number of "lms of same thickness grown at same experimental conditions. No such thickness dependence on structure was studied earlier in the range 50}500 nm. Earlier report [16] on structural studies of "lms in the range 5}100 nm (based on electron di!raction) gives the di!raction pattern only for 5 and 20 nm thick "lms. Most of the descriptions were qualitative, like "lms below 20 nm are discontinuous (island structure) and "lms above 100 nm show translucent spots of increasing size with thickness etc. These qualitative
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Table 1 The X-ray di!raction data for various thicknesses of CdI "lms showing observed lattice spacing d with relative intensity and assigned (h k l) for each thickness. Lattice parameter c is shown at the bottom for each thickness d (nm)
Thickness (nm) P
55
160
250
360
500
1.529
Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) Rel. Int (h k l) c (nm)
(%)
*
*
*
*
(%)
*
*
0.4 (0 0 1) *
*
(%)
100.0 (1 1 0) *
100.0 (1 1 0) 5.0 (1 1 1) *
100.0 (1 1 0) *
0.8 (1 0 2) 100.0 (1 1 0) *
*
*
100.0 (1 1 0) 7.9 (1 1 1) *
14.9 (2 2 0) *
17.6 (2 2 0) *
21.2 (2 2 0) *
20.6 (2 2 0) *
1.1 (3 3 0) *
2.4 (3 3 0) 0.6 (4 0 5) 4.2 (4 4 0) 1.529
1.8 (3 3 0) *
1.8 (3 3 0, 3 1 6) 0.9 (5 0 4) 5.8 (4 4 0) 1.913
0.7557 0.6846 0.6446 0.3447 0.3428 0.3332 0.2287 0.2121 0.1714
(%) (%) (%) (%) (%) (%) (%)
5.3 (1 1 4) 19.0 (2 2 0) 1.9 (2 0 4) 1.6 (3 3 0, 2 1 6) * 7.0 (4 4 0, 4 2 6) 1.599
Fig. 1. Thickness dependence of lattice parameter c of hexagonal structured CdI "lms.
descriptions are well observed in our studies also. However, we are not able to compare the dependence of c on "lm thickness, as it is carried out for the "rst time. This kind of variation with "lm thickness is hard to understand, in general, for a normal
2.9 (4 4 0) 1.891
5.1 (4 4 0, 1 1 11) 1.961
compound. However, CdI is known to be an im portant layer structured material having more than 200 polytype structures [2]. Polytypes di!er only in the sequence of stacking of I}Cd}I layers along c-axis. In our case, the very slow deposition rate might have led to the formation of many polytypes and as a result of admixture of these polytypes a variation of c can be expected. Therefore, it is quite hard to specify the single polytype of the "lms. This may be the reason why none of the earlier reports [11,16,12}15] on "lms speci"ed the structural polytype of the "lms. Usually, the variation of c with various polytypes takes place with the multiplicity of minimum c of 2H polytype of CdI crystal. In the present case, it is neither the transformation of one polytype to the other nor the preferred growth of a single polytype at a particular "lm thickness. The other reason for c-variation is due to the reduction of Van der Waal's gap due to the localization in octahedral voids of overstoichiometric Cd atoms from di!erent layers [18]. However, in our case no appreciable trace of
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Fig. 2. Variation of grain size with thickness of CdI "lms determined from the full width at half maximum of most intense X-ray di!raction peak using Scherrer formula and scanning electron micrograph (SEM).
overstoichiometry was detected in the "lm of any thickness. Therefore, we feel that there is some other mechanism concerned with the polytypism of CdI responsible for the c-variation with "lm thick
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ness. At present, we have no clue about the exact mechanism responsible for the variation of c with "lm thickness. We have also determined the average grain size from the full width at half maximum (FWHM) of the most intense peak using the Scherrer formula [19] as a function of "lm thickness which is compared with the grain size dependence on thickness determined by SEM in Fig. 2. Some SEM pictures are displayed in Fig. 3 for few "lms of di!erent thickness. We have carried out optical absorption/transmission measurements on a large number of samples of di!erent thickness. Some representative optical absorbance curves as a function of wavelength are shown for "lms of di!erent thickness in Fig. 4. The wavy nature of the absorbance away from the fundamental absorption edge is due to the interference fringes arising from the substrate-"lm and "lm-air interfaces. It can be seen from the "gure that these fringes smoothen out as "lm thickness increases. The fringe pattern is much more pronounced in transmission spectra and will
Fig. 3. SEM photographs of CdI "lms for "lms of di!erent thickness: (a) 55 nm (b) 160 nm (c) 250 nm and (d) 360 nm.
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calculated by (2.303A) " , t
Fig. 4. The optical absorption spectra of CdI "lms of di!erent thickness.
be utilized to determine refractive index (n) as a function of wavelength. The steep rise in the absorbance near the absorption edge hints a direct type of transition. The absorption coe$cient was
(1)
where A is the absorbance and t is the "lm thickness, neglecting the re#ection coe$cient, which is insigni"cant near the absorption edge. Dependence of on h near the absorption edge is shown in Fig. 5. In a crystalline or polycrystalline material the nature of optical transitions (direct or indirect) near the absorption edge can be determined by the relation between and the optical energy gap E . Assuming parabolic bands, the relation between and E for a direct type of transition is given by [20] h"Constant(h!E )L and for an indirect type of transition by [20] A(h!E #E )L B(h!E !E )L # , h" exp( /¹)!1 1!exp(! /¹) " "
(2)
(3)
Fig. 5. The absorption coe$cient as a function of h. The inset A shows (h) versus h plot for the determination of E (direct) while inset B shows (h) versus h plot for the determination of E (indirect).
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where E is the phonon energy assisting the transition, is the Debye temperature and A, " B are constants. For a direct type of transition n" or depending on whether the transition is allowed or forbidden in quantum mechanical sense. Similarly, n"2 or 3 for an indirect type of allowed and forbidden transitions respectively. The usual method of determining energy gap is to plot a graph between (h) L and h and look for that value of n which gives best linear graph in the absorption edge region. Obviously, there will be a single linear region in case of a direct type of transition and two linear portions for an indirect type of transition as can be seen from Eqs. (2) and (3). We have plotted (h) L versus h for CdI "lms of various thickness and the best "t was obtained for n" indicating a direct type of allowed transition as shown in the inset A of Fig. 5. In earlier reports [3,4] the absorption data on single-crystals were "tted to n"2 to obtain an indirect energy gap of 3.2 eV by the plot of () versus h. However, in [4] authors clearly indicate that the data "t equally well with n"3 and the exact nature of the transition is not certain. Using the data of versus h in [3,4] we have found that the best "t is obtained for n", that is a direct type of transition. However, of CdI single-crystal was shown to obey the Urbach rule (exponential dependence on h) for temperatures above 80 K [15]. In our case does not show exponential dependence on h near the absorption edge. The excitonic peaks at 5.62 and 6.12 eV have been observed in re#ectivity spectra (at room temperature) of singlecrystal [13] and in absorption spectra of "lms at 80 K [12]. The two groups of excitonic peaks in re#ectivity spectra of single-crystal, one near the absorption edge (at 3.68, 3.58 and 4.8 eV) and the other at 5.7, 6.2 eV have been observed at 30 K [10]. It is also shown that super thin crystalline "lm (2.8 nm) shows excitonic peak at 3.9 eV while comparatively thicker "lm (38 nm) shows excitonic peaks at 5.7 and 6.2 eV at 77 K [14]. At room temperature, however, excitonic peaks are not seen in the absorption spectra. A peak corresponding to a direct band to band transition has been identi"ed in the re#ectivity spectra at 3.8 eV (at room temperature) [3] and at 4.3 eV (at 30 K) [10]. We can see
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that at room temperature both these reports agree very well when we apply the linear temperature coe$cient of energy gap (!1.2;10\ eV [3]) [10]. A direct transition from valence band maxima to conduction band minima at 4.1 eV has been determined from electron energy loss spectroscopy [11]. Band structure calculations [6}11] clearly show the existence of both direct and indirect band gaps of similar magnitudes, the di!erence between the two is about 0.3}0.6 eV. However, the band structure [9] and band gaps [21] of CdI were shown to be insensitive to its polytypes. Therefore, we can realize that both indirect and direct band gaps exist in CdI and they are separated by just 0.3}0.6 eV. Since indirect gap is just less than the direct gap, it lies near the onset of direct gap and can hardly be noticed in (h) versus h plot of much dominated direct transition. Therefore, part of the optical absorption data near the knee or tail of the direct absorption edge have to be re-plotted as (h) versus h to determine indirect gap as shown in the inset B of Fig. 5. We have plotted (h) versus h for the sake of comparison with the reported values of E (indirect) in the literature. However, we still believe that the optical absorption data reveal clearly a direct energy gap showing the best "t to n". Thus determined value of E (indirect) agrees well with earlier experimental results as well as band structure calculations. Our value of E (direct) of 3.6 eV determined for "lms of thickness )250 nm agrees well with the predicted value of 3.8 eV from band structure calculations. However, both energy gaps show thickness dependence as shown in Fig. 6. In general, thickness dependence of energy gap can arise due to one or combined e!ect of the following causes: (a) The change in barrier height due to change in grain size in polycrystalline "lms. (b) A large density of dislocations and (c) Quantum size e!ect. However, "rst one looks reasonable cause in the present case with small contributions from dislocation density as well. Since the "lms in the present study were quite thick, the quantum size e!ect can completely be ruled out. The decreasing energy gap with grain size as shown in Fig. 7 is similar to its thickness dependence and indicates the decreasing barrier height with grain size. The variation of grain
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Fig. 6. Thickness dependence of E (direct) and E (indirect).
Fig. 7. The grain size dependence of direct optical energy gap along with the estimated grain boundary barrier height variation factor (X!fd) as a function of grain size.
boundary barrier height with grain size is given by [21] E "E #C(X}fd), (4) where E is the original barrier height, C is a con stant depending on the density of charge carriers and dielectric constant of the material, X is the barrier width (20}30 nm), d is the grain size and f is a fraction of the order of to depending on the charge accumulation and carrier concentration. We can make the estimation of the change in barrier height as a function of grain size in CdI . We take X"20 nm as the average barrier width, f"
considering the low carrier concentration of CdI and the grain size as a function of "lm thickness from Fig. 2. Thus, the calculated variation factor (X!fd) as a function of grain size is compared with the experimentally observed variation of energy gap with grain size in Fig. 7. The striking qualitative agreement indicates the dominance of barrier height contribution to the observed optical energy gap variation with thickness. As mentioned earlier, the interference fringes observed in the transmission spectra can be used to determine optical constants of the material like refractive index (n), absorption index (k) etc by drawing an envelope on maxima and minima of the fringe pattern as shown in Fig. 8. Detailed description of the procedure is given in [22]. The fringe pattern (number of maxima and minima) show thickness dependence. Inset of Fig. 8 shows the refractive index (n) as a function of wavelength determined for 250 nm thick "lm from the transmission spectra shown with envelope in the "gure. Similarly, n has been determined for all the "lms of di!erent thickness. Accuracy of n decreases near the band edge due to large absorption. The refractive index also shows thickness dependence as shown in Fig. 9. We have compared the thickness dependence of refractive index for three di!erent wavelengths well below the absorption edge. The behavior is similar. The physical interpretation of the variation of refractive index can be ascribed to the variation of both density and electronic structure. The density e!ect on the variation of n can be estimated as (n!1) is proportional to 1/< [23], where < is the unit cell volume. For the present case of hexagonal unit cell < "0.866ac. Since lattice constant a is thick ness independent, we see that (n!1) is proportional to 1/c. Therefore, the thickness dependence of n must show an inverse behavior to that of c, which is indeed true as can be seen by comparing Figs. 1 and 9. This indicates the dominance of density e!ect in the thickness dependence of refractive index. However, small discrepancies for higher thickness side can be understood to be compensated by large electronic structure di!erence observed in thickness dependence of energy gap.
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Fig. 8. The interference fringes in transmission spectra for two di!erent thicknesses of CdI "lms showing the construction of envelopes on maxima and minima of the fringe pattern for the determination of refractive index n. The inset shows thus determined n as a function of wavelength.
Fig. 9. Variation of refractive index n with "lm thickness for three di!erent wavelengths well below the absorption edge.
4. Conclusions The lattice parameter c of hexagonal structured CdI "lms shows a wavy dependence on "lm thick
ness in the range 50}500 nm. The exact mechanism responsible for this dependence is not known at present. However, the polytypism of CdI may be believed to be the main cause in some way. Further experimental investigations are being pursued on similar materials to understand this behavior. The variation of c is very much re#ected in variation of refractive index (n) with "lm thickness. Therefore, we believe that the thickness dependence of n is mainly due to the thickness dependence of c. The optical absorption data clearly show a direct type of interband transition which has already been identi"ed both experimentally and theoretically as discussed. An indirect type of transition also exists along a di!erent direction in the Brillouin zone as revealed by the band structure calculations. The indirect gap can also be determined near the absorption edge by a careful analysis of part of absorption data. The thickness dependences of both optical energy gaps correlate quite well qualitatively with thickness dependence of grain size via the decreasing grain boundary barrier height with
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grain size. Absence of correlation between thickness dependence of energy gap and c parameter could be due to the insensitivity of band structure to polytypes. This is quite possible because only I}Cd}I sandwich contributes mainly to the band structure and their stacking along c-axis has very little e!ect.
Acknowledgements We would like to thank Mr. P.C. Padmakshan, Department of Geology, University of Delhi, for carrying out X-ray di!raction measurements. Also, we would like to thank Dr. P. Arun, Vinod Kumar Paliwal and Hina for their helpful discussions.
References [1] F. Hulliger, in: F. Levy, D. Reidel (Eds.), Structural Chemistry of Layer*Type Phases, publishing company, Dordrecht-Holland, 1976, pp. 31}34, 275. [2] G.C. Trigunayat, Sol. Stat. Ion. 48 (1991) 3}70. [3] D.L. Greenaway, R. Nitsche, J. Phys. Chem. Solids 26 (1965) 1445. [4] P.A. Lee, G. Said, R. Davis, T.H. Lim, J. Phys. Chem. Solids 30 (1969) 2719.
[5] Y. Takemura, T. Komatsu, Y. Kaifu, Phys. Stat. Sol. (B) 72 (1975) K87. [6] J.V. McCanny, R.H. Williams, R.B. Murray, P.C. Kemeny, J. Phys. C 10 (1977) 4255. [7] J. Bordas, J. Robertson, A. Jakobsson, J. Phys. C 11 (1978) 2607. [8] J. Robertson, J. Phys. C 12 (1979) 4753. [9] R. Coehoorn, G.A. Sawatzky, C. Haas, R.A. deGroot, Phys. Rev. B 31 (1985) 6739. [10] I. Pollini, J. Thomas, R. Coehoorn, C. Hass, Phys. Rev. B 33 (1986) 5747. [11] R.D. Bringans, W.Y. Liang, J. Phys. C 14 (1981) 1065. [12] S. Brahms, Phys. Lett. 19 (1965) 272. [13] S. Kondo, S. Matasuoka, T. Saito, Phys. Stat. Sol. (B) 186 (1994) K77. [14] S. Kondo, S. Matasuoka, T. Saito, Phys. Stat. Sol. (A) 165 (1998) 271. [15] H. Nakagawa, T. Yamada, H. Matsumoto, T. Hayashi, J. Phys. Soc. Japan 56 (1987) 1185. [16] R.M. Yu, Philos. Mag. 16 (1967) 1167. [17] S. Kondo, T. Suzuki, T. Saito, J. Phys. D 31 (1998) 2733. [18] I.M. Bolesta, I.V. Kityk, V.I. Kovalisko, R.M. Turchak, Ferroelectrics V192 (1997) 107. [19] B.D. Cullity, Elements of X-ray Di!raction, 2nd Edition, Addison-Wessley, California, USA, 1978, p. 102. [20] N.F. Mott, E.A. Davis, Electronic Processes in Non-Crystalline Materials, Clarendon Press, Oxford, 1979, pp. 273}274. [21] J.C. Slater, Phys. Rev. 103 (1956) 1631. [22] R. Swanepoel, J. Phys. E 16 (1983) 1214. [23] M. Wohlecke, V. Marrello, A. Onton, J. Appl. Phys. 48 (1977) 1748.
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