1A: Introduction to WVASE Data Analysis

Transcription

1 2014 J.A. Woollam Co., Inc A: Introduction to WVASE Data Analysis Nina Hong U Penn, February 2014

2 Session Outline 1. Introduction to Ellipsometry. 2. Classification of Samples. Transparent Semi-absorbing Absorbing Bare substrates, and coated samples. 3. Oscillator Models and Genosc Layer. Types of oscillators. Learning Outcomes: Basic operation of WVASE software. Fundamentals of spectral interpretation J.A. Woollam Co., Inc. 2

3 2014 J.A. Woollam Co., Inc. 3 WVASE Software Opening Screen. Five Empty Windows. Click inside each. How to use these windows?

4 2014 J.A. Woollam Co., Inc. 4 Experimental Data Window Use to open data files. From upper-left. Or right-click to open

5 2014 J.A. Woollam Co., Inc. 5 Graph Window Displays graphs of Experimental Data Select Type to change graphs.

6 2014 J.A. Woollam Co., Inc _CaF2 Substrate.dat Step 1: Select Experimental Data Window. Open data file: 1_CaF2 Substrate.dat Step 2: Select Graph Window Plot Y and D data. Y and D? What are these? What do they mean? Learning Outcomes: Use Experimental Data and Graph Windows. Open data files. Display graphs of Y and D.

7 2014 J.A. Woollam Co., Inc. 7 Ellipsometry Measurement 1. linearly polarized light... E p-plane s-plane 3. elliptically polarized light! p-plane E s-plane plane of incidence 2. reflect off sample... Ellipsometry measures the change in polarization state of light reflected from a surface. Sample properties cause the change. Polarization is rotated, and becomes elliptical. Next: What is polarized light? What is Ellipsometry?

8 2014 J.A. Woollam Co., Inc. 8 Units: Wavelength, Photon Energy, & Wavenumber Y X Light Wave. l E ev Wavelength, l (units in Å, nm, or microns) Photon Energy (ev). Electron Volts Wavenumber (cm -1 ). Used Mid to Far IR. Unit conversions: , E ev, E, l l ev Å nm l m Z cm l m

9 2014 J.A. Woollam Co., Inc. 9 What is Polarization? Shape of the beam as it propagates towards us. (Looking down z-axis). Y Electric field, E(z,t) Y E X X Z Direction of propagation Looking Down Z-Axis: Linear Polarization Polarization state defined by orientation & phase of the E-field vector.

10 2014 J.A. Woollam Co., Inc S P Polarized Light Polarization can be linear, circular or elliptical. Superposition of two orthogonal light beams (components). X wave1 E S P Y E wave2 Z ANIMATIONS FROM ACCESSED JUNE 6, 2012

11 2014 J.A. Woollam Co., Inc What is Polarization? Polarization: Describes shape of electric field along the direction of travel. X wave1 Y wave2 Z Linear: arbitrary amplitudes, in-phase. Circular: equal amplitudes, 90 phase difference. Elliptical: arbitrary amplitudes, arbitrary phases.

12 2014 J.A. Woollam Co., Inc Intensity and Polarization Light Intensity = Size of the Ellipse. One Number Describes Wave Amplitude (E). Intensity is proportional to E 2 Brightness. I E 2 2 ( E x E y 2 ) Less Intense Y E Different Size (Intensity). More Intense Y E X Same Shape! (Polarization) X Polarization = Shape of Ellipse. 2 numbers required: ellipse Azimuth and Ellipticity. Independent of ellipse size. Independent of intensity Ellipsometry always measures 2 numbers: Y & D.

13 2014 J.A. Woollam Co., Inc Light and Polarization For more details 1997, 330 pages 2011, 750 pages 1999, 950 pages

14 2014 J.A. Woollam Co., Inc Ellipsometry: Angle & Plane of Incidence 1. linearly polarized light... E p-plane s-plane plane of incidence Surface Normal Vector 3. elliptically polarized light! p-plane E s-plane 2. reflect off sample Angle of Incidence. Measured from Sample Normal. 2. Plane of Incidence. Defined by incident beam, reflected beam, and sample normal. 3. Polarization Components p- and s- are parallel and perpendicular to plane of incidence.

15 2014 J.A. Woollam Co., Inc tan in e i Ellipsometry Measurement Measures change in polarization of reflected light. in E E in p in s i e in p in s E p-plane s-plane 2 tan out e i E p-plane out E E out p out s i e out p out s plane of incidence s-plane 3 Change in Polarization represented as Ratio of Light-Out to Light-In E E E E out p out s in p in s e i e i out p in p out s in s E E out p out s E E in s in p e i out p out s in p in s

16 2014 J.A. Woollam Co., Inc Ellipsometry Equation For Reflection, use the following Definitions: r p r s E E E E out p in p out s in s D D p s refl p refl s in p in s n,k film Substrate Interference t Rewriting, we get the Ellipsometry Equation: r r p s e i D p D s id tan( Y) e = r p r s Can determine Optical constants & Film Thickness

17 Summary: Ellipsometry Experiment Light is reflected from a surface of interest. The polarization state of incident light is known. The polarization state of reflected/transmitted light is measured. An ellipsometer determines the change in polarization from the sample. Y and D values. 1. linearly polarized light... E p-plane s-plane 3. elliptically polarized light! p-plane E s-plane plane of incidence 2. reflect off sample J.A. Woollam Co., Inc. 17

18 2014 J.A. Woollam Co., Inc Ellipsometer: Block Diagram Every Ellipsometer contains the following components Source Head Light Source Polarization Generator Polarization Analyzer Detector Receiver Head Sample SE also needs wavelength selection.

19 Light Source 2014 J.A. Woollam Co., Inc Woollam Ellipsometers: Configurations Rotating Analyzer VASE, VUV-VASE M-2000 IR-VASE alpha-se RC2 P AR S A P Rotating Compensator C S C A Detector

20 2014 J.A. Woollam Co., Inc General SE References Beginner: 1. H.G. Tompkins, and W.A. McGahan, Spectroscopic Ellipsometry and Reflectometry, John Wiley & Sons, Intermediate: 2. Hiroyuki Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, John Wiley & Sons, Advanced: 3. Ellipsometry at the Nanoscale, Lasurdo and Hingerl, eds., Springer-Verlag, Handbook of Ellipsometry, Tompkins and Irene, eds., William Andrew Publishing, R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light, Elsevier,

21 2014 J.A. Woollam Co., Inc Ellipsometry Conference Proceedings 1. Spectroscopic Ellipsometry, (1993) A.C.Boccara, C.Pickering, J.Rivory, Editors, Elsevier Publishing. 2. Thin Solid Films Vol (1998). R.W.Collins, D.E.Aspnes, and E.A. Irene, Editors, Elsevier Science. 3. Thin Solid Films Vol , (2004). M. Fried, K. Hingerl, and J. Humlicek, Editors, Elsevier Science. 4. Phys. Stat. Sol. (c) 5, No. 5, (2008). M. Schubert, H. Arwin, U. Beck, Editors, Wiley-VCH. 5. Thin Solid Films Vol. 519, Issue 9, (2011). H. G. Tompkins, Editor, Elsevier Science.

22 2014 J.A. Woollam Co., Inc _BK7_SE.dat, Append Transmission. Step 1: Select Experimental Data Window. Open data file: 2_BK7_SE.dat. Append Data 2_BK7_T.dat Step 2: Select Graph Window Plot Y and D data. Plot Transmission data. What do they mean? (Describe polarization ellipse). What is each sensitive to? (Interpret the spectrum). Learning Outcomes: Appending different types of data. Graphing different data types. Sensitivity of different data types to sample properties

23 2014 J.A. Woollam Co., Inc Example 2: Uncoated BK7 Glass Substrate 3 data types. Each sensitive to different sample properties (index, roughness, absorption). 15 Generated and Experimental 180 Generated and Experimental 1.0 Generated and Experimental Y in degrees Model Fit Exp E 55 Exp E 57.5 Exp E 60 Exp E 62.5 Exp E 65 D in degrees Model Fit Exp E 55 Exp E 57.5 Exp E 60 Exp E 62.5 Exp E 65 Transmission Model Fit Exp pt Wavelength (nm) Wavelength (nm) Wavelength (nm) Y: index D: Roughness Transmission: Absorption

24 2014 J.A. Woollam Co., Inc Spectral Features: Transparent Substrate Psi flat and smooth - follows shape of index Delta = 0 or except for surface films Y in degrees Wavelength (nm) D in degrees nm roughness Wavelength (nm)

25 2014 J.A. Woollam Co., Inc _nb.dat. Step 1: Select Experimental Data Window. Open data file: 3_Nb.dat. Step 2: Select Graph Window Plot Y and D data. Plot <n> & <k>. NOTE THE <> BRACKETS! Pseudo -values. What do they mean? Learning Outcomes: Study spectrum of opaque material. Introduction to pseudo <n> & <k>. Sample properties causing change in polarization state.

26 2014 J.A. Woollam Co., Inc Niobium: Psi and Delta Opaque Sample: Metals Experimental Data Experimental Data Y in degrees Exp E 60 Exp E 65 Exp E 70 D in degrees Exp E 60 Exp E 65 Exp E Wavelength (nm) Re-plotted as Pseudo <n>&<k> Experimental Data Wavelength (nm) Experimental Data <n> Exp E 60 Exp E 65 Exp E 70 <k> Exp E 60 Exp E 65 Exp E Wavelength (nm) Wavelength (nm) Opaque Sample. All <n>&<k> angles the same.

27 Psi in degrees Delta in degrees 2014 J.A. Woollam Co., Inc Spectral Features: Opaque Metals Psi stays near 45 Delta away from 0 or 180 Pseudo <n>&<k> same at all angles Wavelength (nm) Wavelength (nm)

28 Substrate Identification Psi Delta <Pseudo n&k> Flat Transparent Approaches 0 at Brewster s angle Flat Close to 0 or 180. <n> has Cauchy shape <k> close to zero Absorbing Shows features of n&k Remains closer to 45 for all angles Shows features of n&k Stays between 0 and 180 <n&k> follow actual n&k. Transparent Substrate Exceptions: 1. <n> separation for variable angle when backside reflections are present. 2. <k> non-zero and Delta away from 0,180 if surface layer (e.g. roughness). What about Semi-Absorbing? 2014 J.A. Woollam Co., Inc. 28

29 2014 J.A. Woollam Co., Inc _Si wafer.dat Step 1: Select Experimental Data Window. Open data file: 4_Si wafer.dat. 40 Experimental Data 180 Experimental Data Y in degrees Exp E 65 Exp E 70 Exp E 75 D in degrees Exp E 65 Exp E 70 Exp E Wavelength (nm) Wavelength (nm) Learning Outcomes: Study spectrum of crystalline material. Note sharp structure below 450 nm. Why? Why so sharp? Identify transparent and absorbing regions.

30 2014 J.A. Woollam Co., Inc Substrate - Semiconductors Psi follows shape of absorption. Delta away from 0 or 180 when absorbing. Sharp features due to absorption Y in degrees D in degrees Wavelength (nm) Wavelength (nm) 1200

31 2014 J.A. Woollam Co., Inc Review: Substrate Types, Spectral Features. Transparent Semi-Absorbing Absorbing Y D Describe unique features of each spectrum.

32 Coated Samples! Thin Film Identification Multiple Reflections at interfaces. Use to estimate film thickness and Index. Can be transparent, Absorbing, or Semi-Absorbing. ~n 0 ~n 1 ~n J.A. Woollam Co., Inc. 32

33 2014 J.A. Woollam Co., Inc Coated Thin Film Examples Transparent Semi-Absorbing Absorbing Y D Coatings add interference effects in transparent regions.

34 2014 J.A. Woollam Co., Inc _SiO2 on Si.dat Transparent Film on Silicon Step 1: Open data file: 5_SiO2 on Si.dat. Step 2: Select Graph Window Plot Y and D data. Note the interference effects. What Causes these oscillations? Y in degrees Experimental Data Exp E Wavelength (nm) Learning Outcomes: Identify spectrum of a coated substrate with transparent film. Observe thin-film interference effects from transparent Film.

35 2014 J.A. Woollam Co., Inc Thin Film on Substrate Multiple reflections in single film lead to an infinite series of reflections. r tot r 01 t 01 t 10 r 12 e i2β t 01 t 10 r 10 r 122 e i4β Phase from different optical paths is tracked by film phase thickness (b) d N 0 N 1 θ i θ 1 r 01 t 01 r 12 r 10 r 12 t 10 t 10 t 10 r 10 r r b 2 d N cos 1 1 l N 2 t θ 12 2 t12 t 12 t tot r tot -2i b -4i b r01 t01r12t10e t01r12 r10t10e... 2

36 2014 J.A. Woollam Co., Inc SE measures polarization change as light interacts with sample. Thin-Film Interference Effects Interference occurs as light recombines after traveling different paths through thin film.

37 SE features Transparent Films Oscillation patterns in ellipsometry data due to thin film interference. Pattern affected by thickness and index. ~n 0 ~n 1 ~n 2 Transparent - oscillations 2014 J.A. Woollam Co., Inc. 37

38 2014 J.A. Woollam Co., Inc Thickness Effects Thicker films have more interference oscillations

39 2014 J.A. Woollam Co., Inc _Dielectric on Glass.dat Very Thick Transparent Film on Glass Step 1: Open data file: 6_Dielectric on Glass.dat. Select Graph Window Plot Y and D data. Thick or thin film? Why small amplitudes? Step 2: Generated and Experimental Wavelength (nm) Learning Outcomes: Identify spectrum of thick versus thin transparent film. Note small amplitude of oscillations. Why? Y in degrees Model Fit Exp Eb 50 Exp Eb 55 Exp Eb 60 Exp Eb 65 Exp Eb 70

40 Thin Film Identification Transparent Absorbing Psi Delta Oscillates Size of oscillations signify index difference between substrate and thin film Number of oscillations indicate thickness Oscillates Number of oscillations indicate thickness No oscillations. Shows features of n&k. Looks like opaque substrate. Shows features of n&k. Stays between 0 and 180. <Pseudo n&k> May separate for multi-angles. Overlapping at multi-angles. VERY THIN LAYER Exceptions: 1. Oscillations will not appear until about nm thickness. 2. Psi may go above 45 for thin coatings. What about Semi-Absorbing? 2014 J.A. Woollam Co., Inc. 40

41 Break! 2014 J.A. Woollam Co., Inc. 41

42 2014 J.A. Woollam Co., Inc _SiNx on Si.dat Semi-Absorbing Film on Silicon Step 1: Open data file: 7_SiNx on Si.dat. Step 2: Select Graph Window Plot Y and D data. Note the interference effects. Discuss IR and UV behavior. Experimental Data Wavelength (nm) Learning Outcomes: Identify spectrum of a UV absorbing film. Observe thin-film interference effects and absorption effects. Y in degrees D in degrees Exp E 65 Exp E 67.5 Exp E 70 Exp E 72.5 Exp E 75 Exp E 77.5 Exp E 80 Experimental Data Exp E 65 Exp E 67.5 Exp E 70 Exp E 72.5 Exp E 75 Exp E 77.5 Exp E Wavelength (nm)

43 2014 J.A. Woollam Co., Inc Semi-Absorbing Films Spectral Identification: What to expect? If light is absorbed before returning to surface, only top reflection is seen (Film appears as opaque substrate).

44 2014 J.A. Woollam Co., Inc WVASE Model Window Add layers. Define fit parameters. Contains optical constants of each layer. What are Optical Constants?

45 2014 J.A. Woollam Co., Inc _SiNx on Si.dat Build Model For Silicon Nitride on Silicon Step 1: Open data file: 7_SiNx on Si.dat. Step 2: Select Model Window. Add Layer: Si_jaw.mat. Add Layer: Si3N4.mat. Add fit Parameter: Film Thickness. Show table of optical constants. 1 si3n nm 0 si_jaw 1 mm Learning Outcomes: Building a model. Adding layers. Adding fit parameters.

46 2014 J.A. Woollam Co., Inc Generate Data Window Generates data from the Model Graphs compare Model Generated and Experimental Data. Thickness is close, so why not a perfect match?

47 2014 J.A. Woollam Co., Inc Optical Constants, Refractive Index, n Light in Different Media: Velocity and wavelength vary in different materials. n = 1 n = 2 Frequency (color) of wave remains the same! Energy of photon remains the same: E( ev ) h v c n f 1,240 l( nm) speed of light Planck s constant v l

48 2014 J.A. Woollam Co., Inc Index of Refraction (n) Index of Refraction, n Describes How Light is affected by Materials: reflection, refraction, phase velocity, absorption f i f r n,k n = 1 n = 2 f t Frequency constant v l c nl

49 Refractive Index, n Determines Wave speed. n = c/n Determines Angle of Propagation. Snell s Law of Refraction: n 1 sin 1 = n 2 sin 2 Example: Pencil in glass of water appears Bent, or Broken. Caused by refraction J.A. Woollam Co., Inc. 49

50 Refractive Index: Transparent Materials Index decreases for longer wavelengths. Higher index for stronger UV absorption. 4 SiO2 Si3N4 Si 3 Index, n Wavelength in nm 2014 J.A. Woollam Co., Inc. 50

51 2014 J.A. Woollam Co., Inc Absorption and Extinction I I z o e Bulk Material Absorption: Deeper propagation gives: = more energy loss. = complete extinction. Absorbing films: Increasing thickness gives: = longer propagation path in film. = wave spends longer time in film. = more absorption by material. = more extinction of wave. 4k l z

52 Absorption: Beer-Lambert-Bouguer Law Absorption coefficient () Extinction coefficient (k) Loss of wave energy to the material. I o z I o I(z) di( z) I ( z) dz z I( z) I o e k 4k l l 4 Pierre Bouguer ( ) D p z Johann Heinrich Lambert ( ) 2014 J.A. Woollam Co., Inc. 52

53 2014 J.A. Woollam Co., Inc Multiple types of optical absorption Common absorption of EM waves: Electronic UV-visible-near IR frequencies Vibrational mid-ir freq Free carriers (conductivity) mostly IR E -e Free carriers accelerated by E-field Rotational (microwave oven)

54 Index of refraction, n 2014 J.A. Woollam Co., Inc Metals: Free-carrier Absorption Metals have more electrons than needed for bonding (covalent bonds = sharing electrons). Electric wave causes free electrons to move. When electrons collide with nearby atoms wave energy is lost as heat n k Nickel Wavelength (nm) Extinction Coefficient, k

55 2014 J.A. Woollam Co., Inc Semiconductors: Electronic Transitions Electrons in energy bands. Gap between valence-conduction band. If light (photon) provides enough energy electron may excite to higher state. Conduction Band Indirect Transition Photon Energy, ev Direct Bandgap Energy, ev Valence Band Momentum, k Real materials have complicated energy bands. See lots of sharp structures in optical constants. S. Adachi, Optical properties of crystalline and amorphous semiconductors, Kluwer Academic Publishers, Boston (1999).

56 2014 J.A. Woollam Co., Inc Infrared Absorption: Molecular Vibrations Infrared light has low energy. Can vibrate molecules (stretching, bending, twisting, rotating). CH2 stretch (symmetric) CH2 stretch (asymmetric) N-H stretch C=O stretch

57 2014 J.A. Woollam Co., Inc Infrared Absorption: Crystal Lattice Vibrations Electric Field causes crystal lattice to vibrate. Longitudinal (LO) or Transverse (TO) Optical Phonons. 180 E k D in degrees TO 2TO TO+LO M. Schubert, 2002 TO+LA Wave Number (cm -1 ) T.E. Tiwald et al., Thin Solid Films, (1998) 661.

58 Optical Constants Complex Refractive Index: ñ(l) = n(l) + ik(l) E z, t = E 0 exp i wt Kz + δ K = 2π λ (n ik) Describes what material does to light wave. Slows down, wavelength changes, refraction, extinction. Alternatively Complex Dielectric Function: e(l) = e 1 (l) + ie 2 (l) D = εe ε = ε χ, ε r = ε ε 0 Describes what wave does to material J.A. Woollam Co., Inc. 58

59 Extinction Coefficient 'k' 2014 J.A. Woollam Co., Inc _SiNx on Si.dat Graphing Optical Constants in WVASE. Using Graph Window. Select Data: Current Layer (Opt. Const.) Note position of current layer arrow > next to layer. Select Global, then Defaults Change plot to e1 & e2 vs ev si3n4 Optical Constants si3n4 Optical Constants 0.30 Index of Refraction 'n' Wavelength (nm) n k Real(Dielectric Constant), e e 1 e Photon Energy (ev) Imag(Dielectric Constant), e 2 Learning Outcomes: Graphing layer optical constants in WVASE. Understand Current layer. Changing defaults settings.

60 Dispersion: Wavelength Dependence Optical Properties vary with wavelength. Transparent regions (normal dispersion). Absorbing regions (anomalous dispersion) J.A. Woollam Co., Inc. 60

61 The Dielectric Function e(w) The Dielectric Function e(e) or e(w), is the Dielectric Polarization as a function of Energy (or Frequency). e dc e transparent absorption absorption Regions of transparency, and regions of absorption. Where do these features come from? 2014 J.A. Woollam Co., Inc. 61

62 2014 J.A. Woollam Co., Inc Kramers-Kronig Relation Real and imaginary parts depend on each other. Optical constants must satisfy KK-relation to be physical. Bumps make wiggles the shape of n & k curves are related. e e 1 2 w w 2 we 2 w 1 P dw w w 2w P 0 P means principal part e1 w 1 dw 2 2 w w Frequency (energy) e 1 e 2 n k Integral equation relates area under e 2 to shape of e 1 and vice-versa.

63 2014 J.A. Woollam Co., Inc Absorbing Regions Absorbing regions occur in different parts of the spectrum. Absorptions have different shapes, depending on cause. Free-carriers Electronic Transitions Molecular Vibrations Lattice Vibrations e Lattice Vibrations Transparent IR Rutile TiO 2 UV to IR Visible Electronic Transitions UV e 1 e e Photon Energy (ev)

64 Dispersion Models: Practical Use. Equations used to describe the optical functions versus wavelength. Transparent Materials: Cauchy, Sellmeier models. Absorbing Materials: Large variety of Oscillator Models: Lorentz, Drude, Gaussian, Tauc-Lorentz, etc. All available within Genosc.mat layer! 2014 J.A. Woollam Co., Inc. 64

65 Transparent Materials: Cauchy Dispersion Equation Describes index (n) as a function of wavelength n(l). For transparent materials only (k=0 or very small). A sets index range n (l) = A +B / l 2 + C / l 4 k (l) = 0 B and C give dispersion shape Augustin-Louis Cauchy ( ) 2014 J.A. Woollam Co., Inc. 65

66 2014 J.A. Woollam Co., Inc _SiNx on Si.dat Using Model For Silicon Nitride on Silicon Select Model Window. Replace Si3N4 layer with Cauchy.mat Set An=2, Bn=0.01. Select Experimental Data Window. Range Select nm. 1 cauchy nm 0 si_jaw 1 mm Generate data. Learning Outcomes: Replacing layers. Define Fit parameters. Cauchy model for index. Select Experimental data range.

67 2014 J.A. Woollam Co., Inc Normal fit WVASE Fit Window Used to optimize model by adjusting fit parameters Or Right-click Perform a Normal Fit to start. What s going on below 450 nm? Beyond 850 nm? How to improve this fit?

68 Dispersion Models: Practical Use. Equations used to describe the optical functions versus wavelength. Transparent Materials: Cauchy, Sellmeier Absorbing Materials: Large variety of Oscillator Models: Lorentz, Drude, Gaussian, Tauc-Lorentz, etc. All available within Genosc.mat layer! 2014 J.A. Woollam Co., Inc. 68

69 2014 J.A. Woollam Co., Inc d_SiNx on Si-genosc.env Silicon Nitride on Silicon Final Model Using Genosc.mat Select Global, then Load New Environment. Open 7d_Sinx on Si-genosc.env Examine Y and D data fits. Are the fits good? Click Genosc layer to open Window. Study oscillators and poles used for fit. Learning Outcomes: Opening Environment Files. Study Genosc fit.

70 2014 J.A. Woollam Co., Inc Extinction Coefficient 'k' 7 7d_SiNx on Si-genosc.env Silicon Nitride on Silicon Final Results 2 srough nm 1 genosc nm 0 si_jaw 1 mm Y in degrees Generated and Experimental Model Fit Exp E 65 Exp E 67.5 Exp E 70 Exp E 72.5 Exp E 75 Exp E 77.5 Exp E Wavelength (nm) 300 Generated and Experimental e1 D in degrees Model Fit Exp E 65 Exp E 67.5 Exp E 70 Exp E 72.5 Exp E 75 Exp E 77.5 Exp E 80 e Wavelength (nm) genosc Optical Constants What is the role of e1 and e2 sections? Index of Refraction 'n' n k Wavelength (nm)

71 2014 J.A. Woollam Co., Inc Ellipsometry Data Analysis Rest of this course 3 days. How to analyze data. Introduce some general concepts.

72 2014 J.A. Woollam Co., Inc Data Content vs Sample Unknowns Ensure data contains information to solve all unknowns. Data Content Sample Unknowns Over-Determined Data Content Sample Unknowns Data Content Sample Unknowns Under-Determined

73 2014 J.A. Woollam Co., Inc Why Analyze Data? Ellipsometry measures two parameters: Y and D. Y and D are a function of film thicknesses and optical constants of materials under study. Desired information must be extracted through a model based analysis using optical physics. Inverse Problem: Result is known instead of Cause. Ellipsometry measures the resulting polarization change rather than sample properties that are the cause. Yin degrees Ellipsometry Measurement Experimental Data Wavelength (nm) Exp Y-E 73 Exp D-E D in degrees Yes Sample Structure 2 srough Å 1 film Å 0 silicon 1 mm

74 2014 J.A. Woollam Co., Inc Generate Data From Model Calculate psi & delta from model. Compare to Experimental data. Y in degrees Generated and Experimental Model Fit Exp E 70 Exp E Wavelength (nm) 80 Generated and Experimental Adjust unknown (fit) parameters to get close to solution. Y in degrees Model Fit Exp E 70 Exp E Wavelength (nm)

75 2014 J.A. Woollam Co., Inc MSE Data Fitting 100 Software adjusts fit parameters to find best match between model and experiment. Yin degrees Wavelength (nm) MSE is Difference. Thickness

76 2014 J.A. Woollam Co., Inc Data Fitting Algorithm D Y D D Y Y N i i i i i i i M N M N MSE exp, exp mod 2 exp, exp mod WVASE32 uses a Mean Squared Error (MSE) to quantify the difference between experimental and calculated model data. A smaller MSE implies a better fit. The MSE is weighted by the error bars on each measured data point, so noisy points are weighted less heavily.

77 2014 J.A. Woollam Co., Inc Minimizing the MSE Levenberg-Marquardt algorithm used to quickly determine the minimum MSE (gives best fit to data). MSE starting thickness (guess) Local Minima BEST FIT Thickness

78 2014 J.A. Woollam Co., Inc Quality of Data Fit Use our Physical Intuition! When fit is complete resulting fit parameters must be evaluated for sensitivity and possible correlation. Compare experimental data with generated data. How low is MSE? Can it be reduced further by increasing model complexity? Are fit parameters physical? How large are 90% confidence limits? Check Correlation matrix. Are parameters correlated? Can some be turned off?

79 Correlation Matrix Example of Correlation Matrix matrix of two-parameter correlation coefficients calculated from the covariance matrix Any off diagonal elements near +/-1 are correlated. Correlated Parameters: Cn.2 and Bn J.A. Woollam Co., Inc. 79

80 General Rules for Data Analysis Find the simplest optical model that fits Experimental Data. User fewest fit parameters. Verify uniqueness of the model. Check correlation matrix, and error bars on parameters. Optical 'constants' for materials are not always constant, and quality of fit can only be as good as the optical constants assumed in the model. May need to fit n & k J.A. Woollam Co., Inc. 80

81 2014 J.A. Woollam Co., Inc Data Analysis Flowchart

82 Summary Session #1 Introduction to Spectroscopic Ellipsometry Polarized light, Ellipsometry experiment. Classification of Samples: Substrate, or coated,? Single-Layer, Multi-Layer? Absorbing, Transparent, Semi-Absorbing? Data Features: Spectral Interpretation. Optical constants. Refraction, absorption, n & k. Dispersion. Oscillator models, Cauchy, Lorentz, Gaussian, etc. WVASE Genosc layer. Fundamentals of WVASE. Basic software operation J.A. Woollam Co., Inc. 82

83 Lunch! 2014 J.A. Woollam Co., Inc. 83