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i THICKNESS AND REFRACTIVE IN DEX MEASUREMENTS OF TRANSPARENT THIN FILMS BY ANDREW HAMMER A Thesis Submitted to the Division of Natural Sciences New College of Florida in partial fulfillment of the requirements for the degree Bachelor of Arts Unde r the sponsorship of Dr. Mariana Sendova Sarasota, Florida May 2010
ii Acknowledgements I would like to thank Professor Mariana Sendova for all of the guidance she gave me during this project. I would also like to thank Dale Musselman for helping me in the laboratory. In addition, I would like to thank Jos Jimenez for teaching me how to use the microspectrometer.
iii Table of Contents Acknowledgements . . . . . . ii Abstract . . . . . . . i v Introduction . . . . . . . 1 1 Spin Coating . . . . . . . 3 1.1 Overview of Spin Coati ng . . . . . 3 1.2 Producing the Films . . . . . 5 1.3 Preparing Solutions . . . . . 6 2 Investigating Thin Films with Atomic Force Microscopy . . 11 2.1 Overview of Atomic Force Microscopy . . . 11 2.2 Measuring Film Thickness by Atomic Force Microscopy . 14 3 Spectra of Thin Films . . . . . . 18 3.1 Reflectance, Transmittance, and Absorbance . . . 18 3.2 Spectra and the Electric Field . . . . 21 3.3 The Microspectrometer . . . . . 22 3.4 Review of Relevant Optics . . . . 25 3.5 Interference in Spectra of Thin Films . . . 27 4 Experimental Results . . . . . . 3 3 4.1 Index of Re fraction and Thickness Measurements . . 33 4.2 Refractive Index of the Substrate . . . . 41 4 3 S h a p e o f the R e f l e c t a n c e S p e c t r a . . . . 4 4 Conclusion. . . . . . . . 4 6 Bibliography . . . . . . . 4 8
iv THICKNESS AND REFRACTIVE IN DEX MEASUREMENTS OF TR ANSPARENT THIN FILMS BY ANDREW HAMMER New College of Florida, 2010 ABSTRACT A method for measuring the thickness and refractive index of a transparent thin film by analyzing the reflectance spectra is developed. An overview of spin coating is presen ted, followed by a method for measuring film thickness by atomic force microscopy. The theory for finding the thickness or the refractive index of a film and refractive index of the substrate is developed, followed by experimental results for polystyrene thin films on glass substrates. Dr. Mariana Sendova Division of Natural Sciences
1 Introduction A dielectric thin film is a thin uniform layer of dielectric material on top of a substrate. For the experiments done in this work, the dielectric material i s polystyrene, and the substrates are glass microscope slides. One application of dielectric thin films is spectroscopic studies of materials such as nanoparticles and carbon nanotubes. One way t he substance of inter est can be deposited in the form of a th in film is by a spin coating techn ique, after which it can then be investigated optically. Two important quantities in studying dielectric thin films are film thickness and the refractive index of the film material The film thickness is the distance from the top surface of the film to the substrate. Ideally the film is uniform, meaning is constant. Thin films can easily be produced such that the thickness does no t vary much across the surface, but it is impossible to eliminate any variation in the film thickness. The refractive index of the film is important because it helps describe how light will behave inside the film. It will be shown that knowing the refrac tive index of the film is necessary for many optical methods of measuring the film thickness. The refractive index of the film depends on the frequency of the light traveling in the film. The fact that the refractive index of the film depends on the freq uency of the light is called dispersion. One familiar example of dispersion is white light traveling through a prism, which is shown in Fig. I.1
2 Fig. I.1: Dispersion of white light by a prism ( http://people.rit.edu/andpph/photofil c/spectrum_8661.jpg) This work consists of four chapters. The first chapter deals with the production of thin films by spin coating. The second introduces atomic force microscopy and gives a method for measuring the thickness of a film. The third chapter deals with the t heory of reflectance and transmittance spectra of thin films T his theory produces a way to measure the refractive index of a film of known thickness or thickness of a film of known refractive index The fourth chapter applies the theory and techniques o f the previous three chapters to experiments with polystyrene thin films. The thickness of one film is measured by an atomic force microscope, and then the refractive index of the film is found by examining the reflectance spectra of the film surface. Th is refractive index is then used to find the thickness of another polystyrene film. In addition, the refractive index of the substrate is estimated.
3 1 Spin Coating 1.1 Overview of Spin Coating Spin coating is the process of applying a liquid solution, consisting of a solid solute mixed into a liquid solvent, to a rotating disk to create a solid film on a substrate. The spin coating process usually goes as follows: A substrate is secured to the spin coater using a vacuum pump. Solution is then appli ed to the substrate. The substrate is then accelerated over the course of a few sec onds to the desired angular velocity After sp inning for approximately one minute, the substrate decelerates to rest over the course of a few seconds. The spinning motion evens out the solution into a uniform layer. During th e entire process, the solvent evaporates, leaving a solid film behind. The spin coating process has a lot of parameters which affect the final film. Table 1 .1.1 categorizes and lists the most releva nt parameters for spin coating. Category Parameters Motion Angular Velocity Angular Acceleration Spin Time Solution Solution Concentration Amount of Solution Applied Materials Used Choice of Solvent Choice of Solute Choice of Substrate Environm ent Temperature Humidity Etc. Table 1 .1.1: Parameters affecting the final film Much experimental and theoretical work has been done to determine the relationship between these parameters and the thickness of the final film. Angular velocity, solution con centration, and solution viscosity have a large affect on the film
4 thickness. Spin time, the amount of the solution applied, and the initial distribution of the solution have little effect on the film thickness. (Norrman, 2005) (Emslie, 1957) For this project, the materials used and the environment are kept constant, while the parameters related to motion and solution concentration are varied. There is a widely know n experimental relationship be tween film thickness and the angular velocity of the substrate. If all of the other variables are fixed, the film thickness behaves as in equation (1.1.1), where is the film thickness, is the angular velocity, and and are constants depending on the other parameters. is positive while is negative. (Andrade, 2006) = (1.1.1) Solution concentration also has an effect on film thickness. The literature pre sents solution concentration in two ways. The first is as a mass ratio, the second as a mass to volume ratio. It will be shown in the section on mixing solutions that these two pieces of information are equivalent. Decreasing the concentration of a so lution, while holding the other parameters fixed, has the effect of producing a thinner film. Experiments by Andrade and Lima have shown that for different solution concentrations, equation (1.1.1) is still obeyed, but with different values of and By varying angular velocity and solution concentration, a wide range of film thicknesses can be produced. It was found experimentally for the films in this work that for small concentrations and high spin speeds, films as thin as 50 were p roduced. For higher concentrations and lower spin speeds, films as thick as 870 were produced.
5 In summary, spin coating provides a way to produce a uniform layer of material on a substrate. Experimental studies have identified which of the parameter s are most relevant in affecting the final film. Two of the most important parameters are spin speed and solution concentration, which both have a substantial affect on film thickness. 1.2 Producing the Films For the experiments discussed in t his work, a spin coater manufactured by Headway Research Inc. (Model PWM32) was used. The solute and subsequently also the film material was polystyrene (Aldrich Ca talog No. 18,242 7). The solvent used to dissolve the polystyrene was toluene. The substrates used w ere Fisher brand pre cleaned microscope slides (Catalog No. 12 550B). The microscope slides were cut into squares with a side length of two inches. The substrates required some preparation to insure that they were clean before spin coating Substrates we re cut and then put in a beaker containing rubbing alcohol. The beaker was then placed in a sonic bath filled with water. The sonic bath was run for twenty minutes to remove any dirt and oil from the substrate, after which the substrates were removed and allowed to air dry. For book keeping purposes, a black marker was used to label each substrate in the corner with an identifying combination of a letter and a number After being labeled, the substrates were ready to be used. Solutions wer e prepared by weighing out the desired amount of polystyrene to mix with a given amount of toluene. To minimize the uncertainty for the value of the solution concentration, a scale accurate to 10 2 grams was used and the toluene was measured with a pipette accurate to 1 After adding the polystyrene to the toluene, the
6 flask was covered with a lid to prevent evaporation of the toluene. Then a magnetic stirrer was used to help dissolve the p olystyrene. Once the substrates were cleaned and the solution was prepared, the films were ready to be deposited. For each film, t he substrate was secured to the spinner with the vacuum pump. 100 of solution was placed on the substrate with a pipette Then the substrate was spun at a constant speed for one minute. The angular acceleration was set at 1000 For safety purposes goggles and gloves were worn because of the toxicity of the toluene. In addition the spinner was covered with a lid while in motion. After spinning, some films had a crack cut into them with a razor blade for purposes of measuring the film thickness with an atomic force microscope. All of the films were allowed to air dry overnight, so the rest of th e toluene could evaporate. After all of the toluene had evaporated, the finished films were ready for experiments 1. 3 Preparing Solutions This section investigates the different ways solution concentration is reported in the literature. It also addresses the process of mixing solutions and changing the concentration of an existing solution. Finally the uncertainty in solution concentrations is calculated. The ideas and calculations in the section are not essential to understanding the rest of this paper, so feel fr ee to skip to chapter 2. There are two predominant ways of reporting solution concentration in the literature. The first is defined as the mass of the solute divided by the total mass of the mixture. Let this concentration be called the weight concentra tion denoted If 1 is the mass of the solute, and 2 is the mass of the solvent, then can be written as in equation (1.3.1).
7 = 1 1 + 2 (1.3.1) The second way to define a conc entration is the mass of solute divided by the volume of solvent. Let this concentration be called the volume concentration denoted L et 2 be the volume of the solute, then can be written as in equation (1.3.2). = 1 2 (1.3.2) The two different ways of talking about solution concentration are equivalent pieces of information. One can easily be found from the other. If the volume of the solute 1 is much smaller than the volume of the solvent 2 then we can use the volume of the solvent and the volume of the mixture interchangeably. = 1 + 2 2 (1.3.3) If the density of the solvent is known, we can write 2 = and substitute back into equation (1.3.1). = 1 1 + (1.3.4) By solving for 1 and plugging the expression into equation (1.3.4), an equation relating the two concentrations is obtaine d. = 1 2 1 = 2 = 1 (1.3.5) Often times when spin coating thin films, one wants to create a solution of a certain concentration, or change the concentration of an existing solution. There are three cases: creating a solution from scratch, diluting a solution, and increasing the concentration by adding more solute. To find the quantities of so lvent and solute to add
8 in each case is an easy algebra problem. By working through this once and recording the results, one does not have to work it out each time one mixes solutions. First the calculations are done for the weight concentration, and th en they are done for the volume concentration. For the first case, note the definition of con centration and then note a fixed volume of milliliters is desired in the end. By solving equation (1.3.4) for 1 an expression is produced which says how ma ny grams of solute should be added to the volume of solvent. 1 = 1 (1.3.6) To dilute an existing solution of initial concentration 0 to a final concentration of < 0 How much solvent should be add ed ? Since solvent is being added the volume of the mixture will be increasing. The mass of solute remains the same. This implies that equation (1.3.6) be used can be solved for to yield equation (1.3.8). 1 = 0 1 0 = ( + ) 1 (1.3.7) = 0 ( 1 ) ( 1 0 ) 1 (1.3.8) Finally the case where the concentration is increase d to > 0 This time it is the volume that is fixed and the amount of solute that is changing. Let 1 be the amount of solute to be add ed Equation (1.3.6) is again used. 1 = 0 1 0 1 + 1 = 1 1 = 1 0 1 0 (1.3.9)
9 T he calculations for volume concentrations are much simpler. To create a solution of concentration and approximate volume 2 simply mix 1 grams of solute where 1 = 2 To dilute a solution, of initial concentration 0 to a final concentration the following two equations must be solved for 2 The res ult is given by equation (1.3.10) 2 = 1 0 2 + 2 = 1 2 = 1 1 0 2 = 1 0 0 (1.3.10) For the case > 0 the following two equations are solved for 1 The result is equation (1.3.11 ) 1 = 0 2 1 + 1 = 2 1 = 2 0 (1.3.11) It appears as though volume concentration is an easier system to go by because the calculations are simpler and the app roximation of equation (1.3.3) is not used. Volume concentration will be used in this work, and from this point on will simply to be referred to as the concentration. That last thing to do is determine the uncertainty in the concentration. The uncertain ty will only be calculated for the case of mixing an original solution. The other cases are similar. If the uncertainty in the 1 is 1 and the uncertainty in 2 is 2 Then the fractional uncertainty for is given by equation (1.3.12). Equation (1.3.12) is valid because the errors in measuring 1 and 2 are both independent and random. (Taylor, 1997)
10 = 1 1 2 + 2 2 2 ( 1 3 12 ) This concludes the calculations dealing with solution concentrations. With the tools of this section, solution concentration can be known to accurate values, an important step when studying thin films produced by spin coating. After the films are made under known conditions, they can be analyzed with the help of an atomic force microscope, which is the subject of the next chapter
11 2 Investigating Thin Films with Atomic Force Microscopy 2.1 Overview of Atomic Force Microscopy An atomic force microscope (AFM) is a device for producing a three dimensional image of a microscopic surface. For an example of an AFM image, see Fig. 2.1 .1. The an analogy between how the AFM produces an image and our own sense experiences. Suppose you are sitting at a table and on top of the table is a blue cup, a n ewspaper and a clear glass vase. Most people would determine the shape and orientation of the objects on the table simply by looking with their eyes. There is another way thou gh that someone could find the same information even with their eyes closed. I nstead of looking, contents. Note that when feeling, one could not determine that the cup was blue, or that the vase was made out of clear glass instead of green glass. Fig. 2.1.1: AFM image of ripples on the surface of a polystyrene thin film. The computer assigns different colors to different heights.
12 More technically, the AFM produces a data set consisting of points in a plane paired with the height of the surface above that point. If ( ) is a point in this plane, then the AFM produces triplets ( ) where is the height of the surface above the point The AFM software sets = 0 to be the lowest point measured on the surface. Fig. 2.1. 2: (a ) Electron micrograph of a cantilever tip (b) Diagram of AFM ( (a) is from http://en.wikipedia.org/wiki/Atomic_force_microscope ) The AFM works by lowering a cantilever tip onto the surface. A laser beam is reflected off the back of the cantilever tip an d continues until it hits a detector (see Fig. 2.1.3) When the cantilever tip is lowered to the surface Van der Waals forces on the cantilever tip from the atoms of the sample bend it slightly causing a small deflection in the laser beam. This deflecti on of the laser beam is picked up by the detector. Internal electronics then move the cantilever tip so that the force on the probe from the sample is constant. By measuring how much the tip has to move as the probe is moved across the surface, the AFM r etrieves the required information to make an image. The AFM sends this information to a computer where the data can be analyzed. The AFM used for this project was made by Quesant. All measurements were in this mode the force betw een the AFM and the sample is Cantilever Tip Detector Laser Surface of Sample a b
13 held constant. There are other modes under which the AFM may operate; however these will not be discussed here. The scans are always over a square region The scan size is given by the length of one of the sides of this sq uare. For this AFM, scan size ranges from a maximum of 40 microns to about 5 As scan sizes become smaller, producing quality images becomes more difficult because of noise effects caused by vibrations from the microscopes surroundings Fig 2.1.4: The atomic force microscope used for this project. The vertical range f or the microscope is 4 When the microscope engages the surface of the sample, it is usually around the midpoint of this range, so it is useful to think of the cantilever tip having 2 to move in either direction. If the microscope is scanning o ver a feature that is too tall, the cantilever tip will not have room to move to keep the force constant. This will cause excessive force on the cantilever tip potentially breaking it. If the surface being scanned has a point that is too far away for the cantilever
14 tip to reach, then the image is not a good representation of the actual surface. No damage is done to the AFM if this happens. For the thickness measurements discussed in the next section, the scan size is usually fai rly large, around 35 40 Because of the relatively large scan size, noise effects are not an issue. The vertical range o f the AFM does become an issue for thickness measurements. 2.2 Measuring Film Thickness by Atomic Force Microscopy An atomic force microscope can be us ed to measure film thickness in a straight forward way. However, there are many disadvantages to this method that make it less than ideal. It is time consuming. It can potentially damage the AFM. It is also partially destructive to the sample, which may not be acceptable in all circumstances. In addition, producing a usable image usually takes multiple attempts. Immediately after a deposited film stops spinning, one can remove it from the spin coater and cut a small crack in the film with a razor knife. The knife is able to cut through the film easily because not all of the solvent has evaporated yet. After letting the film dry, one can then examine the crack under the AFM to see the difference in height between the bottom of the crack and the film sur face. Under the assumption that the knife cuts all the way through the film while not cutting into the substrate, this height is the film thickness. For this project, there were two important applications of the AFM software that were used when measuring several ways the software can make this compensation, and each way works best under
15 certain circumstance same horizontal plane, and then the computer creates another image correcting for the tilt. Of course if those three points do not lie in the same plane, then the computer will produce a worse image than the original, so it is important that the user know that the s the best way for this application is that the film surface is a flat plane. So all the user must do is select three points on the film surf ace away from the crack. Fig. 2.2.1 shows an Fig. 2.2.1: Two AFM images of a crack in a thin film. The left image is the original; the tilt removal th height on the horizontal axis and number of data points recorded for that height on the vertical axis. to determine the height of the film surface above the bottom of the crack. From the color of the film surface in the AFM image, one can tell the general range that the height of the film surface must be in. Then by looking at the height histogram, one can determine more exactly the film thickness. Usually there will be a peak around this height, and half
16 of the width of the peak at half of its maximum value is taken as the uncertainty in the measurement. Fig. 2.2.2 shows the height histogram of the righ t image in Fig. 2.1.1. It can be seen from the AFM image t hat the thickness of the film lays between 500 and 1000 nm. The histogram shows a strong peak around 700 nm, which corresponds to the points on the film surface. The width at half maximum is around 20 nm, so the thickness of the film can be stated as = 7 00 10 Fig. 2.2.2: Height histogram used for a thickness measurement of a polystyrene thin film. The height of the film surface corresponds with the tall peak centered at 700 nm. The disadvantages of this method are many. First off, it may not be acceptable to cut a crack in sample, in which case this method is of no use. Second, since the AFM is scanning over features of varying height, it is possible to damage the AFM if a feature is too tall. Usually when a crack is cut into the film, film ma terial piles up next to the crack
17 (see Fig. the cantilever tip to safely travel over. The opposite problem can also occur. That is the crack may be too low for the cant ilever tip to reach, making the image not usable. case determining the film thickness is not possible, or at best the measurement comes with a large uncertainty. If any of these problems occur, the only thing one can do is withdraw the cantilever tip and attempt another image on a different spot. Engaging the surface with the cantilever tip is a slow process, making measurements time consuming. Despite these disad vantages, a film whose thickness is known can be useful for finding optical properties of the film material. Measuring film thickness by atomic force microscopy is a straig htforward way to produce such a film.
18 3 Spectra of Thin Films 3.1 Reflectance, Transmittance, and Absorbance 1 Fig 3 .1.1 Fig. 3.1.1 : Diagram outlining the effects of shining a beam of light onto a sample There are three types of spectra that are of interest, and those are reflectance, transmittance, and abs orbance. A spectrum is the graph of one of these quantities as a function of the radiation wavelength in vacuum power of the reflected beam divided by the power of the incident beam (see Fig 3.1.1). Reflectance can be expressed in terms of the irradiances of the incident and reflected beam. Recall that the irradiance is the average power per unit area imparted on a surface and that for a sinusoidal plane wave of the form = 0 cos ( + ) the irrad iance of the wave takes the form = 0 2 0 2 where 0 is the amplitude of the wave. Let be the reflectance and and be the reflected and incident irradiances 1 A general reference for this chapter is Optics Fourth Edition by Eugene Hecht and Chapter 9 of Introduction to Electrodynamics Third Edition by Griffiths. Scattered Light Sa mple Incident Beam Reflected Beam Transmitted Beam
19 respectively. Let be the cross sectional area of the bea m. If the beam of light hits the surface at normal incidence, we can write t he reflectance as in equation (3 .1.1). Keeping on can be written as equation (3 .1.2). = = (3.1.1) = ( ) ( ) (3.1.2) If the electric field of the incident and reflected beams are given by polariz ed plane waves with amplitudes 0 and 0 then the reflectance can be expressed as equation ( 3.1.3 ). = 0 ( ) 0 ( ) 2 (3.1.3) Similarly, the transmittance is defined as the power of the transmitted beam divided by the power of the incident beam. At normal incidence the transmittance can be written as eq uation (3.1.4 ) and (3 .1.5 ). If 0 is the amplitude of the transmitted wave, the transmittance can be written as equati on (3.1.6) = = (3.1.4) = ( ) ( ) (3.1.5) = 0 ( ) 0 ( ) 2 (3.1.6) If the sample is a continuous medium, a portion of the incident beam is absorbed by the sample. If the beam travels through a distance through the sample, the intensity of the transmit ted beam is given by equation (3 .1.7 ), where 0 is the irradiance of the
20 transmitted beam at the surface interface and is a constant called the attenuation coefficient which has units of inverse length. The attenuation coefficient is a property of the material that does not depen d on the geometry of the sample. The larger the attenuation coefficient, the larger the amount of energy absorbed by the material. = 0 (3.1.7) Let be defined as minus the argument inside the exponential, that is = Since + 0 = by conservation of energy, 0 can be rewritten as 0 = can then be solved for in terms of the incident, reflected and transmitted irradiances. This also yie lds an expression for the attenuation coefficient. = ln 0 = ln = ln ( ) ( ) ( ) ( ) = 1 ln Multiplying the argument of the l ogarithm by yields an expression for the attenuation coefficient in terms of the reflectance and transmittance. = 1 ln ( ) 1 ( ) (3.1.11) One thing to keep in mind is the pre vious analysis does not account for scattered light. If a portion of the incident radiation is scattered, then the dissipation of energy is not entirely due to absorption. Also it does not account for the multiple reflections that occur off of the surfac e of the sample s which can create interference in the reflectance and transmittance spectra
21 The spectrometer used in this project can give an approximation for = under the assumption that ( ) is negligible. The approximation used is 1 ( ) 1 When this approximation is used, is called the optical density, denoted The spectrometer yields as equation (3.1.12 ). This quantity is also c alled the absorbance spectra. ln (3.1.12) The spectrometer produces ( ) by first measuring and t hen plugging it into equation (3 .1.12 ). Because of this, acquiring the absorbance spe ctra with the spectrometer is equivalent to collecting the transmittance spectra. 3 .2 Spectra and the Electric Field When performing optical experiments, sometimes the light source used does not emit evenly across the spectrum More precisely, 0 ( ) At first this may seem to present a problem, would not an even source make experiments easier? For the applications in this project, the fact that the source emits unevenly is not a problem because the spectra do not depend on 0 ( ) This fact will be illustrated for the case of reflectance ; the cases for transmittance and absorbance are similar. The reflectance does not depend on 0 for measurements made in this work, because 0 is has a factor of 0 in it. This is not necessarily the case for th e reflectance spectra of any sample, but it is the case for the reflectance spectra of transparent thin films. The proof of this has to do with the reflections of electromagnetic waves at an interface, and the a ddition of waves of the same frequency. If the fact that 0 is has a factor of 0 in it is assumed true, then 0 ( ) can be written as 0 = 0 ( ) where ( ) is some function that does not depend on the amplitude of the incident wave.
22 The factors of 0 in ca ncel, and the reflectance spectra does not depend on the amplitude of the incident wave. So even though the source emits unevenly, the reflectance spectra takes this into account in such a way that it is as if the source emits evenly. Another important a spect of the relationship between reflectance and the electric field is the correspondence between extrema. It can be seen from equation (3.1.3) that ( ) is maximized and minimized with 0 ( ) 0 ( ) This can also be shown more formally by taking the derivative of ( ) with respect to 0 ( ) 0 ( ) and setting it equal to 0. = 0 ( ) 0 ( ) 2 (3.1.3) = 2 0 0 0 0 = 0 0 0 = 0 This fact is essential to the analysis of interference later in this work, as it relates the theory deal ing with the electromagnetic waves to the experimentally observable quantity of the reflectance. The same arguments are applicable to the transmittance and absorbance spectra 3 .3 The Micros pectrometer The spectra collected for this project were obtained using a microspectrometer manufactured by C RAIC Because these measurements are an integral part of this work, it is beneficial to take some time to explain the workings of the microspectrometer. A microspectrometer is a combination of a microscope and a spectrometer. The microscope is useful examining the sample surface and for finding the desired location on
23 a sample to collect a spectrum The main components of the spectrometer are a lamp that emits light with wavelengths from the UV all the way to the infrared ( 200 900 ), a s tage for holding the sample, an objective for focusing the light, and a detector which mea sures the irradiance spectra. The spectrometer is hooked up to a computer where the data can be viewed and analyzed using software also prod uced by C RAIC Fig. 3.3.1 : The m icrospectrometer used for the experiments The machine produces the desired spectra by combining the results of three separate measurements called the dark scan the reference scan and the sample scan Each of The dark scan is taken when the lamp is off. The dark scan is taken to account for false positives in the detector, or for any stray lamp The reference scan is taken with the lamp on, an d can be
24 taken with or without an object on the sample stage. The sample scan is again taken with the lamp on, and with the sample on the sample stage. The software then produ ces the desired spectra C by plugging these three scans into equation (3.3.1) = (3.3.1) Ideally, to determine properties of a thin film, one would take the spectra of only the film, without the substrate. This is not possible for the case of spin coated films where t he substrate is essential to the film s existence. To account for this, the reference scan was taken on a substrate without a f ilm. Because the reference scan was taken with an object on the sample stage, the reflectance and transmittance spectra a re really the relative reflectance and relative transmittance spectra. When there is no object on the sample stage during the reference scan, the spectra is referred to as absolute rather than relative. One important feature of the software that was used was its ability to locate maxima and minima of a spectrum. By a simple click of a button, the values of where an extrema occurs are displayed on the screen, ready to be recorded. This feature is very useful, since the alternative is to find the extrema by looking at the numerical data set, which contains approximately 1,000 data points. In comparison to the AFM the microspectr ometer has many advantages. The microspectrometer is simple to use. It does not take long to collect a spectrum. It is not destructive to the sample. Overall, the spectrometer is an effective tool for analyzing a sample optically.
25 3.4 Review of Relevant Optics This section presents the necessary facts and tools that will be needed to analyze the spectra of thin films. T he behavior of waves upon reflection is treated as well as the concept of optical path length Consider a sin u soidal polarized plane wave of a given frequency traveling through a medium with index of refraction 1 and impinging at normal incidence on another medium of index 2 (see Fig. 3.4 .1). One should note that 1 and 2 depend on the frequen cy of the wave, but for any given frequency they can be treated as constants. When the wave hits the interface, there will be a reflected and transmitted wave. Fig. 3.4 .1: Diagram showing an incident, reflected and transmitted electromagnetic waves. In actuality the waves are on top of each other. The transmitted wave does not have a phase shift compared to the incident wave. The reflected w ave though, does undergo a phase shift depending on the relative values of 1 and 2 If 1 < 2 the reflection is called external and the reflected wave under goes a phase shift or radians. Note that it does not matter whether the pha se shift is positive or negative since cos ( + ) = cos ( ) If 1 > 2 the reflection is called internal and the reflected wave does not undergo a phase shift. 2 1 Incident Wave Transmitted Wave Reflected Wave
26 One interesting phenomena that can occur depending on the two mediums is that for one range of frequencies the reflection can be external, and for another range the reflection can be internal. This does not necessarily happen for any two mediums. The concept of optical path length plays an essential role in the theory of interfere nce in spectra of thin films. The optical path length of a wave can be defined as the distance traveled by the wave multiplied by the index of refraction of the medium it is traveling through. Letting be the index of the medium and S the distance traveled, then the optical path length can be writt en as equation (3.4.1). = (3.4.1) If the wave is traveling through vacuum ( = 1 ), then the optical path length of the wave is just the distance traveled. If the wave travels through several m ediums each with refractive index without undergoing any phase shifts, then the is given by equation (3.4.2), where is the distance traveled in the medium. = ( ) (3.4.2) If the wave undergoes a phase shift, due to a reflection for example, then the phase shift must be translated into a distance in vacuum. If is the phase shift in radians, and is the wavelength of wave in vacuum, then the p hase shift can be corrected by adding a term equal to 2 onto equation (3.4.2). If there are multiple phase shifts then the OPL can be stated as equation (3.4.3). = 2 + (3.4.3) Another concept that is essential is the difference in the optical path length s between two waves of the same frequency, denoted If 1 is the optical path
27 length of the first wave and 2 is the optical path l ength of the second wave, then the difference in the optical path length s can be stated as equation (3.4.4). = 1 2 (3.4.4) When examining interference between two waves of the same frequen cy that at the beginning of their optical path are in phase a condition for maximum constructive or destructive interference can be stated in terms of The condition for maximum constructive interference is that the difference in the optical path lengths of the two waves be an integer number of wavelengths. Similarly, the condition for maximum destructive interference is that the difference in the optical path lengths be an odd integer number of half wavelengths. If is an integer and is th e wavelength of the waves in vacuum, this can be written as equations (3.4.5) and (3.4.6). = (3.4.5) = ( 2 1 ) 2 (3.4.6) Equations (3.4.5) and (3.4.6) will be very important in relating extrema in the spectra with quantities related to the film This concludes the overview of the necessary tools that will be needed 3.5 Interference in t he Spectra of Thin Films By looking at the reflectance or transmittance spectra of transparent thin films, quantities such as film thickness and index of refraction can be found. This section will present the relevant theory and corresponding equations tha t are used in the experiments. Only the case of reflectance spectra will be examined. The cases of transmittance and absorbance spectra are similar. All wavelengths of radiation are wavelengths in vacuum.
28 Fig. 3.5.1: Reflection off of a Dielectric Thin Film. In actuality all of the beams are on top of each other. Only the first wave reflected from the substrate is shown. From a theoretical standpoint there are infinitely many reflections. The substrate thickness is not drawn to scale. C dielectric thin film from above at normal incidence The thin film has index and the substrate ha s index (see Fig. 3.5.1 ). and depend on but for any fixed they can be treated as constant. Assume the index of refraction of the surrounding atmosphere is one. When the plane wave hits the top surface, part of the wave will be r eflected and part will be transmitted. When the transmitted wave hits the bottom surface again part of the wave will be reflected while part of it is transmitted. From there the wave travels up and the same thing happens each time it hits an interface, w here in the end there are infinitely many reflected waves. If the difference in the optical path length between the first two waves is such that there is maximum constructive or destructive interference, then the difference between the first reflection a nd all of the other reflected beams is also such that there is maximum constructive or destructive interference. Because of this only the difference in the optical path length s between the first two reflections shown in Fig. 3.5.1 needs to be found to app ly equations (3.4.5) and (3.4.6). Let the difference in the optical path lengths between the reflection from the top surface of the film and the first reflection from the bottom surface of the film be denoted
29 depends on whether < or > If < the two waves both undergo a phase shift upon their reflections, so the phase shift terms in cancel. This leaves as given by equation (3.5.1). If > t hen only the wave that reflects off of the top surface undergoes a phase shift of radians, so is given by equation (3.5.2). = 2 < (3.5.1) = 2 2 > (3.5.2) So how does one determine what is for a given film and substrate if the relative values of and are not known ? Well the trick is to plug equations (3.5.1) and (3.5.2) into equation (3.4.5) for t wo adjacent maxima in the spectra that occur at 1 and 2 and then determine a relation that 1 and 2 must follow for both cases < and > These theoretical relations can then be compared with experiment to determine wha t is. For the case < equations (3.5.3) and (3.5.6) apply to the adjacent maxima. Writing 2 = 1 + 1 under the assumption that the refractive index does not vary much from 1 to 2 the right hand sides of equation s (3.5.3) and (3.5.6) and be set equal to each other. 2 1 = 1 < (3.5.3) 2 2 = + 1 2 < (3.5.6) 1 = + 1 2 < 1 2 = + 1 = 2 3 2 4 3 5 4 < (3.5.7) Equation (3.5.7) is a relationship between adjacent maxima that will apply if < Similarly for the case > another relationship can be found.
30 1 2 = + 3 / 2 + 1 / 2 = 5 3 7 5 9 7 > (3.5.8) By comparing these theoretical results to the experimental values of 1 2 the relative values of and can be found, i.e. whether < or > Also in the event that for one range of wavelengths < and for another range of wavelengths > the crossing point where = can be found. If is known at this wavelength, the n so is T here is a theoretical relationship that and follow as a function of the wavelength This relationship is given by equation (3.5.9) and (3.5.10 ). and are constants that depend on the material of medium. (Al Ani, 1985) Equation (3.5.9) due to the oscillatory electric field of light. (Griffiths, 1999) From equations (3.5.9 ) and (3.5.10) it can be shown that there will either be zero or one crossing point where = = + 2 (3.5.9) = + 2 (3.5.10) To ma ke sure the measurements are only consistent with one t heory ( < or > ) the uncertainty in 1 2 needs to be known. Since the uncertainty in 1 and 2 are random and independent, the uncertainty in 1 2 denoted 1 2 can be written as equation (3.5 .11 ). 1 2 = 1 1 2 + 2 2 2 1 2 (3.5.11) Now it will be shown how the index of refraction and thickness of the film can be found by looking at adjacent maxi ma or adjacent minima in the reflectance spectra. It
31 turns out that the equations for and do not depend on which is valid or whether the adjace nt extrema are maxima or minima. J ust as long as the adjacent extrema are both maxima or both minim a and that the extrema are not on different sides of the = boundary Using the approximation that 2 1 the results from solving the conditions for maximum or minimum interfer ence are given by equations (3.5.12) and (3.5.13 ). This tech nique involves taking the two equations describing the condition for extrema, and then eliminating and solving for 1 or = 1 2 2 1 ( 1 2 ) (3.5.12) 1 = 1 2 2 ( 1 2 ) (3.5.13) The uncertainty in 1 can be expressed in terms of the uncertainties of 1 2 1 2 and The uncertainty in 1 2 is just the sum of uncertainti es of 1 and 2 The result is given by equation (3.5.14 ). 1 = 1 1 2 + 2 2 2 + 1 + 2 1 2 2 + 2 1 (3.5.14) If several measurements of the refracti ve index of the film are taken at a variety of wavelengths, then a computer can be used to fit the data to find the as a function of Equation (3.5.9) states that should be linear in 1 2 After collecting the data points ( ) a computer can be used to perform a linear fit to the points ( 2 ) From the linear fit the constants and in equation (3.5.9) can be determined. Once ( ) is found, it can be used in equation (3.5.12), where 1 = ( 1 ) T hickness measurements of similar films can be done under the assu mption that the refractive indices of the films are similar to the film whose refractive index was measured. If ( ) is considered accurate, then the uncertainty in due to 1 and 2 can
32 be forumulated. Under this assumption a crude approximation for the uncertainty in ca n be written as equation (3.5.15 ) The largest uncertainty for though, is most likely the contribution due to the approximation 1 2 If this approximation fails, then the uncertainty in is significant. = 1 1 2 + 2 2 2 + 1 + 2 1 2 2 (3.5.15) This concludes the presentation of the theory and equations n eeded to analyze the spectra of polystyrene thin films. From these equations the refractive index and thickness of transparent thin films can found. In addition, the relative value of the refractive index of the substrate can be determined. In addition there exists a potential opportunity to give
33 4 Experimental Results The following chapter applies the theory and techniques of chapter s 2 and 3 to polystyrene thin films on glass substrat es. The films were produced as described in chapter 1. The solution concentration used when spinning the films was 46 1 All wavelengths reported are in wavelengths in vacuum. First, the index of refraction of the film is determined and thic kness measurements are performed in section 4.1. In section 4.2, the range of wavelengths where < and the range of wavelengths where > is found. A switching point where = is determined and a rough estimate of is given. In section 4.3, the general shape of the reflectance spectra is examined. 4. 1 Index of Refraction and Thickness Measurements The first step in finding the index of refraction of the polystyrene films was to produce a film of known thickness. The thickn ess of a film was measured using an AFM as described in section 2.2. The film thickness was determined to be 700 10 After the thickness measurement, reflectance spectra of the film were collected using the microspectrometer. Spectra were collected on a spot right next to where the thickness m easurement was taken (see Fig. 4.1.1). All of the spectra turned out to be almost indistinguishable, so no new information was gained by collecting multiple spectra. The reflectance spectr um is shown in Fig. 4.1.2. From the spectrum, maxima and minima could be located. The uncertainty in locating an extrema was 2 This value for the uncertainty is based on the spread in values for repeated measurements. From these values of the extrema, equation (3.5.13) was used to calculate and equation (3.5.14) was used to calculate
34 Fig. 4.1.1: Microscope image of the film surface where reflectance spectra were collected. The spectra were taken where the black square is. The thickness measurement occurred at the crack just to the right of the black square. The crack appears as a thin vertical line. Fig. 4. 1.2: Reflectance spectra of the polystyrene thin film used to determine the refractive index of the film material. The film thickness was measured to be 700 10 80 90 100 110 120 130 140 150 160 170 200 400 600 800 Reflectance 10 2 Wavelength (nm) Spectra Data
35 Table 4.1.1 shows the locations of the extrema along with the calculated values for and A value of for the 327 maximum and the 303 minimum could not be calculated because for both of these extrema the extrema below it l ies on the other side of the = boundary Table 4.1.2 shows the calculated values for ( ) Note that the uncertainty in the measurem ents gets larger as gets smaller. Table 4.1.1: Locations of extrema in the reflectance spectra, along with the refractive index and uncertainty of the refractive index The refractive index was calculated using equation (3.5.13). A graph of the values obtained for ( ) is shown in Fig. 4.1.3 From this graph it can be seen that the uncertainties in ( ) are quite small, except for ( 280 ) where the error is quite large. Because the error for this value is significant, it was not used when doing a linear fit to the graph of vs. 2 as descri bed by equation (3.5.9). Fig. 4.1.4 shows the graph of vs. 2 as well as the equation of the linear fit done by the computer. Maxima ( 1 ) (nm) 1 Minima ( 1 ) (nm) 1 551 1 .11 0.04 690 1.07 0.03 407 1.19 0.06 472 1.14 0.05 327 364 1.29 0.09 280 1. 7 0.2 303 250 265
36 Table 4.1.2: Experimental values for the refractive index of a polystyrene thin film Fig. 4.1.3: Graph of the experimental values for the refractive index given in Table 4.1.2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 200 300 400 500 600 700 800 n (nm) Experimental Data ( ) 280 1 7 0 2 364 1 29 0 09 407 1 19 0 06 472 1 14 0 05 5 51 1 11 0 04 690 1 07 0 03
37 Fig. 4.1.4: Linear fit of versu s 2 T he refractive index should obey a relation of the form = + 2 From the equation of the linear fit shown in Fig 4.1.4, the values of and in equation (3.5.9) can be found, yielding an expression for the refractive index of the film give n by equation (4.1.1). and are rounded to three significant figures. A graph of equ ation (4.1.1) is shown in Fig. 4.1.5. From equation (4.1.1) and Fig. 4.1.5 one can see that di spersive effects are substantial for the polystyrene film. = 0 98 + 38 500 2 (4.1.1) With ( ) given by equation (4.1.1), thickness measurements were then made using equation (3.5.12) and the uncertainty due to 1 and 2 was calculated using equation (3 .5.15). Measurements were taken for a variety of values for 1 to see how the approximation 1 2 is affected with wavelength. y = 38489x + 0.9794 R = 0.9606 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 2E 06 4E 06 6E 06 8E 06 n 2 (nm 2 ) Experimental Data Linear Fit
38 Fig. 4.1.5: Refractive index as given by equation (4.1.1) for wavelengths ranging from 200 to 900 nm. There are two elements at play that affect this approximation. First, as decreases, ( ) together. The first fact seems to work to make the approximation 1 2 worse, while the se cond seems to counteract this effect. It was found experimentally that the extrema being close together did somewhat balance the rapid changes in ( ) Values for 2 1 started out at 0.07 for large wavelengths and increased to 0.10 for small w avelengths. As a consistency check, thickness measurements were performed on the film used to determine the refractive index (Film 2) This is pretty much going in a circle mathematically. However if the thickness does not come out correctly, that is an indicator that something is amiss. In addition thickness measurements were taken on another polystyrene film (Film 1) The values for the thicknesses are presented in Table 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 200 400 600 800 n (nm) Refractive Index
39 4.1.3. The uncertainty in the wavelength was again 2 The graphs of these measu rements are presented in Figs. 4.1.6 and 4.1.7. As with the index of refraction measurements, the uncertainty due to the uncertainties in 1 and 2 gets larger with smaller wavelengths. It seems that these thick ness mea surements give a fairly precise measurement of film thickness. By averaging, one can see from that the thickness of F ilm 2 is around 870 The films examined in these experiments are relatively thick as far as thin film research goes, so it is worthwhile to note that for thinner films the fractional uncertainty in the measurements would be much larger. This concludes the analysis of the spectra to find the refractive index of the film and to measure film thicknesses. It seems that the techniqu es for finding the refractive index of the film and for measuring film thicknesses have some value. These techniques are applicable not only to polystyrene films, but to any transparent dielectric film. Film 2 Film 1 1 1 370 870 70 364 710 50 404 850 50 407 690 30 453 870 40 472 690 30 510 860 30 551 700 20 594 880 30 690 700 10 705 870 20 Table 4.1.3: Thickness measurements of two thin films for different values of 1 The thickness is calcul ated by equation (3.5.12). All values are in units of nm.
40 Fig. 4.1.6: Thickness measurements for Film 1 for different values of 1 Fig. 4.1.7: Thickness measurements for Film 2 for different values of 1 780 800 820 840 860 880 900 920 940 960 300 400 500 600 700 800 h (nm) 1 (nm) Calculated Thickness 640 660 680 700 720 740 760 780 300 400 500 600 700 800 h (nm) 1 (nm) Calculated Thickness
41 4.2 Refractive Index of the Su bstrate Equations (3.5.7) and (3.5.8) state criteria that the quotient 1 2 should satisfy depending on the relative values and As discussed in chapter 3, there are three possibilities for the relative values of and for a given range of wavelengths. In this case the range is 200 < < 900 Those possibilities are: (1) < for all (2) > (3) there exists a wavelength such that > for < and < for > For the polystyrene thin films studied it was found that (3) happen s. It was also determined that 291 < < 295 So for some wavelength between 291 and 295 1 42 < < 1 43 This also shows that = 2 for 295 a nd = 2 2 for 291 To obtain these results, informatio n from seven films was used. The analysis of one of the films will be demonstrated, and then the results from all of the films will be presented. Consider Table 4.2.1 where the maxima, for one of the spectrum used, are li sted in the left column. Table 4.2.1: Maxima for a one of the spectrum are listed, along with the values of 1 2 obtained by dividing a maximum by the one below it. Maxima (nm) 1 2 < > 524 1 35 0 01 1.333 1.400 387 1 24 0 02 1.250 1.28 6 311 1 15 0 02 1.200 1.222 270 1 11 0 02 1.16 7 1.181 243
42 The second column gives 1 2 by dividing the maximum directly to the left of it by the maximum to its lower left. The third and fourth columns are the best theoretical match to t he top entry of the 1 2 column. From this data, it can be seen that the values for < theory match up well with the experimental values for the 524 and 387 maxima, while the values for the > theory do not match up well at all. Since the value for 1 2 for the 387 maximum involves the maximum below it, the 311 maximum must also obey the < theory. In addition, the value for 1 2 for the 311 maximum does not follow the < theory, so the < theory must break down somewhere between 270 and 311 This implies that 270 < < 311 Note that since the theoretical values are the best match to the top entry the > correlation column will not necessa < theory breaks down. This column is included to show that for some of the extrema, one of the theories is consistent with experiment while the other is not. The results for this film can be s ummarized in Table 4.2.2. By doing the same analysis to six more films, was located more precisely ( ) < > 524 387 3 11 270 243 Table 4.2.2: Summary of wavelengths and if < for that wavelength or not.
43 This type of analysis was done for seven films. Producing information for 30 different values of The data was very consist ent. Table 4.2.3 shows the results for the 10 closest wavelengths to the switching point. From Table 4.2.3 one can see that 291 < < 295 ( ) < > 338 327 311 296 284 280 270 267 Table 4.2.3: The 10 closest wavelengths to the = boundary. The closest values of to the switching point are in bold. One last thing should be said about the previous analysis. An attempt to d o a correlation between the theoretical and experimental values of 1 2 starting with the least maximum was made This is the same idea as the analysis presented in Table 4.2.1, but starting from the bottom instead of the top. The values of that did not match up with the < theory seemed to match up with the > theory, however there was too much ambigui ty as to which theoretical value of 1 2 went with the experimental value.
44 For example, the experimental value of 1 2 for some of the maxima was 1 11 0 02 T he theoretical values that fit in this range for the > theory are 1 0952 1 1052 1 1176 and 1 1333 So it was impossible to say what the best correlation was. Fortunately, this part of the analysis is unnecessary as long as equations (3.5.9) and (3.5.10) are true. It was only necessary to determine where the < theory broke down, and below that point the > theory is applicable. 4.3 Shape of the Reflectance Spectra From the Fresnel equations, the reflectance due to a reflected plane wave off of an interface c an be predicted if the refractive indices of both materials are known. The same is true for transmittance. If is the refractive index of the incident material, and is the refractive index of the transmitted material, then the reflectance and transmittance at normal incidence can be written as equations (4.3.1) and (4.3.2) = + 2 (4.3.1) = 4 ( + ) 2 (4.3.2) In se ction 4.1, the refractive index of the film was found as a function of In section 4.2, the refractive index of the substrate was estimated to be around 1 42 for a very small range of wavelengths. Using the approximation that the refractive index of t he substrate does not depend on equations (4.3.1) and (4.3.2) were applied to the relative reflectance spectra used to calculate the refractive index of the film material. All interference terms were ignored Ideally, this computed t h e o r e t ical a v e r a g e for the reflec tance should lie in the middle of the experimental spectra. Fig. 4.3.1 shows the graph of the experimental spectra as well as the theoretical average
45 Fig. 4.3.1: The experimental spectrum is shown in blue. The calculated theoretical average is shown in red. Ideally the red line would lie in the middle of the blue, with the blue oscillating around it. From Fig. 4.3.1, one can see that the theoretical value lies below the experimental for most values of In addition for small values of the the oretical value blows up. This discrepancy between the experimental and theoretical spectra is most likely due to the fact that the refractive index of the substrate is not accurately known. This explains why the theoretical value increases rapidly as decreases: i n a c t u a l i t y the re fractive index of the substrate changes rapidly for smaller wavele ngths, but the theoreti cal a v e r a g e does not take this into account. I n r e a l i t y the experimental reflectance p r o b a b l y does follow some curve similar to the theoretical one (minus the aspect of blowing up for small values of ) If this is the case, th en the overall rises and drops ( not the r i s e s a n d d r o p s o f t h e interference pattern ) can be attributed to the varying refractive indices of the film and the substrate. 50 100 150 200 250 300 200 400 600 800 Reflectance (10 2 ) (nm) Experimental Theoretical Avg
46 Con clusion The experimental results show that there is some worth to the techniques and theory presented. The refractive index of the film was found to obey the Cauchy formula = + 2 F rom the ten extrema observed in the reflectance spectra on ly five usable values for ( ) could be obtained. The linear fit done by the computer could be improved if more experimental values for ( ) could be found. More experimental values for ( ) could not be obtained by doing a similar analysis of the transmittance spectra for the same film, because the extrema of the transmittance spectra occur at the same locations as extrema for the reflectance spectra with the only difference being maxima and minima are reversed. More experimental values for ( ) could be obtained by doing the same analysis to another film of known thickness. The thickness of this film would have to be at least slightly different from the first film used, so that the extrema would occur for different values of The thicknes s measurements from analyzing the spectra had fairl y high uncertainty. The uncertainty could be reduced by accounting for dispersive effects between extrema. Even with a high uncertainty, this method still seemed acceptable for measurements of fairly thic k films ( > 500 ) In many circumstances using the spectrometer is preferable to the AFM. Acquiring a spectrum takes only seconds, while producing a usable AFM image takes around an hour. There are a few situations where the AFM would be preferable. For instance one might be performing a thickness measurement on a film that is very thin ( < 100 ) in while case using the spectrometer might not w ork at all.
47 The experimental results determining the difference in the optical path length were surpris ingly consistent, and the = switching point was located very precisely. The refractive index of the substrate could be determined for this value of but nothing could be said about the refractive index of the substrate over the entire spectrum In conclusion, it should be emphasized that the techniques discussed in this work are not only applicable to polystyrene films, but with minimal adjustments, they are applicable to any transparent thin film under a variety of conditions. This work is an example of how properties of a sample can be extracted from its reflectance spectr um It is likely that there is even more information about the thin films spectra By applying the known laws of optics to the process of collecting a spectrum more information about the geometry and properties of the film can be obtained.
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