Optical Characterisation of Sputtered Silicon Thin

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puter program, called NAKED, to speed up the film characterisation. The envelope .... on the relationship of electric field vectors on either side of each optical interface, .... Outside the region of significant absorption, the extrema of Eqn. 2.7 ...... the angular rotation method developed by Swanepoel (1985), briefly described in.
Optical Characterisation of Sputtered Silicon Thin Films for Photovoltaic Applications A thesis submitted as partial fulllment of the requirement for the Degree of

Master of Engineering Science by

Bryce Sydney Richards at the

Photovoltaic Special Research Centre School of Electrical Engineering University of New South Wales New South Wales 2052 Australia

February 1998

Certicate of Originality I hereby declare that this submission is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by any other person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of a university or other institute of higher learning, except where due acknowledgement is made in the text. I also declare that the intellectual content of this thesis is the product of my own work, even though I may have received assistance from others on style, presentation and language expression.

Abstract The modied envelope method, developed by the present author, is a very useful tool for determining the optical constants and thickness of inhomogeneous thin lms from transmission spectra. Implemented as a Mathematica computer program, the modied envelope method has been dubbed NAKED since it determines the refractive index nf , absorption coecient f , extinction coecient kf , the optical bandgap Eg , and thickness d { essentially revealing a lm's bare optical essentials. Based on the original method of Swanepoel (1984), the authors' modications permit, rstly, the optical characterisation of non-uniformly thick lms which are absorbing over a wide wavelength range, such as amorphous and polycrystalline silicon. The original method could only characterise lms which exhibited a transparent region (e. g. hydrogenated amorphous silicon or crystalline silicon). Secondly, substrate absorption is incorporated into the method, and a more accurate version of an existing envelope curve tting algorithm, ENVELOPE, is developed (McClain 1997). A thin silicon-on-sapphire sample is used to test the method, as the optical properties of the lm is well documented. The results obtained using NAKED show that nf values are accurate to within 1%, while f values are within 5% of published results. After successfully demonstrating the accuracy of the modied envelope method, sputtered amorphous and polycrystalline silicon (both as-deposited and furnace annealed) non-uniformly thick lms on Corning 1737 glass are optically characterised. Results obtained for the Eg of amorphous lms, determined from Tauc plots, were found to be well within the range of published results, while nf and f values were observed to decrease with increased substrate heating. Refractive indices for the annealed polycrsytalline lms were found to be within 0.1 (absolute) of crystalline values over the whole wavelength range. The f values are two to three times greater than crystalline silicon for energies greater than 1.4eV. The increased absorption is attributed to the amorphous material fraction (i. e. grain boundaries), and agreement with this model is obtained using Raman spectroscopy. Increased sub-bandgap absorption is attributed to dangling bond defects in the material (Jackson et al. 1983). The Eg of the annealed material is found to be in the range 0:93 ; 1:13eV.

Acknowledgements Firstly, I would like to thank my supervisor Alistair Sproul, for keeping me on track when frustrations ran high, and for the many lunchtime discussions. A big thank you also goes out my partner in crime, Andreas Lambertz, whose energy and devotion kept the whole thin lm group alive, and who sputtered more lms than you could poke a stick at. I gratefully acknowledge the nancial support, in the form of half-time contracting work, provided by Martin Green and Pacic Solar. Thanks to Mark Keevers for doing an extremely thorough job, proof reading 80% of this thesis - it probably would've been 100% if the words 'time management' meant anything to me! Thanks also to Mark for advice and discussions on all things optical, as well as more down-to-earth topics, and for justifying the name NAKED. Thanks to Mark Gross and Tom Puzzer for their on-line help, and many a cynical chat. I am grateful to Stuart Wenham for delving deep into his memory banks and retrieving the silicon-on-sapphire samples for me - they were better than Aralditing wafers onto glass any day! Anatoli Chtanov provided great insight into the problems faced with optical characterisation and Mathematica - I hope my program does you justice, Anatoli. Thank you to Svetlana Chtanov for the spectroscopic ellipsometry results, Rad Flossman in Applied Geology for polishing that damn hard sapphire, and to Marjorie McClain for her willingness to accept and implement my changes to ENVELOPE. A big cheer goes out to Varian Australia for turning me into a cynic of SCary proportions! I would like to thank Steve Robbo for the limited time we spent biking and talking together - you will always be special to me. Thanks to Paulo Santos and the MPI-FKF crew who got me interested in PV a few years back. Thanks also to John Russel, whom I've never met, but who compiled a CD of 20 NZ classic tunes, "Bliss", which I bought second hand for $2 and proved invaluable company, passing many a long day in front of the computer together. I appreciate all the eorts of my family and friends to remain in touch with me while I serve my sentence on this small o-shore island. Finally, a big thank you to Andrea for her understanding, support, and motivating inuence over the last few years.

Publications from this Work (to date) 1. Richards B, Lambertz A and Sproul A (1997), 'Optical characterisation of sputtered silicon thin lms on glass for solar cell applications', Australia and New Zealand Solar Energy Society(ANZSES): Solar '97 Conference Proceedings, Canberra, paper no. 113.

Contents 1 Introduction

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2 Literature Review

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2.1 Use of Determining Optical Constants . . . . . . . . . 2.2 Review of Methods for Determining Optical Constants 2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Envelope Method . . . . . . . . . . . . . .

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3 Materials and Methods

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3.1 Substrates . . . . . . . . . . . . . . . . . . 3.1.1 Glass . . . . . . . . . . . . . . . . . 3.1.2 Sapphire . . . . . . . . . . . . . . . 3.2 DC Magnetron Sputtering . . . . . . . . . 3.2.1 System Description . . . . . . . . . 3.2.2 Targets and Sputtered Films . . . . 3.3 Crystallisation . . . . . . . . . . . . . . . . 3.4 Spectrophotometry . . . . . . . . . . . . . 3.4.1 System Description . . . . . . . . . 3.4.2 SBW and Energy Level . . . . . . . 3.4.3 The Step! . . . . . . . . . . . . . . 3.4.4 Scattering Samples and Aperturing 3.4.5 Rotating Sample Holder . . . . . . 3.5 Other Characterisation Techniques Used .

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4 Optical Characterisation using the Modied Envelope Method 4.1 Overview of Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 1

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4.2 ENVELOPE Fitting . . . . . . . . . . . . . . . . . 4.3 NAKED - The Modied Envelope Method . . . . . 4.3.1 Envelope Method Extensions . . . . . . . . 4.3.2 NAKED Operation . . . . . . . . . . . . . . 4.4 Modied Envelope Method Verication . . . . . . . 4.4.1 Theoretical a-Si:H Film on Glass . . . . . . 4.4.2 Theoretical c-Si Film on Corning 1737 Glass 4.4.3 Silicon-on-Sapphire . . . . . . . . . . . . . .

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5 Optical Characterisation of Sputtered Silicon Thin Films 5.1 5.2 5.3 5.4

Overview . . . . . . . . . . . Sputtered a-Si Films . . . . As-Deposited poly-Si Films Annealed a-Si Films . . . .

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6 Conclusions

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A Thin Film Optical Theory

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A.1 A.2 A.3 A.4 A.5 A.6

Introduction . . . . . . . . . . . . . . . . . . . . . . Reectance and Transmittance at a Single Interface Single Transparent Layer . . . . . . . . . . . . . . . Single Absorbing Film . . . . . . . . . . . . . . . . Single Film on a Semi-Innitely Thick Substrate . . Thin Film on a Thick Substrate . . . . . . . . . . .

B NAKED: Program Listing2

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The Mathematica (version 2.2.3) le NAKED.ma is included on a disk in the inside rearcover.

Chapter 1 Introduction Polycrystalline silicon thin lms on foreign substrates are an active area of research in the photovoltaics industry in an eort to reduce the cost of solar cells to levels where photovoltaics can compete with existing utility-scale generation. Signicant emphasis by the semiconductor industry has been placed on developing thin silicon lms for thin lm transistors and at-panel displays, however these lms are too thin for photovoltaic applications and the desirable properties of the lm are quite dierent. Various techniques for depositing thin amorphous and polycrystalline silicon lms have been investigated, with chemical vapour based depositions currently proving popular. However, silicon lms produced by direct current sputtering oer safety advantages, avoiding the use of toxic gases, and can be better suited to industrialscale, large-area depositions. Sputtering also oers improved adhesion to the substrate as during the deopsition the atoms have much greater kinetic energy than that of other processes (Materials Research Corporation 1975). A wide range of substrates are also being investigated, including glass, plastic, ceramics and stainless steel. Transparent materials have the advantage that they can serve both as a superstrate or a substrate. As a superstrate, the material provides mechanical support and a protective front surface to the module, while still transmitting light to the semiconducting lm deposited on the rear. For these reasons, this work has focused on sputtered silicon lms on glass. The deposition of both amorphous and polycrystalline silicon is investigated, and also the solid phase crystallisation of the amorphous material in a conventional furnace. The major aim of this work is to determine refractive index and absorption coecient as a function of wavelength, the optical bandgap, and average thickness and thickness variation of the lms. The absorption coecient, especially, is a useful tool for determining the the lm quality and the eect of additional absorption mechanisms, such as free carrier and defect level absorption. Additionally, the transition from amorphous to polycrystalline material can be observed via the large changes in the optical properties of the material. An improved version of the envelope method, originally presented by Swanepoel (1984), was developed and implemented by the author as a Mathematica computer program, called NAKED, to speed up the lm characterisation. The envelope 3

4 method was chosen due to its ability to analyse non-uniformly thick lms, its elegance, and the unique solution of its output. The extensions to the method made by the author, include:

 the inclusion of substrate absorption, which is signicant at certain wave-

lengths. Swanepoel did not include this in his method, but even for glass and especially sapphire the absorption can be large  improving the method so that material which is absorbing across the whole wavelength range can be characterised. Previously, a lm was required to have a transparent region, but both amorphous and polycrystalline silicon exhibit some level of absorption over the whole measured wavelength range and,  the use of a highly accurate envelope-curve tting program called ENVELOPE, written by McClain et al. (1991) with subsequent modications made in collaboration with the author of this work (McClain 1997).

A commercially grown silicon-on-sapphire lm was used as a reference to establish the accuracy of the optical properties determined by the method. This lm was chosen because the optical properties of epitaxially grown silicon-on-sapphire are very similar to bulk crystalline silicon, and are well documented. Having demonstrated the accuracy of the envelope method, amorphous and polycrystalline lms on glass were analysed. Results for the refractive index, absorption coecient, and optical bandgap are compared to other published polycrystalline data, and conclusions about the optical properties of the material reached. Raman spectra also oer information relating to the level of defects and crystallinity of the lm. The non-destructive nature of this method is essential to enable further characterisation and processing of the samples into solar cells. The method could be applied to any absorbing lm on a transparent substrate that exhibits interference in the ultra-violet, visible, and near-infrared wavelength ranges. Another example of this in photovoltaics would be determining the optical properties of anti-reection coatings, although the thickness of the lms would have to be much thicker than is currently standard, in the order of a few microns, to observe the necessary interference.

Chapter 2 Literature Review 2.1 Use of Determining Optical Constants It is important to know the optical properties of a semiconducting material intended for photovoltaic (PV) applications in order to understand and predict the photoelectric behaviour of working devices. A material is considered to have a complex refractive index n~f = nf ; ikf , where nf is called the refractive index and kf the extinction coecient. All of these functions are wavelength  dependent. The absorption coecient f is related to the extinction coecient by: f f = 4k  .

(2.1)

For a light beam incident on a non-scattering, absorbing lm, the amounts of light reected, absorbed, or transmitted by the lm are related by 1=R+A+T .

(2.2)

where R is reectance, A absorbance, and T transmittance. The denitions of these terms are the ratios of reected, absorbed, and transmitted ux to the incident ux, respectively (ASTM Standard 1992). The refractive index largely determines how reective a lm will be, governed by the Fresnel equations (see Appendix A) . This is important in solar cell design as light reected from the front surface is lost and does not contribute to cell output. Knowledge of the refractive index of the lm also assists in the design of suitable anti-reection (AR) coatings to minimise such reection losses. The absorption coecient is a measure of the extent to which photons of energy hc = 1:24 (with  in m) (2.3) Eph = q  are absorbed in the silicon (Si) lm, which is of primary importance to solar cell operation. The photon energy Eph is given in electron-volts (eV), h is Planck's constant 6.62610;34 J. s, c is the velocity of light in a vacuum 3108 m s;1 , and q is 5

2.2 Review of Methods for Determining Optical Constants

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the charge on an electron 1.60210;19 J. eV. A photon absorbed in Si can generate an electron-hole (e-h) pair, the resulting minority carrier can then be collected by the pn junction, and a useful electric current extracted from the device. In practice, however, defects within the polycrystalline silicon (poly-Si), poorly passivated surfaces, and grain boundaries (GB) between adjacent crystallites, act as recombination sites for minority carriers, which prevents them from being collected by the pn junction. The presence of such defects also impacts upon the absorption coecient of the material, and therefore the determination of f is also a useful tool to determine the degree of crystallinity in a given sample. The f can be compared to that of bulk c-Si and the increased absorption at various photon energies attributed to additional absorption mechanisms.

2.2 Review of Methods for Determining Optical Constants 2.2.1 Overview A wide range of techniques can be used and there have been many works published on the determination of the optical constants of thin lms. For material with a refractive index similar to that of silicon (nf  3.4) between 0.5 and 5 m thick,the use of a spectrophotometer along with some spectrum analysis can provide a simple, quick, and non-destructive method for determining a lm's optical constants. All of the experimental data for the methods described below can be collected using a spectrophotometer with a reectance accessory, to obtain transmission and/or reectance data, except for ellipsometry, which naturally requires the use of an ellipsometer. In this section, popular methods that researchers have developed over the years are described, and the advantages, disadvantages, and limitations of each assessed, with the present application, the analysis of non-uniformly thick lms, in mind. These techniques include:

     

reection and transmission at a single location iterative tting methods Kramers-Kronig analysis matrix method ellipsometry and, interference methods, including:

{ angular rotation method { curve-tting methods and, { envelope methods.

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While there is often considerable overlap between these methods, the categorisation is useful for the following overview. Although mainly concerned with nding the refractive indices of single and multilayer lms for lters, Chapter 3 of Palik (1991) provides a good overview of methods that can be used for determining the optical constants of thin lms. The method used in this work for determining the thin lm optical constants is based on the envelope method developed by Swanepoel (1984), and considerably modied and improved to produce, what the present author refers to as, the modied envelope method.

Transmission and Re ection at a Single Location This method is one of the earliest used to analyse a transparent or weakly absorbing thin lm of thickness d upon a thick transparent substrate. A lm which is weakly absorbing has a refractive index much greater than its extinction coecient over the spectral range in question. In this case, several absorbing terms drop out of the analytical equations, and the expressions describing the reection and transmission become greatly simplied (see Appendix A). Determining the optical constants requires measuring the transmission, reection, and thickness at exactly the same point. This requires special modications to the spectrophotometer, and, unless the sample is optically at and parallel, it is vital that each measurement illuminates exactly the same spot in order to avoid errors being introduced from sample inhomogeneities (Lyashenko and Miloslavskii 1964). Ward (1994) highlights several photometric techniques that were used in the 1950s and 1960s. Although considerable simplications can be made if measurements are made at normal incidence (Appendix A), there are often several pairs of nf , kf values which satisfy the equations at any given . Therefore, it is generally not possible to solve equations for nf and kf uniquely (Ward 1994). Various other elaborate methods have been suggested for calculating the optical constants. In 1964, Mal!e (Ward 1994) proposed a graphical method, which used intersections of plotted curves to nd nf and kf , but it was a tedious process and not necessarily very accurate. Earlier, in 1931, Murmann (Ward 1994) had used a similar method, but found that additional knowledge was still required to eliminate unwanted multiple solutions. Section 5.4 of Heavens (1955) presents an excellent outline of other early (pre-1955) techniques implemented to obtain the optical constants and thickness of thin lms.

Iterative Fitting Methods There are an extremely large number of publications (see for example Minkov 1990, Mrafko and Ozvold 1995, Romanov et al. 1996, Swanepoel 1989) which present analytical expressions for the transmission or reection of thin lms on various substrates. The values for the optical constants and thickness are then determined by simply iterating each parameter until agreement with the experimental transmission and/or reection data is achieved. The inherent problem is that a number of

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solutions can accurately t the spectra, and it is often dicult to determine the correct answer (Ward 1994). This is in contrast with the envelope method, described below, which retrieves more information from the spectra by the construction of envelope curves. This provides two additional equations describing the extrema that can instead be solved for a unique solution of nf , kf , average thickness d", and thickness variation #d. To avoid the problems associated with multiple solutions another method was sought for use in this thesis, hence the iterative methods were not pursued at length. Note that a Windows-based computer program, Optikan (Bader et al. 1995), which simply iterates nf , kf , and d" to t experimental transmission and reection data was occasionally used. This proved to be useful for determining the optical behaviour of the substrates, which exhibit no interference fringes in the  range of interest, as well as theoretical sample structures. The capabilities of the software are somewhat limited as it can only handle a maximum of 256 data points, and it is unable to treat non-uniformly thick layers.

Kramers-Kronig Analysis A good summary and application of the technique developed by Kramers, in 1929, and improved two years later by Kronig, is provided in Chapter 3 of Ward (1994). The Kramers-Kronig method uses the spectral distribution of measured or calculated optical data (reection, transmission, or phase change upon reection or transmission) over a very wide range of frequencies to extract the unknown optical information. Regarded as one of the most powerful tools for calculating optical constants, the analysis is based on the assumption that for the complex refractive index, if the variation of the real part with wavelength is known then the imaginary component can be determined, and vice versa. Due to this requirement of one predetermined component, either nf or kf , over a very large  range (tens of microns) the method is not suitable for this current application, where it is desired to nd d, nf and kf simultaneously in the ultraviolet (UV) to near-infrared (NIR) region, hence, it will not be pursued further in this work.

Matrix Method Another method used by researchers is the matrix method. The analysis is founded on the relationship of electric eld vectors on either side of each optical interface, which can be expressed in terms of matrices. By evaluating these matrices, equations describing the transmission and reection of a layer can be derived. The elegance of this method only becomes apparent when multilayer lms are being investigated, as the derived expressions can be simply multiplied by the matrix describing the next layer to solve the experimental reection and transmission data for nf , kf , and d". The technique is popular for investigating AR and highly reective coatings, and various optical lters. One major disadvantage of the matrix method is its requirement for lms to be uniform over large areas (Arndt et al. 1984), and for this reason its application to our problem is not covered in more detail here.

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Ellipsometry Ellipsometry is a commonly used and extremely accurate method for determining thin lm optical constants. The technique involves measuring the change in polarisation of linearly polarised, monochromatic light due to reection from the lm surface at at least three angles of incidence to yield a unique solution. This allows the ratio of the complex Fresnel coecients,  = r~k=r~? = tan exp;i, to be determined, where  expresses the polarisation state of the reected wave, tan is the dierential attenuation, and # represents a dierential phase retardation. Some advantages of ellipsometry are (Arndt et al. 1984): i. only measurements of polarisation angles are involved (no absolute or relative photometric measurements), so it can be highly accurate ii. lm thickness and the optical constants are determined simultaneously and, iii. the substrate refractive index and absorption coecient (if signicant) can be determined while disadvantages include (Arndt et al. 1984): i. the method is very sensitive to lm non-uniformities and, ii. most ellipsometers use lasers as an illumination source, which limits the amount of optical constant data available to a few single wavelengths. Spectroscopic ellipsometers, which have a monochromator built in, exist, however the one we had access to was limited to the 300-800nm range, missing valuable information about the Si material which could be obtained in the infrared (IR) region. Since the disadvantages were critical to our application, and a spectrophotometer with a 175 ; 3300nm wavelength range was available for use, ellipsometry was not investigated further. A very complete description of the application of ellipsometry for measuring the complex refractive indices of materials can be found in Chapter 3 of Palik (1985).

Interference Methods Interference fringes occur when the amplitudes of beams reected from the front and back surfaces of a lm add to produce various degrees of constructive or destructive interference. These are observed as maxima or minima in the reection or transmission spectra. As the number of interference fringes observed is determined by the optical thickness nf d, lms with a lower refractive index will need to be thicker to observe a useful number of interference fringes (within the operating range of the spectrophotometer). In 1964, Lyashenko and Miloslavskii (1964) proposed a simple method which could be applied to weakly absorbing lms where nf  kf . The necessary conditions for the method were:

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i. uniformly thick lms ii. no scattering and, iii. the spectral bandwidth (SBW) of the spectrophotometer (determined by the grating dispersion and slit widths) should be less than the half-width of the interference maximum. The method was based on the observed thin lm interference in the transmission spectrum. First of all, the chromatic order m was determined from two adjacent extrema, m and m+1 , and then nf found using Eqn. A.35, which describes the interference minima. This method resulted in errors for nf and d of around 2%. For weakly absorbing lms, such as tin oxide, values for kf were iterated until a sucient approximation was obtained. At the same time, Valeev (1963) proposed an iterative method that was characterised by the use of nomograms. Importantly, Valeev included the use of an envelope along the transmission minima, which provided additional information about the optical system. The method was applied to cadmium telluride lms on calcium ouride substrates, however the accuracy was not stated. Michailovits et al. (1983) developed the work of Valeev further, including envelope curves for both the interference minima and maxima. An iterative method was used to extract the values for nf and kf for the vanadium oxide lms. The accuracy of the method was estimated to be within 5% for nf , 1% for kf > 0:01, and as high as 10% for kf values around 0.001. Thickness values were accurate to within 5%.

Angular Rotation Method To avoid many of the problems inherent in thin lm optics, Swanepoel (1985) proposed a method based on the shift in interference extrema as the angle of incidence is varied. The technique had the advantage in that it: i. ii. iii. iv.

could analyse inhomogeneous thin lms was less sensitive to SBW variations beam polarisation had no eect and, it determined d and nf to better than 1% accuracy.

The method was successfully applied by another group of researchers (Corrales et al. 1995) in determining the refractive index of arsenic selenide. Unfortunately, no information about f is obtainable with this method, although it could play a useful role in checking nf values determined with another method.

Direct Curve Fitting Attempts by researchers to t parameter sets directly to the equation describing the interference fringes observed in the transmission spectra (Kubinyi et al. 1996) are

2.2 Review of Methods for Determining Optical Constants

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related to the iterative tting methods. As presented in Appendix A, the equation most commonly used to describe the transmission of a uniformly thick, weakly absorbing (nf  kf ), thin lm on a thick transparent substrate of refractive index ns is (Swanepoel 1983) Ax T = B ; Cx (2.4) + Dx2 , where

A B C D x

= = = = = =

16nsn2f ;  2 (nf + 1);2 (nf + 1); nf + n s , 2 cos n2f ; 1 n2f ; n2s , ;  (nf ; 1)2 (nf ; 1) nf ; n2s , 4nf d ,  exp;f d .

Kubinyi et al. (1996) utilise Swanepoel's (1983) envelope method, described fully below, to calculate the optical constants. It is claimed that while the envelope method of Swanepoel is satisfactory for lms of 1m thickness, lms of greater or lesser thickness produce less accurate results (Kubinyi et al. 1996). Kubinyi et al. claim that results obtained from thinner lms will be less accurate with this method since the number of observable interference fringes decrease, thereby reducing the amount of data points extracted. However, this also depends on the spectrophotometer employed, and Kubinyi et al. used a ultraviolet-visible (UV-Vis) spectrophotometer, which had an upper wavelength limited of 900nm. A UV-VisNIR spectrophotometer, such as the Varian Cary 5G as used at the Photovoltaics Special Research Centre (PVSRC) at the University of New South Wales (UNSW), can reach wavelengths up to 3.3m. This yields many more interference fringes, and silicon lms of thicknesses as thin as 400nm have been successfully analysed using the envelope method of Swanepoel. Why Kubinyi et al. reason that lms thicker than 1m were dicult to analyse remains unclear. It was claimed that the interference fringes were too close together, and that the fringe order of interference m became dicult to determine. The author has not encountered this problem to date, and both amorphous silicon (a-Si) and poly-Si lms of up to 3m thickness have been successfully analysed. One possible explanation for the claim of Kubinyi et al. is perhaps the spectrophotometer used had poor wavelength resolution, rendering peak determination dicult. As a result of the perceived limitations of the envelope method by Kubinyi et al., an alternate method is proposed, using a ve parameter set (a1 a2 a3 a4 d) to describe the nf and f (see Equations 2.5 and 2.6). These coecients are then tted to Eqn. 2.4, and the sum-of-squares errors for each parameter at every  minimised. This method is rst applied to the following theoretical model for an hydrogenated amorphous silicon (a-Si:H) lm as used by (Swanepoel 1983): nf () = a12 + a2 (2.5) 5 = 3:5 2 10 + 2:6 ( in nm) ,

2.2 Review of Methods for Determining Optical Constants log10 f () = a32 + a4 6 = 1:5 2 10 ; 8

12 (2.6)

( in nm;1,  in nm) .

To compare Swanepoel's (1983) envelope method with the curve tting method, Kubinyi et al. (1996) simulated and analysed 100 spectra, and the errors for each parameter averaged. Kubinyi et al. claimed that the average mean square errors are typically ten times less when curve tting over the whole spectrum, than when using the envelope method. However, later in the paper a plot comparing the spectrum generated from the now known parameters and the original measured transmission spectrum suggests that there are substantial errors somewhere in the method. This is presented here as Figure 2.1 and the poor agreement obtained using the curve tting method of Kubinyi et al. (1996) is obvious.

Figure 2.1: Plot illustrating the poor t obtained with the direct curve tting method of Kubinyi et al. (from Kubinyi et al. 1996)

It is claimed that the t in the 700-900nm wavelength range is very good, and the reason for the large deviation below 700nm is due to the fact that nf () and f () cannot be modelled with the simple empirical formulae, Eqns. 2.5 and 2.6, as the absorption becomes signicant. The method is then limited by the fact that a fairly good estimate of how the optical constants vary with  must already be known before the method will work successfully! Thus, the method sounded promising, but results in the end can only be regarded as unsatisfactory for the purposes of this thesis. Additionally, the analysis is limited to homogeneous thin lms, which excludes the possibility of characterising non-uniformly thick lms. The latter is a necessity for the majority of thin lm depositions.

2.2 Review of Methods for Determining Optical Constants $LU G

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Q 1  LN 1

*ODVV Q Q8 α8 

Figure 2.2: The thin-lm on innitely thick substrate model used by Manifacier et al. (1976) to develop the envelope method.

2.2.2 The Envelope Method Another group of researchers to take note of Valeev's (1963) envelope curve approach and develop the concept further were Manifacier et al. (1976). The work of Manifacier et al. has resulted in the rst 'classic' paper in the area of determining nf and kf from transmission interference fringes. Swanepoel (1983) made several initial improvements and corrections to the work of Manifacier et al., and then proceeded to develop a more complex method capable of analysing inhomogeneous thin lms (Swanepoel 1984), which has been the starting point of this thesis. A summary of these three key papers, and more recent work, is given in this section.

Manifacier et al. (1976) For the sample structure shown in Figure 2.2, Manifacier et al. derive an equation describing the transmission of light through a weakly absorbing, uniform thin lm (kf2  n2f ) on an innitely thick substrate: 16n n n2 x T = C 2 + C 2 x2 +0 2s Cf C x cos , 1 2 1 2

(2.7)

where C1 = (nf + n0 )(ns + nf ), C2 = (nf ; n0 )(ns ; nf ), x = exp;f d and = 4nf d=. Outside the region of signicant absorption, the extrema of Eqn. 2.7 occur for 4nf d = m , (2.8)  where m is the order number. For the semiconductor on glass case, where nf > ns, the envelopes surrounding the extrema are considered to be continuous functions of  (via nf () and f ()) and are given by 16n0nsn2f x , Tmax () = (C1 + C2 x)2 16n0nsn2f x Tmin() = . (C1 ; C2 x)2

(2.9) (2.10)

2.2 Review of Methods for Determining Optical Constants Solving Eqns. 2.9 and 2.10 for x gives 

14



C 1 ; (T =T )1=2 x = C1 $1 + (Tmax=Tmin)1=2 ] . (2.11) 2 max min The lm's refractive index can then be expressed by rearranging Eqn. 2.9 to yield: 

nf = N + (N 2 ; n20 n2s )1=2 where

1=2

,

2 2 ; Tmin . N = n0 +2 ns + 2n0ns Tmax Tmax Tmin

(2.12) (2.13)

Once nf is known, x can found using Eqn. 2.11. The thickness of the lm is determined using Eqn. 2.8 with two consecutive maxima or minima forcing m = 1: d = 2 (n ( ) 1;2 n ( )  ) . (2.14) f 1 2 f 2 1 Finally, the absorption coecient f can be found using f = ; ln x=d. The error in the refractive index values was determined to be about 2 ; 5%, and, at best, 4% for the thickness d. No estimate of the accuracy was given for absorption coecient values. It was also noted that the SBW of the spectrophotometer should be maintained at a value less than the half-width of the interference maximum to avoid smearing out of the extrema as a result of poorer resolution. This, in fact, creates an upper limit on the thickness that can be measured with the spectrophotometer. Although the method presented by Manifacier et al. (1976) has many advantages over other previously described techniques, there are several disadvantages, for example: i. the envelopes are required to be drawn by hand ii. the substrate is assumed to be of innite thickness iii. the substrate refractive index ns is assumed to be constant over the whole wavelength range iv. the sample must be homogeneous and parallel faced (i. e. uniformly thick) and, v. the variation of nf , kf with  should be small, limiting the analysis to weakly absorbing regions.

Swanepoel (1983) In 1983, Swanepoel presented a very complete paper, making several improvements and corrections to the original work of Manifacier et al. (1976). These will be more fully discussed in the following pages, however, briey, these were: i. modelling the substrate as nitely thick and,

2.2 Review of Methods for Determining Optical Constants

15

ii. a more accurate method for determining the refractive index nf , absorption coecient f , and thickness d. The accuracy of all calculated values, nf , f and d, is determined to be within 1% (for f values > 100cm;1). Swanepoel's (1983) analysis begins with considering the transmission of the bare substrate, given by Ts = n22n+s 1 , (2.15) s or for a known transmission the substrate refractive index, is given by 

1=2

ns = T1 + T12 ; 1 s s

.

(2.16)

The following equation determines the number and position, with respect to the wavelength of the light, of the interference fringes, where the order m is an integer for maxima and half-integer for minima: 2nf d = m .

(2.17)

The sample structure used in Swanepoel's (1983) model is depicted in Figure 2.3, noting that the substrate is now of nite thickness. $LU G

Q  

Q1 )LOP a

Q 1  LN 1

*ODVV Q Q8 α8  7

$LU

Q  

Figure 2.3: The thin-lm on nitely thick substrate model used by Swanepoel (1983) to improve the envelope method.

The expression describing the transmission for such a sample, where nf  kf , as presented earlier is Ax T = B ; Cx cos (2.18) + Dx2 , where

A = 16nsn2f , ;  B = (n;f + 1)2(n; f + 1) n f + n2s , C = 2 n2f ; 1 n2f ; n2s ,

2.2 Review of Methods for Determining Optical Constants

16

;



D = (nf ; 1)2 (nf ; 1) nf ; n2s , = 4nf d , x = exp;f d . As the cosine term takes a value of +1 for a maximum and -1 for a minimum, the extremes of the interference fringes can be written as Tmax = B ; CAxx+ D x2 , (2.19) (2.20) Tmin = B + CAxx+ D x2 . As can be see from Figure 2.4, this means that each extrema has a corresponding value lying on the other envelope curve. Additionally, Swanepoel divides up the transmission spectrum into three regions of absorption: transparent, weak and medium, and strong. 100

Transmission (%)

80

Ts

Tmax Tα

60

Ti

40

Tmin

20 Strong 0 500

Medium 600

Weak

700 Wavelength (nm)

Transparent 800

900

Figure 2.4: Simulated transmission for a 1m thick a-Si:H thin lm on a nitely thick glass substrate. The curves drawn are described in the text (from Swanepoel 1983).

In the transparent region the lm is non-absorbing and f = 0, and the maxima are given by the transmission of the bare substrate Ts. Thus, the refractive indices of the substrate and lm can be found now by substituting x = 1 into Eqns. 2.19 and 2.20, respectively. This yields an equation of the same form as Eqn. 2.16 for the substrate refractive index  1=2 1 1 ns = T + T 2 ; 1 , (2.21) max max

2.2 Review of Methods for Determining Optical Constants and the lm's refractive index can now be determined from   nf = M + (M 2 ; n2s )1=2 1=2 ,

17

(2.22)

2 M = ns 2+ 1 + 2ns T2ns . (2.23) min The lm begins absorbing when the maxima depart from Ts, and f 6= 0. In this region, Swanepoel derives the same expression for nf as (Manifacier et al. 1976) to obtain   nf = N + (N 2 ; n2s )1=2 1=2 , (2.24)

where

2 ; Tmin . N = 1 +2 ns + 2ns Tmax (2.25) Tmax Tmin Swanepoel suggests that the transmission of the bare substrate could initially be measured and the refractive index ns() then obtained over the whole wavelength range using Eqn. 2.21, although this was not implemented. Now that nf () and ns (assumed to be constant) have been calculated, the coecients of Eqn. 2.18 are known. Solving either Eqn. 2.19 or Eqn. 2.20 quadratically for x, and simplifying using Eqn. 2.18 produces (shown here for Eqn. 2.19)   2 2 Emax ; Emax ; (n2f ; 1)3(n2f ; n4s ) 1=2 x = , (2.26) (nf ; 1)3 (nf ; n2s ) 8n2f ns where Emax = T + (n2f ; 1)(n2f ; n2s ) . max It should be noted that Eqn. 2.26 is not equivalent to Eqn. 2.11 as derived by Manifacier et al., due to the dierence in substrate models. In Figure 2.4, Ti is a curve passing through the inection points of the interference fringes and is given by Ti = T2 Tmax+TTmin , (2.27) max min and T is the geometric mean, denoted by p T = Tmax Tmin . (2.28)

where

In the region of strong absorption the interference fringes start to disappear, and there is not enough information contained within the spectrum to calculate nf and x independently. Therefore, Swanepoel recommends extrapolating nf () values found in the other parts of the spectrum and then using Eqn. 2.26 again to nd x. Alternatively, T can be used to determine f values, using the following equation h

where





i1=2

G ; G2 ; (n2f ; 1)6 (n2f ; n4s )2 1=2 x = , (2.29) (nf ; 1)3= (nf ; n2s ) 128n4 n2 G = T 2f s + n2f (n2f ; 1)2 (n2s ; 1)2 + (n2f ; 1)2 (n2f ; n2s )2 . 

2.2 Review of Methods for Determining Optical Constants

18

Instead of calculating the thickness d from any two extrema, Swanepoel proposes a more accurate, graphical method. Supposing that the order number from the rst visible extrema is m1 , Eqn. 2.17 can be written as

l=2 = 2 d (nf =) ; m1 ,

(2.30)

where l = 0 1 2 3 : : :. If l=2 is plotted against nf = and a straight line tted to the points, the line will have a slope 2d and a negative y-intercept of m1 . This method determines the thickness to an accuracy of better than 1%. Once the thickness is determined, the known m and d values can be fed back into Eqn. 2.17 and a more accurate value for nf is obtained. The accuracy achieved for nf is also in the order of 1%. Absorption coecient values were found to be accurate to within 1% in the range of 100cm;1 to 5  104cm;1. Swanepoel notes that relative errors of 1% in ns and nf lead to an absolute error of about 20cm;1 and 100cm;1 in f , respectively. Swanepoel also demonstrated that the innite substrate approximation predicts a 4% larger transmission than the measured substrate alone. This is due to the fact that the model presented by Manifacier et al. (1976) neglects transmission losses arising from reections from the rear surface of the glass. A correction for a too great a SBW is covered as well, which will shrink the amplitude of the interference fringes if not accounted for. In summary, this is a very accurate and relatively straight-forward method. It improves greatly upon the work of Manifacier et al. and the accuracy of nf , f , and d are all 1%. Two major disadvantages still remained however: i. the envelopes were still required to be drawn by hand and, ii. the sample must be homogeneous and parallel faced. The latter of these problems was addressed by Swanepoel the following year.

Swanepoel (1984) The case now being considered is for analysing inhomogeneous samples, such as that illustrated in Figure 2.5. The lm thickness is now expressed as d = d" #d, where d" is the average thickness and #d is the thickness variation. It is assumed that there is no variation in the optical properties throughout the lm, and that thickness varies linearly over the illuminated area. The variation can be in the form of sinusoidal, rectangular, or triangular surface roughnesses, and more random variations can be approximated by the these (Swanepoel 1984). The expression for the transmission through a non-uniformly thick sample is now an integration of Eqn. 2.18 over dmax = d" + #d to dmin = d" ; #d. Substituting x and into Eqn.2.18, the integral appears as 1

Td = d ; d max min

Z dmax

dmin

A exp;f d  dd0 .(2.31) 4nf d ;  d ; 2  d f f B ; C cos  exp +D exp 0

0

0

0

2.2 Review of Methods for Determining Optical Constants

$LUQR  ΔG

/LJKW ΔG

)LOP a Q I = Q I − LN I

GDYJ

6XEVWUDWHQV

GV

$LUQR 

19

7ΔG

Figure 2.5: A non-uniformly thick, absorbing thin-lm on a transparent substrate.

To illustrate the eect of the thickness variation, consider a transparent lm with the following properties: #d = 1000nm40nm, nf = 3:5, ns = 1:51. Equation 2.31 was evaluated for the non-uniformly thick, transparent lm (above), as well as a uniformly thick lm with the same optical properties. The compressive eect of thickness variation on the transmission spectra can be seen in Figure 2.6.

Transmission (%)

100 80 60 40 Δd=0nm

20 0

Δd=40nm

Envelope curves for Δd=40nm 600

700

800 900 1000 Wavelength (nm)

1100

1200

Figure 2.6: Simulated transmission for a 1m thick transparent thin lm, with and without a 40nm thickness variation (from Swanepoel 1984).

A slightly more complex comparison is plotted in Figure 4.4 in Section 4.3.1. The transmission maxima of the uniformly thick lm meet the bare substrate transmission Ts, described by Eqn. 2.15. For the non-uniformly thick lm, the extrema never reach Ts even though the modelled c-Si lm is transparent for  > 1100nm. Considering only the interference free absorption, given by the geometric mean T and Eqn. 2.28, thickness variations in the lm, if not corrected for, can lead to the false assumption that the lm is more strongly absorbing than the uniform lm and

2.2 Review of Methods for Determining Optical Constants

20

has a higher absorption coecient. As the integral Td is dicult to solve analytically, an approximation is to consider x as having an average value over the integration range (Swanepoel 1984), of

x" = exp;f d .

(2.32)

This is an excellent approximation, provided that #d  d" and that 0 < #d < =4nf . The cosine term in Eqn. 2.31 has the dominant shrinking eect on the spectra, as for each d0 the frequency of the fringes will be dierent, and integrating over the range dmin to dmax will average out the extrema. This simplies the integral to "

Td = 4n #d p ax 2 arctan 1 ; bx f



!



1 + bx tan 4nf dmax p  1 ; b2x

 !# 1 + b 4 n d x f min ; arctan p 2 tan ,  1 ; bx

where

" ax = B +AxD x"2

and

" , bx = B +C xD x"2

(2.33) (2.34)

and the envelope expressions are then (M&arquez et al. 1995)

Tmax x Tmin x





 p a arctan p1 + b tan 2nf #d , (2.35) = 2nf #d 1 ; b2  1 ; b2 

  a 1 ; b 2 n # d f = 2n #d p . (2.36) arctan p tan  1 ; b2 1 ; b2 f

In the transparent region of the transmission spectra the coecients become A C . a= B+ and b = (2.37) D B+D Thus we now have two independent equations to solve simultaneously for the two unknowns #d and nf , yielding a unique solution (Swanepoel 1984). Where the lm becomes absorbing, Eqns. 2.35 and 2.36 can be solved using the coecients ax and bx, replacing the transparent region coecients a and b. As #d is now known, there are again two equations to solve uniquely for two unknowns, nf and x. The average thickness d" is found using the same method as described in the earlier work of Swanepoel (1983) in the previous section. Once d" has been determined f can be found easily from x, and kf can be calculated. If the calculated f values near the strongly absorption region appear noisy, since they are calculated from the dierence between the two envelope curves and can be aected by experimental errors, these few points can be determined using the geometric mean T , or solely Tmax using Eqn. 2.26 as Swanepoel recommends. Swanepoel found that the average thickness, refractive index, and absorption coecient of the lms to be within 0.1%, 0.1%, and 5%, respectively. Although

2.2 Review of Methods for Determining Optical Constants

21

the absorption coecient values are less accurate than that from the uniform lm analysis, it is essential to be able to deal with lm inhomogeneities. A variation in refractive index for a uniformly thick lm is also modelled by Swanepoel. The optical pathlength nf d occurs as a product in the phase angles, and therefore the eect of #nf d and nf #d are equivalent, with #nf = 0:14 corresponding to #d = 40nm for the curves in Figure 2.6. Thus, it is actually impossible to determine whether #nf or #d is causing the shrinking of the transmission spectra (Swanepoel 1984). Although Swanepoel does not mention it, the author believes that no change in f should be observed. Referring to Eqns. 2.32 to 2.36 it can be seen that x always uses the average thickness d", and that x would be the same for a uniformly thick lm d with a non-uniform refractive index #nf .

Bah et al. (1993) Bah et al. (1993) propose a simple, approximate method which is designed to model the transmission of a non-uniformly thick, thin lm on a transparent substrate. The analysis relies upon the following: i. the transmission extrema values only change with optical thickness nf d" and, ii. the magnitude of the transmission minima are less aected by thickness nonuniformities than those of the maxima. Light transmission through a non-uniformly thick lm d = d" #d is given by (Bah et al. 1993) Z d+d ~T = 1 T (d ) dd , (2.38) 2#d d;d yielding the following equations for the upper and lower envelopes around the extrema

T~max

1=2 = 2 (Tmax Tmin ) arctan

T~min

1=2 = 2 (Tmax Tmin ) arctan

" "

#

Tmax Tmin

1=2

tan 2 ,

(2.39)

Tmin Tmax

1=2

tan 2 ,

(2.40)

#

where Tmax and Tmin are the transmission values for a plane-parallel lm of thickness d", and = 4nf #d=. At normal incidence, Tmax and Tmin can be expressed as (Bah et al. 1993)

; R23 )(1 ; R31) x Tmax = (1 ; x (R (1R; )R1=122 ))(1 , 2 ; R ((R )1=2 ; x (R )1=2 )2 12 23 31 23 12 ; R23)(1 ; R31 ) x Tmin = (1 + x (R (1R; )R1=122))(1 , 2 ; R ((R )1=2 + x (R )1=2 )2 12 23 31 23 12

(2.41) (2.42)

2.2 Review of Methods for Determining Optical Constants

22

where R12 , R23 , and R31 are the reectances at the air-lm, lm-substrate, and substrate-air interfaces, respectively, and x = exp;f d . Expanding the trigonometric functions in Eqns. 2.39 and 2.40 one obtains #Tmax = Tmax ; T~max = x2 Tmax ; Tmin , Tmax Tmax 3 Tmin #Tmin = T~min ; Tmin = x2 Tmax ; Tmin , Tmin Tmin 3 Tmax and combining these



#Tmin = Tmin #Tmax Tmax

2

.

(2.43) (2.44) (2.45)

Using Eqns. 2.43, 2.44, and 2.45, Bah et al. (1993) derive the relation

Tmin = T~min ; #Tmax



T~min Tmax

!2

,

(2.46)

for the region of negligible absorption where x = 1. In this case the expressions describing the extrema then simplify from Eqns. 2.41 and 2.42 down to

Tmax = n22n+0 nns2 , 0 s 4n n2 n Tmin = (n2 + n20)(fn2 s+ n2 ) , 0 s f f

(2.47) (2.48)

where n0, nf , and ns are the air, lm, and substrate refractive indices, respectively. Since Tmax is independent of nf , its value can now be determined and subsequently used in Eqn. 2.46 to nd Tmin and thus nf . However, Eqn. 2.46 is only strictly true when Tmin = T~min (i.e. #Tmin = 0) so Bah et al. (1993) make the assumption that there is no change in the magnitude of the minima due to thickness variations. There is a denite change in the minima though, as shown in Figure 2.6, and it is exactly this few percent change in transmission that contains the important information to enable the optical constants to be correctly determined! The absorption coecient of the lm is found assuming that the reectivity from the substrate-air interface R31 is zero. This is not valid for a substrate of nite thickness, and for glass this value would typically be around 4%. Using Swanepoel's (1983) model for an a-Si:H lm of thickness d = d"  #d and a substrate refractive index of ns = 1:5 the lm's refractive index nf is found to be accurate to 0:01 (absolute) (Bah et al. 1993). However, the discrepancy between theoretical and calculated f values is around 30% at the shortest  (600nm) calculated, while at the longest wavelength (1200nm) the error in f can exceed 100%. Thus, the large number of approximations and the large error in f , which became apparent after testing the method briey, did not encourage the present author to investigate the method further.

2.2 Review of Methods for Determining Optical Constants

23

More Recent Advances Several research groups, working in conjunction with Swanepoel, have continued to develop and apply the envelope method (M&arquez et al. 1995, Myburg and Swanepoel 1987, Minkov and Swanepoel 1993). Myburg and Swanepoel (1987) investigated the parameter #d and found it to be an important quantity embodying the eect of the following four factors: i. the product of the average refractive index n" f and thickness variation #d over the illuminated area ii. the product of the average thickness d" and refractive index variation #nf over the illuminated area iii. eect of slit width and, iv. eect of #nf with depth (perpendicular to the plane of the substrate). It was recommended that the variation of refractive index nf be determined by measuring the lm at several points on the sample, and that the slit width be kept to a value of 1nm or less. The SBW, which depends on the slit width and the dispersion angle of the grating (Oriel Corporation 1994), is actually the more important parameter here, rather than the physical slit width alone. It is unclear what SBW was used in the work of Myburg and Swanepoel. As a mathematical correction for the last factor does not exist, Myburg and Swanepoel assumed that there is no variation of nf along the optical axis. In this work Myburg and Swanepoel monitored the stability of a-Si:H lms over time, and reported a signicant increase in thickness and decreases in the values of #d, nf , and f . Values of the optical bandgap Eg were determined by plotting (f h )1=2 versus (vs.) energy, E or h , where  = 1=, commonly referred to as a Tauc plot. This is implies that p

f h / h ; Eg ,

(2.49)

and assumes that the density of electron states near the edges of the conduction and valence bands have a parabolic distribution. It has been shown that p 3

f hnf / h ; Eg

(2.50)

gives a better t to experimental data for a-Si and a-Si:H than the Tauc plot (Swanepoel et al. 1985, Klazes et al. 1982). In 1992, Minkov, rstly, showed that using envelope curves to obtain the optical constants guaranteed that the solution is always singular (Minkov 1992a). In a second paper (Minkov 1992b), the eect of angle of incidence and polarisation of the light was investigated. It was found that using s-polarised light incident at an angle of 30 provides more accurate results. It was also noted that transmission can usually be measured to a greater accuracy than reection by most spectrophotometers due to calibration problems. In the strong absorption region, the transmission spectra yield more accurate kf values, although nf will be less accurate than could

2.2 Review of Methods for Determining Optical Constants

24

Envelopes i () Polar- d (nm) #ns (%) #d (%) #nf (%) #nf (%) isation (strong) (weak) True 0 Unpol. 1000 0.3 0.2 1.8 0.6 True 0 Unpol. 550 0.2 0.2 1.8 0.6 True 30 s 1000 0.2 0.1 1.8 0.6 True 30 s 550 0.2 0.2 1.8 0.6 Computer 0 Unpol. 1000 0 0 4.6 0.4 Computer 0 Unpol. 550 0.8 0.6 2.7 1.2 Computer 30 s 1000 0.1 0.1 5.8 0.3 Computer 30 s 550 0.9 0.8 4.6 1.2 Table 2.1: Comparison of errors in refractive index and thickness values for both computer and hand-drawn envelopes, various angles of incidence, light polarisations, and lm thicknesses.

be obtained using reection spectra. Minkov and Swanepoel (1993) developed the envelope method further by implementing a computer program to t envelope curves to the transmission and reection spectra. It was claimed that the ENVELOPEy algorithm of (McClain et al. 1991), a modied form of which was employed by the author of this work, was not suitable and an alternative method was developed. Minkov and Swanepoel's (1993) algorithm used a parabolic curve t in the absorbing region of the spectra, and a linear t in the transparent region. A comparison between computer tted envelopes and the "true", which are assumed to be hand-drawn, envelopes show agreement to within 2%. This algorithm is suitable for a-Si:H lms but not for a-Si or poly-Si lms which can be absorbing across the whole wavelength range, and do not necessarily have a at linear region in the NIR. Additionally, lms that exhibit free carrier absorption in the NIR region could not be analysed with this algorithm. Error analysis using the computerised algorithm yielded the results shown in Table 2.1 from the transmission spectra (Minkov and Swanepoel 1993). Surprisingly, Minkov and Swanepoel found the error in absorption coecient values to be less than 0.2% for all cases. Therefore, it was noted from Table 2.1 that the error in the refractive index of the lm was signicantly greater than all other errors. The errors in analysis of the thinner lm are higher due to the fewer extrema in the spectra, which had an upper limit of  = 900nm. M&arquez et al. (1995) used Swanepoel's (1984) method to characterise wedge-shaped amorphous arsenic trisulphide thin lms using a UV-Vis-NIR spectrophotometer in the range 0:3 ; 2m. Diuse reectance measurements showed that negligible amounts of light were scattered from the lms, removing any uncertainty in this regard. A second version of Swanepoel method was used by M&arquez et al. (1995) for obtaining the optical constants (Swanepoel 1984). It is not described in the section above, however it achieves exactly the same results. The dierence to the initial y ENVELOPE, written in capitals, refers strictly to the computer program.

2.2 Review of Methods for Determining Optical Constants

25

method is that eect of #d on the extrema is calculated and the output is a set of extrema for a uniform lm of the same average thickness. The ENVELOPE algorithm (McClain et al. 1991) was then used to t envelope curves to the corrected extrema. The author noted that some of the corrected maxima lie above Ts, the bare substrate transmission level, by up to 1.8% (absolute) which would introduce serious errors in the analysis. M&arquez et al. found that the accuracy of d" values for three samples were 1.5%, 2.4%, and 0.8% when compared with surface proler (Dektak IIA) results, which is higher than expected from the method. The accuracy of nf and kf results were not supplied. Laaziz and Bennouna (1996) examined the experimental eect of #d and SBW using 1 ; 2m thick a-Si lms. The method used models the inverse transmission, instead of the transmission, of a uniformly thick lm, as an interference-free term and an oscillatory term. For non-uniformly thick lms two correction terms are added, and along with non-zero SBW corrections the equation used appears as follows   1 = expd +R10 R2 exp;d $1 ; R2R3 A2 ] + T (1 ; R10 )(1 ; R2 )(1 ; R3 )A p

 2 R10 R2 $1 ; R2R3 A2 ] sin #  (1 ; R10 )(1 ; R2 )(1 ; R3 )A # 2

sin

4 



4nf d

3

2  5 cos(1 + 2 ; ) 4nf d 2

,

(2.51)

where = 4nf d=, and 1 and 2 are the phases of the Fresnel reection coecients. The method nds nf , d", and #d, and the work conrms the work of other authors in the eects that #d and non-zero SBW have on the interference fringes. It is of concern to the author that envelope curves were simply tted using a high order polynomial, as the envelopes do not seem to be a good t to the transmission extrema in the gures of Laaziz and Bennouna (1996). Rizk et al. (1996) published a paper investigating the optical properties of sputtered amorphous and subsequently annealed silicon thin lms. Using Swanepoel's (1983) method for uniformly thick lms, the optical bandgap Eg for a-Si was found to be 1.25eV, increasing to around 1.40eV for the microcrystalline silicon (c-Si) lms formed upon annealing. Finally, it should be noted that more varied uses for the envelope method have been found. Bovard et al. (1985) used transmission measurements, performed in a vacuum, to determine the optical constants and thickness of titanium dioxide while the material is being deposited onto glass. This permits the study in variations of the optical constants with changes in the deposition conditions, and also identies any deposited layers that are unstable (e.g. oxygen decient). After considering all of the above methods, the envelope method of (Swanepoel 1984) came the closest to being able to characterise lms for the present application. The author decided to investigate this method further, and to try and implement

2.2 Review of Methods for Determining Optical Constants

26

changes which would permit the envelope method to characterise a-Si and poly-Si lms, which are absorbing across the whole wavelength range. The extensions to the envelope method developed by the author are presented in Chapter 4.

Chapter 3 Materials and Methods 3.1 Substrates There are a large range of materials that can be used as substrates for thin lm solar cells. The most common of these is glass, although stainless steel, sapphire, and ceramics are also popular. Glass and sapphire substrates were used in this work, particularly because of their transparent nature, enabling them to become the protective top surface of the module after encapsulation. In this section, the reasons these substrates were chosen will be discussed, along with their most important features, namely: i. ii. iii. iv.

transmission range refractive index structural characteristics - thermal, mechanical and chemical and, cost.

The transmission range and refractive index determine the amount of light absorbed or reected by the substrate, and therefore what fraction of the incident light can be potentially utilised by the solar cell. The thermal characteristics of the substrate are important when the material undergoes high temperature deposition processes. In our case, a good match of the thermal expansion coecient will minimise the number of silicon lms that peel or ake o the substrate. Signicant mechanical characteristics are strength and resistance to shock, while chemical characteristics, such as resistance to corrosion from gases and water vapour in the atmosphere, aect the suitability of the material to harsh environments. Finally, in order to produce a solar cell that can generate electricity for as few jc/kWh as possible, cost naturally plays an important role.

3.1.1 Glass Glass is formed from molten silicon dioxide (SiO2), which when cooled rapidly, forms a transparent amorphous solid. The glass softens again gradually when heated, 27

3.1 Substrates

28

without exhibiting any sharp melting point. Ordinary glass has an approximate composition of 76% SiO2 and the remainder being calcium and sodium oxides, however several other oxides can be added to give the glass dierent properties. For example, the addition of boron oxide produces borosilicate glass which expands very little upon heating, while alumina improves the chemical resistance (Gillespie et al. 1989).

Glass Properties Most of the glass now used for photovoltaic research was originally developed as a substrate for at panel displays. One original glass developed for this purpose is Corning 7059 bariaborosilicate glass and this has been extensively used in thin lm solar cell research. Bariaborosilicate glass is borosilicate glass with a baria content, used instead of lime or lead to give the glass a higher refractive index. More recently, Corning have developed a new boroaluminosilicate glass, coded 1737, which has several advantages over 7059 (Lapp et al. 1994): i. the boroaluminosilicate glass can be annealed at higher temperatures due to a higher strain point, 666C for 1737 versus 600C for 7059 ii. for deposition temperatures up to 350C, commonly used for a-Si deposition, 1737 does not require pre-annealing to ensure minimal thermal shrinkage. However for poly-Si depositions an annealing step is still needed to reduce thermal shrinkage. iii. 1737 has a thermal expansion coecient almost the same as that of silicon, giving silicon lms deposited onto 1737 glass a greater resistance to thermal shock iv. the improved chemical durability of 1737 glass permits harsher cleaning methods to be used with no danger of damaging the surface and, v. the lower density of 1737 glass still provides a sti front surface for PV modules while reducing the weight For the above reasons, the author purchased 250 50mm50mm1.1mm pieces of 1737 glass from Corning for our thin lm silicon depositions.

Optical Characterisation of 1737 Glass Initially, it was necessary to optically characterise the glass to obtain the most accurate results from the envelope method analysis. Three dierent sets of data for ns and ks were obtained and the results from each compared. Firstly, Corning supplied refractive index data for seven wavelength values between 436nm and 656nm (Corning Inc. n.d.), and a more complete data set for both ns and s values was obtained from Corning upon request (Corning Inc. 1997a). Secondly, transmission and reection measurements were performed on the glass, and the optical constants determined with Optikan (Bader et al. 1995). Finally, the ns of the glass in the

3.1 Substrates

29

range 300-800nm was measured using spectroscopic ellipsometry. A comparison of the results is illustrated for the refractive index in Figure 3.1 and absorption coecient in Figure 3.2. -4

Refractive index ns

-6

n1737 = 1.65599 - 6.62248*10 λ + 1.36881*10 λ

1.54

-9

3

- 1.59892*10 λ + 1.0667*10

1.52

+ 7.56102*10

-20

6

-12

λ - 5.62614*10

4

2

λ - 3.97249*10 -24

λ

-16

λ

5

7

1.50 Corning Ellipsometry Reflection and transmission

1.48 1.46 500

1000

1500 2000 Wavelength (nm)

2500

-1

Absorption coefficient αs (cm )

Figure 3.1: Refractive index of Corning 1737 glass determined by various techniques. The solid line is a polynomial t to the data acquired from Corning (Corning Inc. 1997a), ellipsometry, and reection and transmission.

10

-6

10

-7

10

-8

10

-9

10

-10

10

-11

Corning Reflection and transmission

500

1000

1500 Wavlength (nm)

2000

2500

Figure 3.2: Absorption coecient s : The solid line is an interpolated t to the data acquired from Corning (Corning Inc. 1997a).

The results achieved with transmission and reection experiments, and subsequent analysis using Optikan, are not quite consistent with the other results. This is because the method is less sensitive than ellipsometry, and there are a wider range of possible ns and ks values which provide good agreement with the experimental

3.1 Substrates

30

data. However, as the reection and transmission data for ns spanned the desired wavelength range, a seventh-order polynomial was tted to all three data sets, and this is displayed in Figure 3.1. Initially it was thought to be doubtful that the long wavelength (> 2000nm) ns data dropped so far below 1.5, and all possible experimental errors were considered. Upon further reading, it was found that a local maximum in s at a certain wavelength is accompanied by a point of inection in ns very close to the same wavelength, and at longer wavelengths ns can decrease to values near, or even below, unity (Jenkins and White 1957). This is shown in Figure 3.3, where a reduction in n for a transparent sample occurs near the absorption band. Additionally, the limited validity of the Cauchy expression n = n0 + n1=2 + n2 =4 is indicated in Figure 3.3, where n0, n1, and n2 are constants. Since it was desirable to optically characterise the 1737 glass into the NIR, a higher order polynomial, displayed in Figure 3.1, was required to model the dispersion in the glass and this expression is used in the modied envelope method analysis.

Figure 3.3: Dispersion of a transparent material (similar to quartz), showing a reduction in n in the NIR due to increased absorption (from Jenkins and White 1957).

It is not stated how the s data from Corning was measured, but as the sample used was 3mm thick (Corning Inc. 1997a) the results are likely to be more accurate than those obtained by reection and transmission experiments. This is because there is so little absorption in 1.1mm of 1737 glass, the determination of s by reection and transmission is limited by the noise level. It is interesting to note that the reection and transmission data follows the same trend as the Corning data, and while it does not appear to be noisy, it is oset by a factor of 103. This cannot be explained by the author. However, a thicker sample with increased absorption should result in more accurate s values, regardless of the method used, and to model the glass absorption, interpolated values from Corning's data are used, obtained using the Mathematica function 'interpolation'. The properties of the glass in the IR were

3.1 Substrates

31

Transmission (%)

100 80 60 40 20 0

2

3 4 Wavelength (μm)

5

Figure 3.4: Transmission of 1737 glass in the IR, showing several strong absorption peaks beyond 2.6m.

also investigated using Fourier transform infrared (FTIR) spectroscopy after strong absorption beyond 2.6m was observed with spectrophotometry. As illustrated in Figure 3.4, no light is transmitted at  > 5m, and there are several absorption peaks between 2m and 5m. The peak at 2.8m is found in all silicate glasses and is due to the hydroxyl stretch of the Si-OH group. Absorption at 3.7-3.8m is attributed to an overtone of a boron lattice vibration (B-O stretch), however overtones of the Si-O stretch are also found at these wavelengths (Corning Inc. 1997b).

Glass Cleaning Various cleaning methods were tested for their eectiveness on the glass. Initially, an ultrasound bath containing a beaker of soapy water with the samples mounted upright was implemented. However, this was found to be unsatisfactory as several visible spots remained on the sample after one hour of operation. A method used since then involved cleaning the grease o the surfaces with a lint-free cloth and liquid soap, rinsing in deionised (DI) water, placing the glass in a beaker of RCA1 (hydrogen peroxide : ammonium hydroxide : DI water in the ratio 1:1:4) and boiling for 20 minutes. This removes any organic compounds on the surface of the glass. This step is followed by another DI water rinse, and then blow-drying the plates with a nitrogen gun. The glass has essentially no visible marks on it after this procedure, as long as care was taking during the drying stage. Later, an oxygen plasma etch was used instead to remove contaminants from the front surface of the glass. Placing the glass in the plasma for 10 minutes proved successful, and although only the surface which will have a Si lm deposited onto it is cleaned by this technique it is much quicker and avoids the problem of stains appearing during drying.

3.1 Substrates

32

3.1.2 Sapphire Sapphire, crystalline aluminium oxide (Al2O3 ), was also used as a substrate in this work. Commercially grown epitaxial silicon-on-sapphire (SOS) samples were available and proved extremely useful as a \test sample" in determining the accuracy of the envelope method. Thus, sapphire was not used for deposition purposes, however its properties make it a very interesting substrate material due to (Griot 1995/1996): i. it being an extremely hard and strong material. This means that substrates need only to be a few hundred m thick to provide adequate mechanical support ii. it is chemically inert and insoluble in almost everything iii. wide-range transmittance (between 150nm and 6000nm) iv. it has a relatively good lattice constant match to silicon, and allows epitaxial growth directly o the substrate (Schl(otterer 1976) and, v. it has exceptional thermal conductivity. One disadvantage of sapphire is that, due to its hexagonal crystalline structure, it exhibits anisotropy in many of its properties, and is birefringent. In a birefringent or double refracting material, light travelling parallel to the optical axis experiences no double refraction and passes straight through the material. Light travelling perpendicular to the optical axis experiences one refractive index (ne) for light polarised in the plane of the optical axis (the extra-ordinary or e-ray), while light polarised perpendicular to the to the optical axis (the ordinary or o-ray) sees a dierent refractive index (no). Optical databooks publish data sets for both polarisations (Palik 1985), however for our purposes the dierence in optical properties remains relatively small and birefringence causes only a refractive index change of 0.0085 for  > 400nm values (Oriel Corporation 1990). For ns 1:77, this eect causes less than 0.5% variation in ns. As the SOS samples were manufactured for the electronics industry, the rear (nondeposited) surface of the substrate was roughened to act as a getter (Green 1995). Thus, the substrate had to be polished in order to specularly transmit light. This was performed for the author by the polishing unit in the School of Applied Geology. Figure 3.5 shows the refractive index found from reection and transmission data using unpolarised light, and widely accepted published data (Palik 1985, Malitson 1962). The refractive index can be described by the following Sellmeier dispersion relation (Malitson 1962): 2 2 2 1 : 023798  1 : 058264  5 : 280792  nAl2 O3 ; 1 = 2 ; 0:061448212 + 2 ; 0:11069972 + 2 ; 17:926562 . 2

(3.1)

There is relatively little published data on the absorbing properties of sapphire in the visible and NIR spectrum (Palik 1991). There were only two works found which

3.1 Substrates

33

1.80 Refractive index ns

1.78 1.76 1.74 1.72 Palik Malitson Reflection and transmission data

1.70 1.68 1.66 500

1000

1500 Wavelength (nm)

2000

2500

-1

Absorption coefficient αs (cm )

Figure 3.5: Refractive index data of sapphire: The solid line is the Sellmeier relation (Eqn. 3.1) derived by Malitson (1962).

10

1

Reflection and transmission data Mitchell

-4

αsapphire (λ) = 10 ^ [ -6.87209 + 0.05325 λ - 1.73977*10 λ

10

-7

3

-17

6

-31

9

-10

4

2 -13

+ 2.90085*10 λ - 2.77298*10 λ + 1.57483*10 λ

0

-21

7

-25

- 5.22965*10 λ + 9.3343*10 λ - 6.86665*10 λ

5

8

- 3.25339*10 λ ]

10

-1

400

800

1200 1600 Wavelength (nm)

2000

2400

Figure 3.6: Absorption coecient data for sapphire: The solid line is a polynomial t to reection and transmission data while published UV and visible data from Mitchell et al. (1960) is shown for comparison.

contained absorption coecient values for the visible spectrum (Levy 1961, Mitchell et al. 1960). The data from Mitchell et al. (1960) extended the furthest into the visible and this is plotted in Figure 3.6 along with data obtained from reection and transmission experiments, and Optikan analysis. As the data from Mitchell et al. (1960) did not span over the desired range, the Optikan data set was chosen to represent the sapphire absorption and was tted with a ninth-order polynomial, as seen in Figure 3.6. This equation, along with Eqn. 3.1 are used to model the optical properties of the sapphire substrate.

3.2 DC Magnetron Sputtering

34

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3ODVPD5HJLRQ

6XEVWUDWHV 6XEVWUDWH +ROGHU

'&%LDV 6XSSO\

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Figure 3.7: Schematic of typical DC sputtering system (from Materials Research Corporation 1975).

3.2 DC Magnetron Sputtering 3.2.1 System Description Sputtering is a process which involves the transfer of almost any material from a target to almost any substrate. The deposition of material is achieved by bombarding the target surface with gas ions accelerated by a high voltage. The transfer of momentum from the gas ions to the target results in atoms being ejected from the target. These travel through the vacuum chamber and are deposited onto the substrate as a thin lm. When the sputtering target is negatively charged, electrons are ejected into the gas, argon in this case, near the target. The applied voltage accelerates the electron towards the positively charged anode, gaining energy from the electric eld. If the electron collides with a gas molecule it will transfer part of its energy and leave behind an ion and another free electron. This cumulative eect is called a selfsustaining glow discharge or plasma, as the area where ionisation occurs glows. The diagram in Figure 3.7 shows the layout of a simple direct current (DC) sputtering system. A DC or radio frequency (RF) power supply can be used depending on the type of material to be deposited. The sputtering system at UNSW was always used with a DC power supply, which is suitable for depositing conducting materials, including doped silicon. Due to the often high gas pressures used, DC biasing is designed to negatively bias the substrate to help eliminate the substantial amounts

3.2 DC Magnetron Sputtering Target Dopant Resistivity No. Type ().cm) 1 n: P 0:002 ; 0:08 2 p: B 0:005 ; 0:02 3 p: P 2:5 ; 7:5 4 n: P 0:08 ; 0:12 5 n: P 3:6 ; 5:0 76 p: B 0:2 ; 0:5 102 n: P 150

35 Doping Level Growth Thickness ; 3 (cm ) (mm) 17 19 1:1  10 ; 3:6  10 CZ 1 2:5  1018 ; 1:9  1019 CZ 3 15 15 1:8  10 ; 5:7  10 CZ 3 16 17 6:2  10 ; 1:1  10 CZ 0.4 14 15 9:1  10 ; 1:2  10 Unknown 0.6 3:2  1016 ; 9:6  1016 CZ 0.33 3:0  1013 FZ 0.3

Table 3.1: Range of doping levels and types of sputtering targets available.

of gas that can be incorporated into the lm. As the lm is deposited, it is lightly cleaned by ions which remove any weakly trapped gas atoms resulting in a lm with better adherence and greater density (Materials Research Corporation 1975). Additionally, the substrate may be heated up to around 600C, so poly-Si lms may be deposited or in situ annealing performed, if desired. The sputtering system uses a rotary pump to start evacuating the chamber and a turbomolecular pump to achieve a relatively high vacuum, down to 1  10;6Torr, in the chamber. Inside the chamber, a shutter is used so that the rst 30 seconds of sputtered material from the target strike the shutter and not the substrate. This "pre-sputtering" removes any impurities that were present on the target surface from being incorporated into the lm. Silicon can typically be sputtered at deposition rates of around 200* A s;1 using a DC power of 100W and an argon pressure between 0.5Pa and 5Pa.

3.2.2 Targets and Sputtered Films Special sputtering targets and 85mm diameter wafers with a wide range of doping levels were available for use as targets, as given in Table 3.1. The two most commonly used dopants are available, boron (B) and phosphorus (P), and the majority of the targets are produced by the Czochralski (CZ) process. It should be noted that, rstly, the same doping level of the sputtered lm as the target will not necessarily be achieved as not all the dopants may be activated in the lm. Secondly, it is only possible to deposit one lm at a time before the chamber has to opened. It is suspected that the oxidation that occurs before the lm can be placed in a furnace may inhibit the crystallisation process, however this has not been demonstrated yet. Due to the low conductivity it was dicult to sputter highly resistive lms such as target 102, a oat-zone (FZ) wafer. Ideally, RF sputtering would be used to deposit such lms.

3.3 Crystallisation

36

3.3 Crystallisation After depositing a lm, its transmission was usually measured immediately with the spectrophotometer. Following this, lms were placed into a nitrogen purged furnace at atmospheric pressure and annealed for a period of 60-80 hours at around 610C. As mentioned previously, in situ annealing in the sputtering chamber is possible, however this was not performed due to the high demand for the sputtering system. The a-Si to poly-Si transition can be observed as the lm colour changes from reddish-brown to yellow.

3.4 Spectrophotometry The main focus of this work is the determination of the optical constants of the Si lms. The machine used for all measurements was a Varian Cary 5G UV-VisNIR double-beam spectrophotometer, illustrated in Figure 3.8. Unfortunately, this spectrophotometer is not very stable or reliable and it was constantly being repaired throughout the year, and was out of operation for a total of ve months during 1997. This meant that a lot of the initially planned work had to be discarded due to a lack of time, and the majority of the author's time was spent simply trying to maintain the system. This section will include a brief description of how the spectrophotometer functions in transmission mode as well as highlighting several potential measurement problems. Transmission mode was preferred over the use of an integrating sphere for collecting reection data due to the larger errors and noise associated with the latter.

Figure 3.8: The unreliable Varian Cary 5G UV-Vis-NIR double-beam spectrophotometer.

3.4 Spectrophotometry

37

3.4.1 System Description The Cary 5 optics are based around a double monochromator with the optical arrangement shown in Figure 3.9. Double-monochromators have better resolution and stray light rejection than single single-monochromators. The Cary uses a tungsten halogen lamp as the visible and NIR source, and a deuterium lamp as the UV source. An automated carousel swaps the lamp being used at a specied wavelength l , typically 300nm. Light from the lamp passes into the rst monochromator via mirrors N1, N2, N3 and N4, then through the lter mechanism, designed to keep stray light from entering the second monochromator. The collimating mirror N5a directs the beam onto the diraction grating where it is dispersed into its constituent wavelengths. The gratings are double sided and ruled at 1200 lines mm;1 for the UV-Vis spectrum and 300 lines mm;1 for the NIR, relying on a motor driven chain to ip between them at the detector change-over wavelength dc. The light enters the second monochromator, and is reected o the second collimating mirror N5b which directs the light through the exit slit. A chopper, spinning at 30 revolutions per second splits the light into sample and reference beams. The chopper produces three states during each revolution, namely sample, reference, and dark. The rst state uses the mirror section of the chopper to reect light into the detector compartment via mirrors N9S and N10S. In the reference state, light passes straight through the chopper and o mirrors N8 and N9R to the detector compartment. The dark state is where a black metal plate blocks the beam, and during this period the monochromator changes wavelengths. Light passing through the sample or along the reference beam path is detector by a photomulitiplier tube (PMT) in the UV-Vis region and a lead sulphide (PbS) detector in the NIR. The detectors are changed in unison with the gratings, typically at dc values of 800 ; 850nm. The principle advantage of a double-beam spectrophotometer is that uctuations in lamp intensity do not aect the transmission measurements. This is because a reference is constantly being taken throughout the scan, and the nal transmission value does not rely only on a previously measured reference scan as a single-beam machine does. Before a transmission measurement can be performed, it is necessary to assign a value to certain parameters, which include: i.  scan range, e.g. 300 ; 2600nm ii.  interval, usually set to 1nm steps for an accurate transmission spectra iii. averaging time - the time that the monochromator remains at each . Typically set to 0.33s for a noise-free spectrum, this results in the average of ten readings being recorded at each  iv. SBW - this parameter and its various options will be discussed at length in the following section v. slit height - options include full or reduced, producing a 13.4mm or 9.1mm high beam image in the focal plane, respectively and,

3.4 Spectrophotometry

Figure 3.9: Varian Cary 5G spectrophotometer optical layout.

38

3.4 Spectrophotometry

39

vi. dc and l - typically 850nm and 299nm (to avoid unnecessary UV lamp usage), respectively. The Cary must be left on for at least two hours to warm-up, as the PbS detector is sensitive to temperature variations and is constantly receiving blackbody radiation emitted from the black plate in the chopper. Before transmission measurements can begin, a `zeroline' is performed with a black metal plate blocking the sample beam, followed by a baseline where no sample is present. These two scans must always be performed as they dene the 0% and 100% transmission levels.

3.4.2 SBW and Energy Level The SBW is determined by the physical slit width and the dispersion angle of the grating, and setting the SBW can be performed in several ways. To resolve sharp absorption peaks, the SBW should ideally be as small as practically possible, around 1nm. However, a low SBW reduces the beam size and the intensity of the beam. This signal reduction poses no problem to the PMT due to its high internal gain, but the PbS detector has a poor sensitivity just above the detector change-over wavelength dc and at  > 3m and the experimental data becomes extremely noisy at low SBWs. The method implemented by Varian to improve the signal-to-noise (S/N) ratio was to introduce another parameter confusingly termed the `energy level', EL, which aects the amplication of the signal caused by the reference beam striking the detector (Varian Australia OS/2 software version). In the NIR, the EL is set to be a constant value of usually 3.0 to minimise non-linearity errors, as this EL value used for calibration purposes. This results in the SBW being constantly varied at each , presumedly, in order to maintain a high enough light intensity reaching the PbS detector. A plot of how the SBW varies with  at various ELs is plotted in Figure 3.10. In the normal mode of operation, called `auto', the SBW will be large and possibly at its maximum value of 20nm at the limits of the PbS detector, and then suddenly jump to a low constant value throughout the UV-Vis spectrum. Often it is desirable to have a constant SBW, and that the illuminated area on the sample remains constant at all wavelengths. This is important for interference fringe measurements as the thickness variation in samples will generally be greater over a larger area, and if the SBW is of the same order of magnitude as the period of the interference fringes then compression of the fringes will result. The requirements of the envelope method are that the same area on the sample is illuminated at all wavelengths, the thickness variation can be approximated as being linear, and the SBW is kept to a small constant value (Swanepoel 1984). This an is extremely dicult criteria to fulll for a UV-Vis-NIR spectrophotometer covering such a large  range and including a grating change-over. After exhaustive experimentation, the solution in this case has been to use the settings SBW=2nm in the visible and EL=3.0 in the NIR. This minimises any step that may appear in the transmission spectrum due to non-linearity errors in detector calibration and produces 'noise-free' (more than adequate S/N ratio) data. There have not been any problems observed with the magnitude of the SBW compressing the interference

3.4 Spectrophotometry

40

20 EL = 1 EL = 3 EL = 10

SBW (nm)

16 12 8 4 0

500

1000

1500 2000 Wavelength (nm)

2500

3000

Figure 3.10: Variation of SBW with  for a range of energy levels in 'auto' mode, where the SBW is xed in the UV-Vis region. A signicant step at the detector change-over wavelength dc = 850nm can be seen.

fringes, and the period of the fringes at 850nm is typically of the order of 100nm. In the worst case, this may result in an interference fringe in the 850-900nm  range being slightly compressed, but since the spectra extend both well above and below this region the error induced would be minimal.

3.4.3 The Step! All UV-Vis-NIR spectrophotometers that use two detectors, so their hapless users are faced with a multitude of problems when trying to avoid the appearance of a 'step' in transmission or reection data at the detector/grating change-over. This investigation was complicated by the fact that the detector and grating change-over wavelengths could not be de-coupled. The possibility of a step was already mentioned in the previous section, where a sudden change to a larger SBW illuminates a greater area of the sample which may be of a dierent average thickness, thereby changing the value of the transmission. Other possibilities, all investigated by the author, include: i. dierent optical paths for visible and NIR light ii. misaligned detectors iii. polarisation eects iv. blackbody radiation and, v. birefringent samples. It is believed that the Cary is slightly optically misaligned, and that the sample beam has a slight horizontal oset when changing between the gratings. This was determined by measuring a non-uniformly thick sample in several positions, and

3.4 Spectrophotometry

41

D

E

Figure 3.11: Measurement positions of a sputtered silicon on glass sample with darker rings representing greater thickness and the white region the illuminated area.

Transmission (%)

100 Measured at (a) Measured at (b)

80 60 40 20 0

500

600 700 800 Wavelength (nm)

900

Figure 3.12: Evidence of a step due to horizontal beam oset at the detector change-over, dc = 800nm.

nding that when the beam was aligned perpendicularly to the visible interference fringes from in the lm no step resulted, while a beam aligned within a single fringe produced a 5% step, Figures 3.11 and 3.12. A change in SBW at dc can be ruled out as the cause of the step as it is the average optical thickness nf d", and thus the period of the fringes, which has changed. This can be explained by observing the curve in Figure 3.12, just above 800nm where it has reached a local maximum, and then noticing that just below 800nm it reaches another maximum. This occurs when d" has changed, therefore beam is no longer centred on the same location. Misalignment of detectors, a problem also experienced with the Cary in 1997, will produce a step at dc if the dierence between the visible and NIR signal levels is too great for the Cary to correct for. Various electrically erasable programmable read-only memory (EEPROM) chips which store tabulated values required by the software to correct for the step were updated, although no improvement was noticed. Extensive alignment of both detectors was performed by Varian Australia and the problem has since disappeared.

3.4 Spectrophotometry

42

Polarisation can often cause a step since the PMT is more sensitive to linearly polarised light than unpolarised light. The baseline takes into account any polarisation of the light from the lamps, however if the sample introduces any polarisation change an increase in transmission in the visible region is possible. Although this is not usually a problem for normally incident light, it is a real concern for light incident at angles greater than 30, and in this situation a depolariser should be placed between the sample and the detector. An Oriel broadband (250 ; 2500nm) depolariser (Oriel Corporation 1990) was purchased and tested, but no reduction in the step was observed. Blackbody radiation emitted from the chopper is problematic when small (less than 1nm) SBWs are used, as this radiation comprises a signicant fraction of the total light striking the detectors. This was not an issue in the transmission measurements performed though, as a SBW2 was always employed. To date, there has not been a step which can be attributed to being caused by the use of a highly birefringent sample, for example sapphire. Such a sample could produce a step, either due to the emerging beams striking an increased area on the detectors, or polarisation eects. The sapphire samples measured in this work were less than 0.4mm thick, so any shift in the path taken by a beam would be small and thus the beam is essentially striking the same area of the detector. This may not be the case for samples of the order of a few millimeters thick.

3.4.4 Scattering Samples and Aperturing Light striking a slightly rough interface will be partly specularly transmitted and reected, and partly diusely transmitted and reected. The fraction of scattered light and its angular distribution depend not only the refractive index, but also the degree of surface roughness and angle of incidence (Tao 1994). In transmission mode, light that is not specularly transmitted will not be collected and is therefore lost. Accurate envelope method analysis cannot be performed on samples which are scattering. A helium-neon (He-Ne) laser, incident at 20 on the lm, was used to check any degree of scattering in the sputtered samples. The use of the integration sphere to collect nearly all the diusely transmitted light may be useful for scattering samples, however this has not been performed and one would need to consider the eect that scattering has on the interference fringes. It would also be necessary to determine what fraction of the scattered light passed out the front of the sample, not being collected. Aperturing of the sample in the focal plane using a sheet of black card was performed when it was absolutely necessary to ensure that the same area of the sample was illuminated at all . This also introduces the question of scattering from the edges of the aperture, however samples measured in this manner have been successfully analysed. The method for checking the overall accuracy of the results will be described later in this chapter.

3.5 Other Characterisation Techniques Used

43

3.4.5 Rotating Sample Holder A new sample holder, shown in Figure 3.13, was designed and constructed in order to measure the transmission at various angles of incidence. An manual x-y translation stage with 13mm of travel was employed to be able to position the front surface of a sample (up to 5mm thick) about the axis of rotation. The rotation stage could be read to an accuracy of greater than 0.1. The motivation for this was to use the angular rotation method developed by Swanepoel (1985), briey described in Section 2.2.1. Due to the extended length of time that the Cary was out of action, this method has been developed but not tested.

Figure 3.13: Rotating sample holder designed to t inside the Cary sample compartment.

3.5 Other Characterisation Techniques Used Various other characterisation techniques were used to analyse the samples. These include FTIR and Raman spectroscopy, sheet resistance meter, and a thickness proler.

FTIR Spectroscopy As mentioned previously, FTIR spectroscopy was used to observe the absorption peaks in the 1737 glass in the IR region of the electromagnetic spectrum. A Nicolet Model 520 machine was used to perform this experiment on two clean glass plates.

Thickness Proling A Dektak IIA thickness proler was used to determine the thickness of the sputtered silicon samples. To perform a measurement it is necessary to etch a hole through

3.5 Other Characterisation Techniques Used

44

the silicon lm to the substrate. This can be achieved with either a drop of boiling potassium hydroxide being placed on the lm, or by using the oxygen plasma etching machine with a hole drilled in a mask placed over the front surface. The thickness was not measured for all samples due to the destructive nature of the method, as samples could not then undergo further optical characterisation.

Sheet Resistance Meter To measure the sheet resistivity s ()=2) of the sputtered lms, a four-point-probe (FPP) was used, where a known current is passed through two outer probes and the voltage drop measured across two inner probes. From s , the resistivity of the lm is f = d s ().cm), where d is the lm thickness.

Raman Spectroscopy The Raman spectra of the samples were measured with a Renishaw Model 2000 machine located in the School of Materials Science at UNSW. Raman spectra are useful for analysing the degree of crystallinity of the poly-Si and any stress or strain that may be present in the lm. An IR laser,  = 780nm, was used as the source so that the beam was approximately uniformly absorbed throughout the whole thickness of the lm. Typical Raman spectra for a-Si, poly-Si, and c-Si are shown in Figure 3.14. Raman lines are usually expressed as a frequency shift in wavenumbers, having the units of inverse centimeters, where 1cm;1 = 10000/ (m), relative to the laser line at 0cm;1. The Raman peaks arise from the interaction of a photon with an optical phonon(s), resulting in a phonon(s) either being absorbed or emitted in the process. The Raman spectrum of a-Si is closely related to the phonon density of states (DOS). The three broad peaks of a-Si at 150cm;1y, 300cm;1, and 480cm;1 correspond to interactions of a photon with a transverse acoustic (TA), longitudinal acoustic (LA), and transverse optical (TO) phonons, respectively. An estimate of the degree of disorder in the material can be obtained from the ratio of TO to TA peak heights (Baba et al. 1994), and another measure is the width of the TA peak at half-maximum (Beeman et al. 1985). In crystalline material, energy and momentum must be conserved, and since a photon has close to zero momentum the phonon must also have this same value. A single optical phonon near the zone centre satises this criterion, and this gives rise to a high intensity peak at 520cm;1, along with other smaller peaks at 300cm;1 and 960cm;1, corresponding to multiple phonon processes (Green 1995). The broadening of the 520cm;1 peak on the low frequency side is attributed a relaxation of the zero phonon momentum rule, due to the presence of grain boundaries in the material. This means that phonons with momentum greater than zero can participate in a Raman scattering event. Thus, phonons of energy less than 64meV (520cm;1) are involved, leading to a broadening of the peak on the low wavenumber side (Benrakkad et al. 1994). For very small grain sizes a shift in the peak to y The TA peak at 150cm;1 appears to be closer to 180cm;1 in Figure 3.14, due to the cut-in

of the laser-line absorption lter

Norm. Raman Intensity (a.u.)

3.5 Other Characterisation Techniques Used

45

1.0

TO LA

0.8 0.6

TA a-Si poly-Si c-Si

0.4 0.2 0.0

200

300 400 500 -1 Frequency Shift (cm )

600

Figure 3.14: Typical Raman spectra for a-Si, poly-Si, and c-Si.

lower wavenumbers is also observed (Fauchet and Campbell 1990), however stress or strain in lms is another mechanism which has the same eect.

Chapter 4 Optical Characterisation using the Modied Envelope Method Since the optical characterisation methods described in Chapter 2 were limited in scope, it was necessary to develop a method that was capable of characterising a-Si and poly-Si lms, which are absorbing over the whole wavelength range. In this chapter, the development, details, and tests of the method will be described. In the next chapter, the method is applied to the experimental samples of a-Si and poly-Si on glass. The modied envelope method developed by the author, is an extension to the envelope method presented by (Swanepoel 1984), and is implemented as a Mathematica (version 2.2.3) program. Written by the author, the complete program listing is included in Appendix B. The program has been dubbed 'NAKED' as it nds the lm's refractive index nf , absorption coecient f , extinction coecient kf , the optical bandgap Eg , and thickness d { essentially revealing a lm's bare optical essentials. Although this is actually the method used to obtain results, it is, in itself, the major result of the present author's work and is therefore presented its own chapter.

4.1 Overview of Procedure Figure 4.1 is a ow diagram showing the necessary steps for nding the optical constants and thickness of a thin lm. Parallelograms and diamonds in the diagram are used to indicate that an input or a decision, respectively, from the user is required, while T is an abbreviation for transmission. An improved version of the original ENVELOPE (McClain et al. 1991) algorithm is used to accurately t envelope curves to the transmission spectra (McClain 1997). Several executions of the ENVELOPE program are necessary before the tightest tting tolerances can be found, but this procedure does not take more than ve minutes. The output le from ENVELOPE is then used as the primary input to NAKED, along with ns and s data for the substrate. If the correct #d has been supplied then the transmission extrema calculated from the determined optical constants and thickness should lie exactly on the original transmission spectrum. If this is not the case, the program 46

4.2 ENVELOPE Fitting

47

will have to be re-executed and a lower value for #d used. This is explained fully in Section 4.3.2.

4.2 ENVELOPE Fitting The ENVELOPE algorithm is normally run in a DOS window operating with Windows 95. Modications to the program were performed by the author in conjunction with McClain (1997), and several important improvements were made: i. to obtain accurate results the maximum number of data points was increased from 1200 to 3000, permitting analysis of spectra with a 1nm wavelength interval ii. in the original version, each data point was replaced by the average of itself and its two neighbouring data points (McClain 1997). This was designed to handle noisy data sets, however it often resulted in the extrema being intersected in two places by the 'tangential' envelope curve. The section of code performing this operation was commented out and the program re-compiled. iii. the y-axis (transmission) on graphical output was altered so that the scale would always be between 0% and 100% and, iv. the size of the array which performs the nal smoothing of the envelope curves was increased so that smoother ts of the envelopes to the transmission spectra could be obtained. Previously, a problem was encountered with the accuracy of the generated envelope curves, and tting errors of up to 0.3% (absolute transmission) were observed. Increasing the array size so that lower nal smoothing tolerances could be used greatly improved the situation. A comparison of two sets of envelope curves tted with the old and new versions is given in Figure 4.2. The rst step taken in tting the envelope curves involves the experimental data from the Cary being sorted by ascending wavelength and manually deleting the data for transmission values less than 0.05%. The latter is required since the uctuations in the signal at this level due to noise are treated by the ENVELOPE algorithm as being extrema. Both the initial and nal tolerances should be decreased to values as low as possible to obtain the most accurate envelope curves possible. When accurate envelope curves are achieved, the output le (*.tan extension) containing the tangential points is saved. This format has three columns of data, , Tmax, and Tmin , and is used as the input to NAKED. It is necessary to manually delete the data associated with the rst and last extrema as the transmission values are inaccurate due to the lack of two neighbouring extrema to provide an accurate t (McClain et al. 1991). A PostScriptTM graphic of the transmission spectra and envelope curves can also be generated by ENVELOPE.

4.2 ENVELOPE Fitting

48

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(19(/23(DOJRULWKP WRILWHQYHORSHFXUYHV

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1R

0:9eV of the a-Si lms also decrease as the temperature is raised, while at lower energies many structures observed in the f curves remain unexplained. The range of optical bandgaps, determined from Tauc plots, for the a-Si samples was 1:41 ; 1:54eV and in good agreement with at least two other published works

80 (Kanicki 1992, Schoenfeld et al. 1994). As-deposited sputtered poly-Si was also investigated, however no trends were observed here since two of the samples deposited at temperatures above 500C were unable to be characterised due to a signicant fraction of the transmitted beam being scattered. Furnace annealing (about 600 for 60 ; 80 hours) of several a-Si samples was performed in an environment of nitrogen at atmospheric pressure. The nf for all annealed samples were determine to lie within 0.1 (absolute) higher than c-Si over the  range where extrema can be resolved. Above 1.4eV, the f of the annealed samples is between two to three times greater than c-Si, while dangling bond densities were lowest in samples 06-06-4 and 05-01-3 in the sub-bandgap region (Jackson et al. 1983). Jackson et al. (1983) explained that the reduction in f , relative to a-Si, is primarily due to a decrease of amorphous material in the lm. Using Raman spectroscopy, samples 06-06-4 and 05-01-3 were found to exhibit the lowest amount of a-Si features and also had the narrowest 520cm;1 peak widths. Thus, the lms that exhibited the most crystalline properties were those deposited as a-Si without substrate heating. The optical bandgaps of the poly-Si samples were found to be in the range 0:93 ; 1:13eV. These values are in good agreement with, rstly, the value of 0:97  0:03 for poly-Si presented by (Jackson et al. 1983), and secondly the 1.12eV value for c-Si. Thus, the modied envelope method, realised through the computer programs ENVELOPE and NAKED, has successfully provided extensive optical information over a wide wavelength range for the non-uniformly thick, defected a-Si and poly-Si sputtered lms.

Appendix A Thin Film Optical Theory A.1 Introduction This appendix will discuss thin lm optical theory and its application to semiconducting lms on foreign substrates. As the eld of thin lm optics is very wide, the following models of increasing complexity will be presented, with the goal being to eventually be able model a thin absorbing lm on a nitely thick, foreign substrate:

   

a single interface a single layer or lm a single lm on a foreign substrate of semi-innite thickness a single lm on a thick foreign substrate.

These cases will be discussed for transparent and absorbing lms, obliquely and normally incident light, and all interfaces are assumed to be optically at and parallel. Finally, equations will be derived that accurately describe the thin lm interference fringes that are experimentally observed in reection and transmission spectra (Stenzel 1996).

A.2 Re ectance and Transmittance at a Single Interface An optical medium can be described by its complex refractive index n~ = n ; ik, where the real component n is called the refractive index. The imaginary component k is called the extinction coecient, and is related to the absorption coecient  by , (A.1) k =  4 where  is the wavelength of the light in a vacuum. 81

A.3 Single Transparent Layer

 ⊥

82  __

7__

θ

L

θ

U

θ

W

7⊥

3 3

L

W

9__

t⊥

Figure A.1: Fresnel coecients illustrated by incoming waves with Eelds normal to and parallel the plane of incidence.

When light is incident on an interface, the reectance and transmittance can be described using the Fresnel coecients. The Fresnel coecients for a single interface between two non-absorbing media, shown in Figure A.1, are i ; ni cos t , rk = nnt cos (A.2) t cos i + ni cos t cos i tk = n cos2n i + , (A.3) t i ni cos t i ; nt cos t , (A.4) r? = nni cos i cos i + nt cos t cos i t? = n cos2n i + , (A.5) i i nt cos t where r and t are the electric eld coecients for reection and transmission, respectively. The transmittance of an interface is T =1;R , (A.6) where the reectance at the interface is given by R = r2. For normally incident light, i = t = 0, of any polarisation 2  n i ; nt (A.7) R = n +n . t i For absorbing media, n~ instead of n in the Fresnel coecients, and then the reectance is given by R = r r.

A.3 Single Transparent Layer A single lm of transparent (non-absorbing) material with two at interfaces is shown in Figure A.2, where T12 and R12 are the transmission and reection coefcients for the interface between medium 1 and 2, respectively. The reectance is

A.3 Single Transparent Layer

83

dened as follows, noting that it is a geometrical series,

R = R12 + T12 R21 T21 + T12 R213 T21 + = R12 + T12 R21 T21 = R12 + T112;R21RT2 21 21 2 R 12 : = 1 + R12 And similarly for transmittance,

1 X j =1

2 j ;1 (R21 )

(A.8)

T = T12 T21 + T12 R212 T21 + = T12 T21

1 X j =1

2 j ;1 (R21 )

= 1T;12 TR212 21 1 ; R 12 . = 1 + R12

(A.9)

It should be noted that Eqns. A.8 and A.9 are independent of thickness, and do not include interference eects due to phase dierences between the reected and transmitted rays. For transparent media T + R = 1, and for the special case of normally incident light

R12 T12 θ



n1 = R21 = nn2 ; 2 + n1

2



,

(A.10)

n1 = 1 ; R12 = 1 ; nn2 ; 2 + n1 #

L

θ

W

#

2

%

%

= 4n1 n2 2 . (n2 + n1)

(A.11)

0HGLXP33 0HGLXP33 0HGLXP33

Figure A.2: Reectance and transmittance of a single layer.

It should be emphasised that T12 is not the same as jt12 j2, as jt12 j2 = 4n21 = (n1 + n2 )2, whereas the transmittance T is dened as being the ratio of the transmitted power

A.4 Single Absorbing Film

84  ~ E



 H~  = n = (cE 2), then, as

to the incident power. Dening intensity as I = shown in Eqn. A.11, 2 n2 jt j2 = 4n1 n2 . I n 2 cos t 2 jE2 j T12  I cos  = 2 = n1 jE1 j n1 12 (n2 + n1 )2 1 i

(A.12)

For a transparent layer (n2 = n) in air (n1 = 1), the reectance and transmittance for the situation depicted in Figure A.2, substituting Eqns. A.10 and A.11 into Eqns. A.8 and A.9, are ;

2

2 2 nn;+11 2 R ( n ; 1) 12 R = 1 + R = ; n;1 2 = n2 + 1 , 12 1 + n+1

;



1 ; nn;+11 2 1 ; R 12 T = 1 + R = ; n;1 2 = n22+n 1 . 12 1 + n+1

(A.13) (A.14)

A.4 Single Absorbing Film To improve the model developed so far, to be able to deal with absorbing lms, such as semiconductors, the earlier values of transmittance and reectance between medium 2 and 1 become, for a lm of thickness d T21 ! T21 exp;d and R21 ! R21 exp;d : Using the relation T12 = 1 ; R12 , which is still valid, the equivalent equations to Eqns. A.8 and A.9 are, in the absorbing case (not including interference eects) ;d ;d R = R12 + T12 R21 exp T;21de 2 1 ; (R21 exp )   R12 1 ; exp;2d (2R12 ; 1) , (A.15) = 2 exp;2d 1 ; R12 ;d T = T12 T21 exp;2d 2 1 ; (R21 exp ) R12 )2 exp;d . (A.16) = (11 ; ; R122 exp;2d

Layers and Thin Films Figure A.3 shows the transmission of a 1.1mm thick plate of Corning 1737 aluminosilicate glass that was measured by the author with various spectral bandwidths (SBW) (see Section 3.4.2) using a spectrophotometer. With a large SBW of 5nm (# = 2:5nm), we observe that the transmission is constant at T = 91.8%, and rearranging Eqn. A.14 for n r 1 n = T + T12 ; 1 , (A.17)

A.4 Single Absorbing Film

85

Transmission (%)

92.4 92.0 91.6 91.2 2000

2001

2002 2003 Wavelength (nm)

2004

2005

Figure A.3: Comparison between the experimentally measured transmission spectra for various spectral bandwidths: SBW = 5nm (solid), SBW = 2nm (dashed), SBW = 0.5nm (dotted).

gives a value of n = 1:52 which is a reasonable value for the glass in the infrared (IR) region. However, the theory developed up until now cannot correctly predict the spectra for the lower SBWs. Previously, the model was only capable of nding the intensity of reected and transmitted beams, and the phase shift of the light was ignored. The constructive or destructive interference, observed only when the SBW is small enough to resolve the fringes, occurs when the lm or layer is optically coherent. A lm is considered to be coherent if its optical thickness is much smaller than the coherence length of the light. The coherence length is determined by the light source and the SBW, and, for example, for a tungsten lament lamp it is of the order of a mm (Tao 1994). The coherence length of the light of the monochromator is given by (Tao 1994) 2 (A.18) lc = # 21 . When the distance between the front and rear surfaces is large, many interference fringes can occur over the given  range. However, when the pathlength dierence is greater than lc, the phases of the waves are randomised and cannot interfere with each other. In this situation, waves behave as incoherent sources, and the intensities, rather than the amplitudes, of the waves should be added (Rancourt 1996). Therefore, if the optical path length nd through the lm is much less than lc then the lm is coherent, creating the following condition for observing the maximum thin lm interference 2 . #  2nd (A.19) In the previous example, where # = 0.5nm, n = 1.52, d = 1.1mm, and  = 2000nm, this condition is not met, as 0.5nm is 6 0.38nm. Practically however,

A.4 Single Absorbing Film

86

all lms that have the same order of thickness as the wavelength will exhibit some interference (Stenzel 1996). Before implementing this in the 1737 glass transmission case, the eect of thin lm interference in a simple non-absorbing lm, as shown in Figure A.4, will be determined. Multiple reections are neglected for the moment, and light is normally incident, however it is drawn obliquely to clarify the diagram. 979H[SLω3/ . 7

3

3



3

Figure A.4: Thin lm interference in a non-absorbing lm. As we would like to check whether the period observed in Figure A.3 is correct, the two main reected beams of A.4 are added together, obtaining i2!nd=c E = r12E0 ;+ t12 r21 t21 E0 exp   r12E0 1 + a exp;i2 ,

(A.20)

where is the phase change in radians after one pass across a lm of thickness d. From the diagram in Figure A.5, the phase change can be found geometrically. The path dierence of the two beams multiplied by 2= (for n1 = 1) gives θ

[

L

/

θ

U

θ

U

D 

θ

U

D 

G

Figure A.5: Geometric derivation of an expression for the phase change .

2 = 2 (n2 a ; x) = 2v (n2 a ; x) , where a = cos2d , x = L sin i, and L = 2d tan r giving, r

A.5 Single Film on a Semi-Innitely Thick Substrate

87



where



2n2d ; 2d sin  sin  = v cos r i  r cos r p = 2vd 1 ; sin2 r , p sin r = sinn i , and n2 cos r = n2 1 ; sin2 r .

(A.21)

2

It should be noted that possible absorption is taken into account here when the complex refractive index is used, and that for case of normally incident light and negligible absorption Eqn. A.21 reduces to = 2nd=. The reected intensity is then given by ;



I / jE j2 1 + a2 + 2a cos 2 ,

(A.22)

which is a periodically changing function. Since neighbouring maxima and minima must be separated by  radians, =)

  4   1



 ;  1  nd =  min max jmin ; max j = 4nd .

(A.23)

For the observed minima min = 2003.0nm and maxima max = 2002.4nm, n = 1.52, and rearranging Eqn. A.23 for d, we get a value of 1.096mm, which lies well within Corning's thickness tolerances, 1.10.1mm. Alternatively, for a known d, this method can be used to get an estimate of n in regions of minimal dispersion (small variation of n with ).

A.5 Single Film on a Semi-Innitely Thick Substrate This case is depicted in Figure A.6, where incoming light is incident on medium 2, a thin lm, and some of the light is transmitted through to medium 3, a substrate of semi-innite thickness d. The reectance and transmittance for this case can be written as

r123 = r12 + t12 expi r23 expi t21 + t12 expi r23 expi r21;expi r23 expi t21 +  = r12 + t12 r23 t21 expi 1 + r21 r23 exp2i + 2i (A.24) = r12 + 1t12;rr23 t21r exp i , 21 23 exp t123 = t12 expi t23 h+ t12 expi r23 expi r21 expi t23 + i  ; 2 ; i i i i i = t12 exp t23 1 + r21 exp r23 exp + r21 exp r23 exp + i t23 = 1 ;t12r exp , (A.25) 2i 21 r23 exp

A.5 Single Film on a Semi-Innitely Thick Substrate

88

7

θ

  L

0HGLXP33

9

7

9

7

/

7

θ

V

9



0HGLXP33  ILOP 0HGLXP33  VXEVWUDWH

9

Figure A.6: Thin lm on a semi-innitely thick substrate.

where the phase dierence in absorbing media will be complex. The reected and transmitted intensities from the electric eld strength are found using the following identity from the Fresnel formulae t12 t21 = 1 ; r212 , (A.26) simplifying Eqn. A.24 to r23 exp2i . r123 = 1r12; + (A.27) r21r23 exp2i Using Eqn. A.21, we can write the transmittance, reectance, and absorptance as s jt j2 , T = nn3 cos (A.28) 123 1 cos i

R = jr123 j2 ,

(A.29)

A = 1;T ;R .

(A.30)

Practically, the incident medium is very often air (n1 = 1) and the substrate is essentially non-absorbing (n3 is real). For this system with a non-absorbing lm, Eqns. A.28 and A.29 simplify to n3 cos s t2 t2 (A.31) T = 1 + r2 nr12cos+ i2r12 23r cos , 12 23 12 23 2 r232 + 2r12r23 cos 2 . R = 1r12+ + (A.32) r122 r232 + 2r12 r23 cos 2 Examining the period of oscillation again, the model presented in Eqn. A.23 can now be improved to include absorption and phase dierence. For neighbouring extrema    1  1  (A.33) ;   = p 2 1 2  min max 4d n2 ; sin i

A.5 Single Film on a Semi-Innitely Thick Substrate

89

and for any two extrema   1  1 #m   p , (A.34)  ;   = a b 4d n22 ; sin2 i where #m ; 1 = no. of extrema observed between extrema a and b =) #m = 1 for neighbouring extrema: For a material with minimal dispersion and normally incident light, the  of the mth extrema is at (A.35) m = 4nm2d for m = 1, 2, 3, : : : . However, under strong absorption Eqns. A.33 and A.34 are not valid, as a constant n is assumed, and in this case  s  1 n22    a a

s

; sin2 i ; 1 n ; b b 2 2

   2 sin i  

= #4m d .

(A.36)

Thus, for a spectrum with normal dispersion, the interference fringes will appear increasingly closer together as j increases, and as the lm becomes more absorbing.

Half and Quarter Waveplates In thin lm optics, two very important cases exist for half-wavelength /2 and quarter-wavelength /4 lms. A /2 lm can be realised with even m-values in Eqn. A.34, as then optical thickness nd of the lm is half that of the incident wavelength. The oscillating term is a maximum, cos 2 = 1, and from Eqn. A.31 2 2 n t 3 12 t23 T=n (A.37) 2 , 1 (1 + r12 r23 ) which for normally incident light becomes (A.38) T = 4n1n3 2 . (n1 + n3 ) This is the same result as would be obtained using Eqn. A.11 for the interface n1 : n3 , so a /2 lm is optically totally inactive. To determine whether this corresponds to a maxima or a minima in a spectrum, the denominator in Eqn.A.31 should be examined. When this, or the r12 r23 product, is small there will be a transmittance maximum. This is true for n1 < n2 > n3, which is the case for a silicon lm on glass in air. However, when n1 < n2 < n3 , for example an antireection (AR) coating (e.g. for TiO2 n 2.4) on silicon in air, the r12r23 product is smaller, and a minima will be observed at the same  in the transmittance spectrum. A /4 lm will occur when m has odd values, the term cos 2 = -1, and for normally incident light, from Eqn. A.31 2 2 T = nn3 t12 t23 2 1 (1 ; r12 r23 )

A.5 Single Film on a Semi-Innitely Thick Substrate

90

4n1n22 n3 . (n1n3 + n22 )2

=

(A.39)

This result is especially interesting for lms where n2 = pn1 n3 , as the transmittance is unity, and the reectance must be zero. This is the principle of an AR coating.

Single Absorbing Film in Air Before developing the model to predict the transmittance and reectance of a lm on a thick substrate, a single absorbing lm is treated, including the eects of thin lm interference in the analysis. For normally incident light in this two-media case, the reectance and transmittance simplify to i , (A.40) t = 1 ;t12r t21r exp 2i 12 21 exp r21 exp2i . r = 1r12; + (A.41) r12 r21 exp2i For an absorbing material, is complex and has the form = 0 + 00. The transmitted intensity is then jt12 j2 jt21 j2 exp;2 T= , (A.42) 1 + jr12 j2 jr21 j2 exp;4 $< (r12 r21 cos 2 0) ; = (r12r21 sin 2 0)] where < and = denote the real and imaginary components, respectively. It is quite complex to expand the applicability of Eqns. A.15 and A.16 to be valid in the case of obliquely incident light, because the angle of refraction r and the refractive index n~ 2 become complex quantities. This will not be performed here, although from Eqn. A.41 the expression for a incoherent thick lm can be derived by assuming the phase of the incident light as being randomly distributed 0

00

Z1

jt12 j2 jt21 j2 exp;2 1 + jr12j2 jr21j2 exp;4 $< (r12 r21 cos 2 0) ; = (r12 r21 sin 2 0)] 0 p~ ;sin i 2 2 ; 4vd = n j t j j t j exp 12 21 p , (A.43) = 2 1 ; jr12j jr21 j2 exp;8vd = n~ ;sin i

T = 1

0

d 0

00

2

2 2

2

2 2

using the identity  Z  ap; b tan x2 + c dx 2  p = arctan . a + b cos x + c sin x a2 >b2 +c2 a2 ; b2 ; c2 a2 ; b2 ; c2 A similar expression for the reected intensity is

p p . ;8vd = n~ ;sin i

(A.44)

(A.45) R = jr12 j2 + jt12 j jr212j jt21 j2 2 2 2 1 ; jr12 j jr21 j exp For normally incident light, Eqns. A.43 and A.45 simplify to Eqns. A.16 and A.15, respectively, if the phase terms are neglected, which is valid for an incoherent lm. 2

2

2

exp;8vd = n~22;sin2 i

A.6 Thin Film on a Thick Substrate

91

A.6 Thin Film on a Thick Substrate The nal goal is now to derive equations that can describe the reectance and transmittance of an absorbing thin lm on a transparent thick substrate in air, shown in Figure A.7. The complex refractive index of the substrate and lm are indicated with n~ s and n~f , respectively, and ds is the substrate thickness. Substituting Eqns. A.24 and A.25 into Eqns. A.45 and A.43, respectively, the transmittance and reectance of this system is

jt123 j2 jt31 j2 exp;2=(s) , 1 ; jr321 j2 jr31j2 exp;4=(s) 2 2 2 ;4=(s ) R = jr123 j2 + jt123 j jr31 j2 jt3212j exp;4=(s) . 1 ; jr321 j jr31 j exp T =

θ

(A.46) (A.47)

$LU3  

L



/

)LOP 3  = 3 

/

6XEVWUDWH 3  = 3 8

V

1

$LU3   

Figure A.7: Diagram of a thin lm upon a thick substrate, d  ds.

These equations are valid for absorbing lms and substrates, as well as obliquely incident light, while polarisation must be taken into account with the correct choice of Fresnel coecients. The most popular form of this equation, for an absorbing lm (nf , kf ) on a transparent substrate (ns) was presented by Swanepoel (1983) 0 T = B 0 ; CA0xx+ D0x2 , (A.48) where

;



A0 = 16ns n2f + kf2 ,   ;   B 0 = (nf + 1)2 + kf2 (nf + 1) nf + n2s + kf2 , ; ;  ;  C 0 = n2f ; 1 + kf2 n2f ; n2s + kf2 ; 2kf2 n2s + 1 2 cos ,  ;  ; ;  ;kf 2 n2f ; n2s + kf2 + n2s + 1 n2f ; 1 + kf2 2 sin ,   ;   D0 = (nf ; 1)2 + kf2 (nf ; 1) nf ; n2s + kf2 , = 4n f d , x = exp;f d .

A.6 Thin Film on a Thick Substrate

92

Eqn. A.48 can be simplied by neglecting terms containing kf , since kf  nf ns over the whole spectrum (Swanepoel 1983), and the resulting expression is extensively used throughout the remainder of this work: Ax (A.49) T = B ; Cx + Dx2 , where

A B C D x

= = = = = =

16nsn2f , ;  2 (nf + 1);2 (nf + 1); nf + n s , 2 cos n2f ; 1 n2f ; n2s , ;  (nf ; 1)2 (nf ; 1) nf ; n2s , 4nf d ,  exp;f d .

Appendix B NAKED: Program Listingy NAKED - A Modified Envelope Method to Determine n, a, k, Eg, d ± delta d of Inhomogeneous, Absorbing Films  %U\FH5LFKDUGV 3KRWRYROWDLFV6SHFLDO5HVHDUFK&HQWUH 7KH8QLYHUVLW\RI1HZ6RXWK:DOHV __o WHO .HQVLQJWRQ _-\PointSize[0.01], Frame->True, GridLines-> Automatic,PlotRange->All]; SetOptions[Plot,PlotRange->All,Frame->True,GridLines->Automatic]; tanpt[[i,1]]}, {i,1,leng}]; tanptsoutput = TableForm[tanpts,TableSpacing->{0}, TableHeadings->{{}, {"\n nm", " Uncorrected: \n TM", "\n tm"," Corrected: \n TM","\n tm","\n s"}}] (* display table of corrected vs measured T values *)

nm 2182. 2050. 1933. 1830. 1738. 1655. 1580. 1513. 1451. 1395. 1343. 1296. 1252. 1211. 1173. 1138. 1106. 1075. 1047. 1020.

Uncorrected: TM tm 0.893468 0.332229 0.893124 0.333607 0.88873 0.333708 0.881635 0.332401 0.87092 0.330484 0.85633 0.327948 0.837099 0.324639 0.813773 0.320186 0.784841 0.314844 0.752976 0.309265 0.719539 0.303024 0.685353 0.296032 0.649361 0.288412 0.613136 0.280465 0.578214 0.272335 0.54476 0.264308 0.513496 0.256301 0.481974 0.24783 0.45222 0.239301 0.421606 0.230036

Corrected: TM tm 0.898755 0.334195 0.893943 0.333913 0.88945 0.333978 0.882078 0.332568 0.871435 0.330679 0.85696 0.328189 0.837682 0.324865 0.814303 0.320395 0.785968 0.315296 0.754533 0.309904 0.72067 0.3035 0.686099 0.296354 0.650141 0.288758 0.614008 0.280864 0.579128 0.272765 0.545672 0.264751 0.514381 0.256743 0.482796 0.248253 0.452966 0.239696 0.422268 0.230397

s 1.4841 1.48827 1.49088 1.49239 1.49326 1.49382 1.49426 1.49467 1.49514 1.49566 1.49625 1.49688 1.49756 1.49828 1.49901 1.49973 1.50044 1.50117 1.50185 1.50254

95 995. 972. 950. 930. 910. 892. 875. 859. 843. 828.

0.391011 0.35976 0.325178 0.29006 0.25102 0.213352 0.175637 0.140493 0.108245 0.079792

0.220146 0.209482 0.197316 0.18371 0.167414 0.149846 0.130754 0.109672 0.088428 0.070876

0.39159 0.360262 0.325601 0.290409 0.251294 0.213559 0.175782 0.140589 0.108305 0.0798271

0.220472 0.209774 0.197572 0.183931 0.167597 0.149991 0.130862 0.109747 0.0884771 0.0709072

1.5032 1.50382 1.50443 1.505 1.50558 1.50611 1.50662 1.50711 1.5076 1.50807

Divide transmission spectrum up into different regions of absorption: i)

Transparent region (where maxima should be limited only by transmission of substrate for a uniformly thick film)

(* Calculation to find thickness variation delta d *) a := 16*n^2*s; b := (n+1)^3*(n+s^2); c := 2*(n^2-1)*(n^2-s^2); d := (n-1)^3*(n-s^2); (* coefficients for transmission expression *) at := a/(b+d) /. {n->n2, s->tanpts[[i,6]]}; bt := c/(b+d) /. {n->n2, s->tanpts[[i,6]]}; (* want to solve minima and maxima for variables n2 (initial guess at nf) and delta d (wavelength independent) *) For[i=1; n2d={}, i{0,36},FrameLabel->{lam, "Thickness Variation (nm)"}];

ii)

Weak and Medium Absorption Region

(* Using delta d calculate initial guesses for nf and x in abs. region *) deltad = 15.3; ax := a*x/(b+d*x^2) /. {n->n3, s->tanpts[[i,6]]}; bx := c*x/(b+d*x^2) /. {n->n3, s->tanpts[[i,6]]}; (* coefficents including absorption term x=Exp[-alpha.d] *)

96 n3xfind = Table[FindRoot[{ tanpts[[i,4]] == tanpts[[i,1]]/(2*Pi*n3*deltad) * ax/((1-bx^2)^0.5)* ArcTan[(1+bx)/(1-bx^2)^0.5 * Tan[2*Pi*n3*deltad/tanpts[[i,1]]]], tanpts[[i,5]] == tanpts[[i,1]]/(2*Pi*n3*deltad) * ax/((1-bx^2)^0.5)* ArcTan[(1-bx)/(1-bx^2)^0.5 *Tan[2*Pi*n3*deltad/tanpts[[i,1]]]]}, {n3,3.5},{x,1}],{i,1,leng}]; n3x = Table[{tanpts[[i,1]], n3xfind[[i,1,2]], n3xfind[[i,2,2]]},{i,1,leng}]; (* n3x are nf and x values from solving the two eqns. for max. and min. *)

iii) Strong Absorption Region Clear[a,b,c,d,e,f,g,h,w,x,lambda]; (* clears values of variables used - useful if have to recalculated firstm *) strext = {}; lmint = {}; lmfit = {}; lmdata = {}; strabs = leng-4; (* strabs = number of points up until the strong absorption region which are not influenced by errors in T. Points at wavelengths shorter than this can be considered to be interference free and calculated using the mean of the extrema. *) ndata = Table[{n3x[[i,1]],n3x[[i,2]]}, {i,1,strabs}]; nform = NonlinearFit[ndata, f/x^4 + g/x^2 + h, x, {{f,1000},{g,3},{h,2}}]; nfit = nform[[1,2]]/lambda^4 + nform[[2,2]]/lambda^2 + nform[[3,2]]; (* Since no nf data is available in the region of strong absorption there is no means of finding nf here. Therefore, nf must be extrapolated into this region using a Cauchy fit (= nfit) to data at longer wavelengths *) n4 = Table[ReplacePart[n3x[[i]],nfit /. lambda->n3x[[i,1]],2], {i,strabs,leng}]; n5 = Table[n3x[[i]],{i,1,strabs-1}]; n5 = FlattenAt[AppendTo[n5, Table[n4[[i]],{i,1,Length[n4]}]], strabs]; (* replaces nf values found previously in strong absorption region with extrapolated values from nfit. n5 contains the new values of {Wavelngth (nm), nf, x} *)

Determining the order number and average thickness (* Plotting the equation l/2=2*d*(n/lambda)-m1 , where l = 0,1,2,3... , yields a straight line of slope 2*d (yielding the thickness) and y-intercept m1 (the first observable extrema in IR). m1 has an integer value for a maxima, and half-integer value for a minima *) lmdata = Table[{n5[[i+1,2]]/n5[[i+1,1]], N[i/2]},{i,0,leng-1}]; lmint = Fit[lmdata, {1,w},w]; firstm = -7.5; (* firstm = m1 (the order number of the first observable extrema in IR *) straight = 20; lmnonlin = NonlinearFit[Table[lmdata[[i]],{i,1,straight}],f*w + firstm, w, {f,1000}]; lmfit = firstm + lmnonlin[[1,2]]*x; (* fits line w. slope 2d and y-int. firstm to the first 'straight' points*) DisplayTogether[ListPlot[lmdata], Plot[lmfit,{x,0,0.006}, PlotStyle-> RGBColor[1,0,0]], AxesOrigin->{0,0}, FrameLabel->{FontForm["N/l", {"Symbol",10}],"l/2"}]; d2 = N[lmfit[[2,1]]/2]; (* determine thickness d2 from slope *) Print["Film thickness: ",NumberForm[d2,4,NumberPoint->"."]," ",nm];

Film thickness:

2157. nm

97 Improve nf values using the order values m and thickness d (* nf values can be improved using the equation 2*nf*d = m*lambda *) n6 = Table[{n5[[m,1]], N[(n5[[m,1]]*((m-(2*firstm+1))/2)/(2*d2))], n5[[m,3]]}, {m,1,leng}]; ntest = Table[{n6[[i,1]],n6[[i,2]]},{i,1,leng}]; Clear[f,g,h]; range = {Ceiling[tanpts[[1,1]]/100]*100,Floor[spectrum[[1,1]]/100]*100}; (* sets standard x-range to plot *) nform = NonlinearFit[ntest, f/x^4 + g/x^2 + h, x, {{f,1000},{g,3},{h,2}}]; nfit = nform[[1,2]]/lambda^4 + nform[[2,2]]/lambda^2 + nform[[3,2]]; DisplayTogether[ListPlot[ntest], Plot[nfit, {lambda,range[[1]],range[[2]]}, PlotStyle->RGBColor[1,0,0]], FrameLabel->{lam, "Ref. Index nf", "Extrapolation of n"," "}, PlotRange->{range,Automatic}];

Determining x in the strong absorption region i)

In the region where extrema exist

a := 16*n^2*s; b := (n+1)^3*(n+s^2); c := 2*(n^2-1)*(n^2-s^2); d := (n-1)^3*(n-s^2); (* Either Eqns. 12 and 13, or 18 and 19 from Swanepoel (1983) are used to calculate the absorption in the interference free region *) (* em := 8*n^2*s/tanpts[[i,4]]+(n^2-1)*(n^2-s^2); *) (* xstrabs = Table[(em-(em^2-(n^2-1)^3*(n^2-s^4))^0.5)/((n-1)^3*(n-s^2)) /. {n->n6[[i,2]], lambda->n6[[i,1]]}, {i,strabs,leng}]; *) ta := (tanpts[[i,5]]*tanpts[[i,4]])^0.5; (* ta is the geometric mean of the maxima and minima *) g := 128*n^4*s^2/ta^2+n^2*(n^2-1)^2*(s^2-1)^2+(n^2-1)^2*(n^2-s^2)^2; xstrabs = Table[(g-(g^2-(n^2-1)^6*(n^2-s^4)^2)^0.5)^0.5/((n-1)^3*(n-s^2)) /. {n->n6[[i,2]], lambda->n6[[i,1]]}, {i,strabs,leng}]; n6x = Table[ReplacePart[n6[[i]],xstrabs[[i+1-strabs]],3],{i,strabs,leng}]; nx = Table[{n6[[i,1]],n6[[i,2]],n3x[[i,3]]},{i,1,strabs-1}]; nx = FlattenAt[AppendTo[nx, Table[{n6x[[i,1]], n6x[[i,2]], n6x[[i,3]]}, {i,1,Length[n6x]}]],strabs];

ii)

In the interference free region

(* Using nfit, extrapolate n into strong absorption region and calculate x from interference-free T data *) strongpts = Table[{spectrum[[i,1]], spectrum[[i,2]]/100}, {i,(Last[n6][[1]] spectrum[[1,1]]),1,-1}]; (* find x in the region below where the first extrema is resolved (with ENVELOPE)*) (* em := 8*n^2*s/strongpts[[i,2]]+(n^2-1)*(n^2-s^2); *) (* xstrong = Table[{strongpts[[i,1]], nfit /. lambda->strongpts[[i,1]], (em-(em^2-(n^2-1)^3*(n^2-s^4))^0.5)/((n-1)^3*(n-s^2)) /. n->nfit /. lambda->strongpts[[i,1]]}, {i,1,Length[strongpts]}]; *) ta := strongpts[[i,2]]; g := 128*n^4*s^2/ta^2+n^2*(n^2-1)^2*(s^2-1)^2+(n^2-1)^2*(n^2-s^2)^2; xstrong = Table[{strongpts[[i,1]], nfit /. lambda->strongpts[[i,1]], (g-(g^2-(n^2-1)^6*(n^2-s^4)^2)^0.5)^0.5/((n-1)^3*(n-s^2)) /. n->nfit /. lambda->strongpts[[i,1]]}, {i,1,Length[strongpts]}];

98 (* simply use the same expression as used above with the transmission curve as the input, ignoring any interference *) nx = FlattenAt[AppendTo[nx, xstrong], Length[nx]];

Display s, n, alpha, and k values at all wavelengths nak = Table[{nx[[i,1]], s /. lambda->nx[[i,1]], nx[[i,2]], N[-Log[nx[[i,3]]]/(d2*1 10^-7)*nx[[i,1]]*1 10^-7/(4*Pi)], 1239.75/nx[[i,1]], -Log[nx[[i,3]]]/(d2*1 10^-7)},{i,1,Length[nx]}]; nakoutput = TableForm[nak,TableSpacing->{0},TableHeadings->{{}, {"nm","s","n","k","eV","a (1/cm)"}}] For[imcount=leng, Equal[Abs[Im[nak[[imcount,6]]]],0] && imcountRGBColor[1,0,0]], ListPlot[nplot], FrameLabel->{lam, "Refractive Index, n"},PlotRange-> {range,{3.4,Ceiling[nak[[nakleng,3]]*10]/10}}];

101 10 7.94496 10 207709. n(lambda) = 3.749 + ------------ + ------4 2 lambda lambda

Clear[h,j,k]; (* Stop plotting nak data set before points reach infinity!*) alphaev = Table[{nak[[i,5]], nak[[i,6]]},{i,1,infcount}]; (* alpha data in cm-1 *) evrange=N[{Floor[nak[[1,5]]*100]/100, Ceiling[nak[[nakleng,5]]*100]/100}]; alpharange = {1, N[Ceiling[Log[10,nak[[infcount,6]]]*10]/10]}; DisplayTogether[LogListPlot[acsi, PlotJoined->True, PlotStyle-> RGBColor[1,0,0]], LogListPlot[alphaev], FrameLabel->{"Energy, eV",FontForm["a",{"Symbol",10}]}, PlotRange->{evrange,alpharange}, AxesOrigin->{1,1}]; (* alpha data displayed as a log alpha vs. eV plot is more useful *)

(* From alpha the type of transition can be determined: - direct transitions have a alpha^2 dependence, while - indirect transitions have an alpha^0.5 dependence. *) alphahalf = Table[{nak[[i,5]], Sqrt[nak[[i,6]]]},{i,1,infcount}]; csihalf = Table[{acsi[[i,1]], Sqrt[acsi[[i,2]]]},{i,1,csileng-1}]; alpha2 = Table[{nak[[i,5]], nak[[i,6]]^2},{i,1,infcount}]; csi2 = Table[{acsi[[i,1]], acsi[[i,2]]^2},{i,1,csileng-1}]; DisplayTogether[ListPlot[csihalf, PlotJoined->True, PlotStyle-> RGBColor[1,0,0]], ListPlot[alphahalf], PlotRange->{evrange,{0, Ceiling[alphahalf[[Length[alphahalf],2]]/10]*10}}, FrameLabel->{"Energy, eV","alpha^0.5"}];

102 DisplayTogether[ListPlot[csi2, PlotJoined->True, PlotStyle-> RGBColor[1,0,0]], ListPlot[alpha2], PlotRange->{{1,evrange[[2]]}, {0,Ceiling[alpha2[[Length[alpha2],2]]]}}, FrameLabel->{"Energy, eV", "alpha^2"}];

kdata = Table[{nak[[i,1]], nak[[i,4]]},{i,1,infcount}]; DisplayTogether[ListPlot[kcsi, PlotJoined->True, PlotStyle->RGBColor[1,0,0]], ListPlot[kdata], FrameLabel->{lam,"Ext.Coeff., k"}, PlotRange-> {range,{0,0.25}}];

(* The optical bandgap Eg can be determined from the x-intercept of a stright line fit to the high energy region *)

103 tauc = Table[{alphaev[[i,1]],Sqrt[alphaev[[i,2]]*alphaev[[i,1]]]}, {i,1,infcount}]; egplot = Table[{alphaev[[i,1]],(nak[[i,3]]*alphaev[[i,2]]* alphaev[[i,1]])^(1/3)}, {i,1,infcount}]; DisplayTogether[ListPlot[tauccsi, PlotJoined->True, PlotStyle-> RGBColor[1,0,0]], ListPlot[tauc], FrameLabel->{"Energy (eV)", "(a·E)^0.5","Tauc Plot",""}, PlotRange->{evrange,{0, tauc[[infcount,2]]+25}}]; DisplayTogether[ ListPlot[egplot], ListPlot[Egcsi, PlotJoined->True, PlotStyle->RGBColor[1,0,0]], FrameLabel->{"Energy (eV)", "(n·a·E)^(1/3)","Eg Plot",""}, PlotRange->{evrange,{0, egplot[[(infcount),2]]+5}}];

Check the solution found (* use determined parameters (d, delta d, n, alpha) as inputs to see if the transmission expression generates points which lie on the original T spectrum *) thickmin = d2 - deltad; thickmax = d2 + deltad; (* max and min thicknesses for integration *) a := 16*n^2*s; b := (n+1)^3*(n+s^2); c := 2*(n^2-1)*(n^2-s^2); d := (n-1)^3*(n-s^2); For[i=1; trans={}, i nak[[i,1]], s->nak[[i,2]], n->nak[[i,3]], alpha->nak[[i,6]]*10^-7}, {thick,thickmin,thickmax}]} ; (* generates T points by numerical integration *) trans = AppendTo[trans, transpts]; ];

104 transfit = Table[{trans[[i,1]], 200*s/(s^2+1) /. lambda->trans[[i,1]], 100*tanpts[[i,4]], 100*tanpts[[i,5]], trans[[i,2]]},{i,1,leng}]; transfit = FlattenAt[AppendTo[transfit, Table[{strongpts[[i,1]], 200*s/(s^2+1) /. lambda->strongpts[[i,1]], " ", 100*strongpts[[i,2]], trans[[leng+i,2]]},{i,1,Length[strongpts]-nakleng+infcount}]],leng+1]; (* Print table showing: i) substrate limited T; ii) maxima TM found by ENVELOPE; iii) minima found by ENVELOPE; iv) the extrema generated from the optical constants and thickness *) transout = TableForm[transfit,TableSpacing->{0},TableHeadings->{{}, {" nm"," Ts"," TM"," tm"," T pt"}}] WriteString[nakfile,"\n\n"]; Write[nakfile, Format[transout,OutputForm]]; WriteString[nakfile,"\n\n"];

nm 2182. 2050. 1933. 1830. 1738. 1655. 1580. 1513. 1451. 1395. 1343. 1296. 1252. 1211. 1173. 1138. 1106. 1075. 1047. 1020. 995. 972. 950. 930. 910. 892. 875. 859. 843. 828. 827 826 825 824 823 822 821 820 819 818 817 816 815 814 813 812 811 810 809 808 807 806 805 804 803 802 801

Ts 92.6824 92.5844 92.523 92.4875 92.4668 92.4537 92.4434 92.4336 92.4226 92.4102 92.3963 92.3814 92.3653 92.3484 92.3312 92.314 92.2972 92.28 92.2638 92.2475 92.2319 92.2171 92.2026 92.1892 92.1755 92.1629 92.1508 92.1392 92.1274 92.1162 92.1155 92.1147 92.1139 92.1132 92.1124 92.1117 92.1109 92.1102 92.1094 92.1086 92.1079 92.1071 92.1063 92.1056 92.1048 92.1041 92.1033 92.1025 92.1017 92.101 92.1002 92.0994 92.0987 92.0979 92.0971 92.0963 92.0956

TM 89.8755 89.3943 88.945 88.2078 87.1435 85.696 83.7682 81.4303 78.5968 75.4533 72.067 68.6099 65.0141 61.4008 57.9128 54.5672 51.4381 48.2796 45.2966 42.2268 39.159 36.0262 32.5601 29.0409 25.1294 21.3559 17.5782 14.0589 10.8305 7.98271

tm 33.4195 33.3913 33.3978 33.2568 33.0679 32.8189 32.4865 32.0395 31.5296 30.9904 30.35 29.6354 28.8758 28.0864 27.2765 26.4751 25.6743 24.8253 23.9696 23.0397 22.0472 20.9774 19.7572 18.3931 16.7597 14.9991 13.0862 10.9747 8.84771 7.09072 7.77183 7.54234 7.29633 7.04583 6.79384 6.54884 6.31181 6.09128 5.88625 5.69982 5.53154 5.3773 5.24612 5.13164 5.03147 4.94165 4.8685 4.79241 4.72642 4.65949 4.5884 4.5125 4.43009 4.337 4.23913 4.12909 4.00846

T pt 33.4763 89.4004 33.3748 88.2032 33.0318 85.6851 32.4456 81.4204 31.519 75.4274 30.3038 68.5943 28.8717 61.4112 27.3165 54.5868 25.6766 48.2918 23.9714 42.2421 22.0806 36.0499 19.7262 28.9676 16.6301 21.1857 12.7789 13.9084 8.87974 8.12169 8.39933 8.09299 7.74793 7.38671 7.02099 6.66684 6.33046 6.0239 5.74794 5.50623 5.29816 5.11936 4.97701 4.8642 4.77751 4.71203 4.67204 4.63778 4.61917 4.60339 4.58435 4.55824 4.52025 4.4632 4.39114 4.29494 4.17602

105 800 799 798 797 796 795 794 793 792 791 790 789 788 787 786 785 784 783 782 781 780 779 778 777 776 775 774 773 772 771 770 769 768 767 766 765 764 763 762 761 760 759 758 757 756 755 754 753 752 751 750 749 748 747 746 745 744 743 742 741 740 739 738 737 736 735 734 733 732 731 730 729

92.0948 92.094 92.0932 92.0924 92.0917 92.0909 92.0901 92.0893 92.0885 92.0877 92.087 92.0862 92.0854 92.0846 92.0838 92.083 92.0822 92.0814 92.0806 92.0798 92.079 92.0782 92.0774 92.0766 92.0758 92.075 92.0742 92.0734 92.0726 92.0718 92.071 92.0702 92.0694 92.0686 92.0678 92.0669 92.0661 92.0653 92.0645 92.0637 92.0629 92.062 92.0612 92.0604 92.0596 92.0587 92.0579 92.0571 92.0563 92.0554 92.0546 92.0538 92.0529 92.0521 92.0513 92.0504 92.0496 92.0487 92.0479 92.047 92.0462 92.0454 92.0445 92.0437 92.0428 92.042 92.0411 92.0403 92.0394 92.0385 92.0377 92.0368

3.88561 3.75627 3.6273 3.48765 3.35817 3.23202 3.10856 2.99284 2.88091 2.77933 2.6829 2.59511 2.51369 2.43763 2.36874 2.30008 2.23479 2.16973 2.10805 2.04418 1.98052 1.9135 1.85047 1.78331 1.71507 1.64765 1.58098 1.51369 1.45193 1.38886 1.32894 1.27183 1.21698 1.16653 1.11439 1.06754 1.02375 0.980809 0.940306 0.901067 0.864301 0.82713 0.791382 0.757283 0.723333 0.691153 0.658377 0.627243 0.596724 0.56671 0.538966 0.512451 0.484412 0.459802 0.435826 0.413552 0.392177 0.370583 0.351674 0.333306 0.314781 0.297683 0.281442 0.264991 0.251928 0.23579 0.22475 0.210344 0.19797 0.186435 0.175436 0.164331

4.0443 3.89687 3.74335 3.57459 3.41513 3.25978 3.10966 2.97108 2.8409 2.72591 2.62106 2.52948 2.44859 2.37688 2.31543 2.25686 2.20332 2.15088 2.10175 2.04958 1.99597 1.93673 1.87901 1.81436 1.746 1.67624 1.60547 1.53287 1.4653 1.39631 1.33087 1.269 1.21038 1.15722 1.10357 1.05622 1.01285 0.971117 0.932383 0.895276 0.860773 0.825824 0.792065 0.759597 0.726833 0.695367 0.662822 0.631514 0.600485 0.569727 0.541128 0.513728 0.484856 0.459542 0.435024 0.412387 0.390831 0.369227 0.350432 0.332279 0.314036 0.29724 0.281288 0.265089 0.252226 0.236217 0.225256 0.210851 0.198437 0.186828 0.175732 0.164518

106 728 727 726 725 724 723 722 721 720 719 718 717 716 715 714 713 712 711 710 709 708 707 706 705 704 703 702 701 700 699

92.036 92.0351 92.0342 92.0334 92.0325 92.0316 92.0308 92.0299 92.029 92.0281 92.0273 92.0264 92.0255 92.0246 92.0237 92.0229 92.022 92.0211 92.0202 92.0193 92.0184 92.0175 92.0166 92.0157 92.0148 92.0139 92.013 92.0121 92.0112 92.0103

0.153455 0.14423 0.135871 0.127172 0.117534 0.112169 0.104299 0.0975719 0.0908326 0.0852437 0.0814315 0.0739168 0.0689101 0.0642004 0.0597158 0.0553093 0.0501304 0.0472533 0.0425856 0.0395955 0.0396866 0.0350849 0.0325719 0.0303991 0.0275961 0.0263936 0.0245279 0.0224181 0.0194409 0.0176281

0.153562 0.14424 0.135825 0.127083 0.117429 0.11207 0.104217 0.0975471 0.0907794 0.0852941 0.0814184 0.0739473 0.0690235 0.0641994 0.0597314 0.0554963 0.0501494 0.0473273 0.0421335 0.0399223 0.0397346 0.0352209 0.0334976 0.0308689 0.0280494 0.0269731 0.025872 0.0225868 0.0224817 0.017333

If[Mod[firstm,1]== 0.5, transerror = Sum[(transfit[[i,4]] - transfit[[i,5]])^2+ (transfit[[i+1,3]] - transfit[[i+1,5]])^2,{i,1,leng,2}] + Sum[(transfit[[i,4]] - transfit[[i,5]])^2, {i,leng+1, Length[transfit]}], transerror = Sum[(transfit[[i,3]] - transfit[[i,5]])^2+ (transfit[[i+1,4]] - transfit[[i+1,5]])^2,{i,1,leng,2}] + Sum[(transfit[[i,4]] - transfit[[i,5]])^2, {i,leng+1, Length[transfit]}]]; Print["Sum-of-squares error: ", transerror]; WriteString[nakfile,"Sum-of-squares error", transerror]; Close[nakfile]; Sum-of-squares error: 2.01567 (* If the generated points are a good fit to the original T spectrum then the analysis is complete. If not, it is likely that the films is absorbing over the whole wavelength range and the estimate for delta-d is too large. Reduce delta-d and repeat the analysis. *) DisplayTogether[ListPlot[spectrum, PlotJoined->True, PlotStyle-> RGBColor[1,0,0], PlotStyle->Thickness[0.005]], ListPlot[trans], Plot[200*s/(s^2+1),{lambda,range[[1]],range[[2]]},PlotStyle-> RGBColor[0,1,0]], PlotRange->{range,{0,100}}, FrameLabel->{lam, "Transmission (%)"}];

107

DisplayTogether[ListPlot[spectrum, PlotJoined->True, PlotStyle-> Thickness[0.001]], ListPlot[trans], PlotRange->{{range[[2]]+100, range[[2]]+250},{0,10}}, FrameLabel->{lam, "Transmission (%)"}];

Bibliography Arndt D, Azzam R, Bennett J, Borgogno J, Carniglia C, Case W, Dobrowolski J, Gibson U, Tuttle Hart T, Ho F, Hodgkin V, Klapp W, Macleod H, Pelletier E, Purvis M, Quinn D, Strome D, Swenson R, Temple P and Thonn T (1984), `Multiple determination of the optical constants of thin-lm coating materials', Applied Optics 23 (20), 3571{3596. ASTM Standard (1992), `Standard test method for solar absorptance, reectance, and transmittance of materials using integrating spheres', ASTM Standard E 903-82, 465{473. Baba T, Matsuyama T, Sawada T, Takahama T, Wakisaka K, Tsuda S and Nakano S (1994), `Polycrystalline silicon thin lm solar cell prepared by solid phase crystallization (SPC) method', 24th IEEE Photovoltaics Specialists Conference, Hawaii pp. 1315{1319. Bader G, Ashrit P and Truong V (1995), `Transmission and reection ellipsometry studies of electrochromic materials and devices', SPIE 2531, 70{81. Bah K, Czapla A and Pisarkiewicz T (1993), `Simple method for optical parameter determination of inhomogeneous thin lms', Thin Solid Films 232, 18{20. Beeman D, Tsu R and Thorpe M (1985), `Structural information from the Raman spectrum of amorphous silicon', Physical Review B 32(2), 874{878. Benrakkad M, Perez-Rodriguez A, Jahwari T, Samitier J, Lopez-Villegas J and Morante J (1994), `Raman characterization of polycrystalline silicon: Stress prole measurements', Materials Research Society: Polycrystalline Thin Films: Structure, Texture, Properties and Applications Symposium pp. 609{614. B(oer K (1990), Survey of Semiconductor Physics, Van Nostrand, New York. Bovard B, Van Milligen F, Messerly M, Saxe S and Macleod H (1985), `Optical constants derivation for an inhomogeneous thin lm from in situ transmission measurements', Applied Optics 24(12), 1803{1807. Bustarret E and Hachicha M (1990), `A discussion of electronic optical absorption spectra of nanocrystalline silicon thin lms', Materials Research Society Symposium Proceedings 164, 211 {216. Corning Inc. (1997a). Personal communications with D. Thurstans (Australia) and Eric Caillot (Advanced Materials Division, U.S.A.). 108

BIBLIOGRAPHY

109

Corning Inc. (1997b). Personal communications with D. Thurstans (Australia) and P. Bocko (Advanced Display Products division, U.S.A.). Corning Inc. (n.d.), `Corning 1737 glass: Material information booklet'. Corrales C, Ramirez-Malo J, Fernandez-Pena J, Villares P, Swanepoel R and M&arquez E (1995), `Determining the refractive index and average thickness of AsSe semiconducting glass lms from wavelength measurements only', Applied Optics 34(34), 7907{7913. Dyer T, Marshall J, Pickin W, Hepburn A and Davies J (1993), `Polysilicon thin lms and devices produced by low-temperature (600) furnace crystallisation of hydrogenated amorphous silicon (a-Si:H)', Materials Research Society Symposium Proceedings 283, 691{702. Fauchet P and Campbell I (1990), Critical review of raman spectroscopy as a diagnostic tool for semiconductor microcrystals, in P. Fauchet, K. Tanaka and C. Tsai, eds, `Materials Issues in Microcrystalline Semiconductors', Materials Research Society Symposium Proceedings (vol. 164), Materials Research Society, pp. 259{264. Gillespie R, Humphreys D, Baird N and Robinson E (1989), Chemistry, 2nd edn, Allyn and Bacon, Massachusetts. Green M (1995), Silicon Solar Cells: Advanced Principles and Practice, Photovoltaics Special Research Centre, University of New South Wales, Sydney. Griot M (1995/1996), `Optics catalogue'. Heavens O (1955), Optical Properties of Thin Solid Films, Butterworths Scientic Publications, London. Hulth&en R (1975), `Optical constants of epitaxial silicon in the region 1{3.3 eV', Physica Scripta 12, 342{344. Jackson W, Johnson N and Biegelsen D (1983), `Density of gap states of silicon grain boundaries determined by optical absorption', Applied Physics Letters 43(2), 195{197. Jenkins F and White H (1957), Fundamentals of Optics, 3rd edn, McGraw-Hill, New York. Kamins T (1988), Polycrystalline Silicon for Integrated Circuit Applications, Kluwer Academic Publishers, Boston. Kanicki, J., ed. (1992), Amorphous and Microcrystalline Semiconductor Devices, Volume II: Materials and Device Physics, Artech House, Boston. Klazes R, Broek M, Bezemer J and Radelaar S (1982), `Determination of the optical bandgap of amorphous silicon', Philosophical Magazine B 45(4), 377{383. Kubinyi M, Benk(o N, Grofcsik A and Jeremy Jones W (1996), `Determination of the thickness and optical constants of thin lms from transmission spectra', Thin Solid Films 286, 164{169.

BIBLIOGRAPHY

110

K(uhl C, Druminski M and Wittmaack K (1976), `Proles of the optical absorption constant and interface composition in epitaxial silicon lms', Thin Solid Films 37(3), 317{321. K(uhl C, Schl(otterer H and Schwidefsky F (1974), `Optical investigation of dierent silicon lms', Journal of the Electrochemical Society 121, 1496{1500. K(uhl C, Schl(otterer H and Schwidefsky F (1976), `An optically eective intermediate layer between epitaxial silicon and spinel or sapphire', Journal of the Electrochemical Society 123, 97{100. Laaziz Y and Bennouna A (1996), `On some important care to take when making spectrophotometric measurements on semiconductor thin lms', Thin Solid Films 277, 155{161. Lambertz A (1997). Personal communications with A. Lambertz (present address: Forschungszentrum J(ulich, Institut f(ur Schicht- und Ionentechnik, Germany). Lapp J, Moat D, Dumbaugh W, Bocko P and Anma M (1994), New substrate for advanced at panel displays, in `Corning Product Information', Originally Presented at SID International Symposium, San Jose, California. Levy P (1961), `Color centers and radiation-induced defects in al2o3', Physical Review 123(4), 1226{1233. Lubberts G and Burkey B (1984), `Optical and electrical properties of heavily phosphorus-doped epitaxial silicon layers', Journal of Applied Physics 55(3), 760{763. Lubberts G, Burkey B, Moser F and Trabka E (1981), `Optical properties of heavily phosphorus-doped polycrystalline silicon layers', Journal of Applied Physics 52(11), 6870{6878. Luft W and Tsuo Y (1993), Hydrogenated Amorphous Silicon Alloy Deposition Processes, Marcel Dekker, New York. Lyashenko S and Miloslavskii V (1964), `A simple method for the determination of the thickness and optical constants of semiconducting and dielectric layers', Optical Spectroscopy 16, 80{81. Malitson I (1962), `Refraction and dispersion of synthetic sapphire', Journal of the Optical Society of America 52(12), 1377{1379. Manifacier J, Gasiot J and Fillard J (1976), `A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin lm', Journal of Physics E: Scientic Instruments 9, 1002{1004. M&arquez E, Ramirez-Malo J, Villares P, Jimin&ez-Garay R and Swanepoel R (1995), `Optical characterization of wedge-shaped thin lms of amorphous arsenic trisulphide based only on their shrunk transmission spectra', Thin Solid Films 254, 83{91.

BIBLIOGRAPHY

111

Materials Research Corporation (1975), The Basics of Sputtering, Materials Research Corporation, Orangeburg, New York. McClain M (1997). Personal communications. McClain M, Feldman A, Kahaner D and Ying X (1991), `An algorithm and computer program for the calculation of envelope curves', Computers in Physics 5(1), 45{ 48. McKenzie D, Savvides N, McPhedran R, Botten L and Nettereld R (1983), `Optical properties of a-Si and a-Si:H prepared by DC magnetron techniques', Journal of Physics C: Solid State Physics 16, 4933{4944. Michailovits L, Hevesi I, Phan L and Varga Z (1983), `Determination of the optical constants and thickness of amorphous V2O5 thin lms', Thin Solid Films 102, 71{76. Minkov D (1990), `Computation of the optical constants of a thin dielectric layer from the envelopes of the transmission spectrum, at inclined incidence of the radiation', Journal of Modern Optics 37(12), 1977{1986. Minkov D (1992a), `Optical characterization of a thin layer using polarized light', South African Journal of Physics 15(3/4), 65{72. Minkov D (1992b), `Singularity of the solution when using spectrum envelopes for the computation of the optical constants of a thin dielectric layer', Optik 90(2), 80{84. Minkov D and Swanepoel R (1993), `Computerization of the optical characterization of a thin dielectric lm', Optical engineering 32(12), 3333{3337. Mitchell E, Rigden J and Townsend P (1960), `The anisotropy of optical absorption induced in sapphire by neutron and electron irradiation', Philosophical Magazine 5, 1014{1027. Mott N and Davis E (1971), Electronic Processes in Non-Crystalline Materials, Clarendon Press, Oxford. Mrafko P and Ozvold M (1995), `Determining optical constants of thin lm on substrate from transmission and reection data', SPIE 2648, 72{76. Myburg G and Swanepoel R (1987), `Stability as regards lm thickness, homogeneity and optical properties of thin a-si:h lms', Japanese Journal of Applied Physics 26(3), 341{346. Myburg G and Swanepoel R (1988), `On the uniformity of thin a-Si:H lms prepared in an rf-glow discharge system', Japanese Journal of Applied Physics 27(6), 899{907. Oriel Corporation (1990), `Optics catalogue, vol. iii'. Oriel Corporation (1994), `Optics catalogue, vol. ii'.

BIBLIOGRAPHY

112

Palik E (1985), Handbook of Optical Constants (Vol. 1), Academic Press Inc., Orlando. Palik E (1991), Handbook of Optical Constants (Vol. 2), Academic Press Inc., Boston. Peter K, Welleke G, Prasad K, Shah A and Bucher E (1994), `Free-carrier absorption in microcrystalline silicon thin lms prepared by very-high-frequency glow discharge', Philosophical Magazine 69(2), 197{207. Rancourt J (1996), Optical Thin Films: User Handbook, SPIE Optical Engineering Press, Washington. Rizk R, Achiq A, Madelon R, Gourbilleau F and Cruege F (1996), `Optical, electrical and structural studies of microcrystallized sputtered silicon', Solid State Phenomena 51-52, 243{248. Romanov D, Victoria N and Kalameitsev A (1996), `Full-optical characterization of thin lms in photovoltaic cells', Materials Research Symposium Proceedings 426, 587{592. Saitoh T, Hori N, Suzuki K and Iida S (1991), `Optical characterisation of very thin hydrogenated amorphous silicon lms using spectroscopic ellipsometry', Japanese Journal of Applied Physics 30(11B), L1914{1916. Schl(otterer H (1976), `Interface properties of Si on sapphire and spinel', Journal of Vacuum Science and Technology 13(1), 29{36. Schoenfeld O, Hempel T, Zhao X, Sugano T and Aoyagi Y (1994), `Optical, electrical and structural investigations of the transition from amorphous to microcrystalline silicon', Japanese Journal of Applied Physics 33, 6082{6085. Stenzel O (1996), Das Dunnschichtspektrum, Akademie Verlag, Berlin. Swanepoel R (1983), `Determination of the thickness and optical constants of amorphous silicon', Journal of Physics E: Scientic Instrumentation 16, 1214{1222. Swanepoel R (1984), `Determination of surface roughness and optical constants of inhomogeneous amorphous silicon lms', Journal of Physics E: Scientic Instrumentation 17, 896{903. Swanepoel R (1985), `Determining refractive index and thickness of thin lms from wavelength measurements only', Journal of Optical Society of America A 2(8), 1339{1343. Swanepoel R (1989), `Transmission and reection of an absorbing thin lm on an absorbing substrate', South African Journal of Physics 12(4), 148{156. Swanepoel R, Swart P and Aharoni H (1985), `Inuence of argon partial pressure on the electrical and optical properties of sputtered hydrogenated amorphous silicon', Thin Solid Films 128, 191{203. Tao G (1994), Optical Modeling and Characterisation of Hydrogenated Amorphous Silicon Solar Cells, PhD thesis, University of Delft.

BIBLIOGRAPHY

113

Tsai C and Fritzsche H (1979), `Eect of annealing on the optical properties of plasma deposited amorphous hydrogenated silicon', Solar Energy Materials 1, 29{42. Valeev A (1963), `Determination of the optical constants of weakly absorbing thin lms', Optica Spektrosk. (USSR) 15, 269{274. Varian Australia (OS/2 software version), `Varian on-line help', Orangeburg, New York. Ward L (1994), The Optical Constants of Bulk Materials and Films, 2nd edn, Institute of Physics Publishing, Bristol. Zheng T (1997). Personal communications with T. Zheng (Photovoltaics Special Research Centre, UNSW.).