Part 1: Theory of Ellipsometry .................................. 1
1. Polarized Light and Ellipsometry ............................ 3
1.1. A Quick Guide to Ellipsometry ......................... 4
1.1.1. Light Waves and Photons ....................... 4
1.1.2. Polarization of Light ......................... 6
1.1.3. Ellipsometric Configurations .................. 9
1.1.4. Null Ellipsometry ............................ 12
1.1.5. Photometric Ellipsometry and Polarimetry ..... 13
1.2. Maxwell and Wave Equations ........................... 19
1.2.1. Linear Local Response ........................ 20
1.2.2. Linear Non-Local Response .................... 22
1.2.3. Dipole Moment, Susceptibility and
Inductions ................................... 23
1.2.4. Relationships Between Optical Constants ...... 24
1.2.5. Wave Equation for Monochromatic Fields ....... 26
1.2.6. Plane Waves in Isotropic Medium .............. 29
1.3. Representations of Polarization ...................... 31
1.3.1. Representation by Ellipsometric Angles ....... 32
1.3.2. Special Cases: Linear and Circular
Polarization ................................. 35
1.3.3. Orthogonal Polarization States ............... 37
1.3.4. Representation by Complex Numbers ............ 37
1.3.5. Light Intensity, Detection of Polarization
State ........................................ 40
1.4. Propagation of Polarized Light ....................... 45
1.4.1. Jones Vectors ................................ 45
1.4.2. Jones Matrices ............................... 48
1.4.3. Quantum Mechanical Description, Partial
Polarization ................................. 53
1.4.4. Stokes Vectors ............................... 56
1.4.5. Mueller Matrices ............................. 59
1.5. Reflection and Transmission of Polarized Light at
Planar Interfaces .................................... 67
1.5.1. Matching Plane Waves at a Planar Interface ... 67
1.5.2. Fresnel Coefficients ......................... 72
1.5.3. Special Values of the Angle of Incidence ..... 74
1.5.4. Ratio of Amplitude Reflectivities ............ 76
1.5.5. Propagation Matrices, Stratified
Structures ................................... 80
1.5.6. Substrate-Film-Ambient System ................ 85
1.6. References ........................................... 90
2. Optical Physics of Materials ............................... 93
2.1. Introduction ......................................... 93
2.2. Propagation of Light in Solids ...................... 102
2.2.1. Optically Isotropic Solids and the Complex
Dielectric Function ......................... 102
2.2.2. Optically Anisotropic Solids and the
Dielectric Tensor ........................... 110
2.2.3. Dispersion Relationships .................... 124
2.3. Classical Theories of the Optical Properties of
Solids .............................................. 125
2.3.1. Semiconductors and Insulators: the Lorentz
Oscillator Model ............................ 125
2.3.2. Metals: The Drude Free Electron Model ....... 129
2.3.3. Plasmons .................................... 132
2.3.4. Optical Sum Rules ........................... 136
2.4. Quantum Mechanical Theories of the Optical
Properties of Solids ................................ 137
2.4.1. Quantum Theory of Absorption and
Dispersion .................................. 138
2.4.2. Direct Interband Transitions in Solids ...... 146
2.4.3. Band Structure and Critical Points in
Solids ...................................... 150
2.4.4. Indirect Interband Transitions in Solids .... 153
2.4.5. Intraband Transitions in Metals ............. 157
2.5. Modeling the Optical Properties of Solids ........... 159
2.5.1. Classical Lorentz Oscillator Models ......... 159
2.5.2. Classical Drude Models ...................... 172
2.5.3. Generalized Quantum Mechanical Models ....... 178
2.5.4. Specialized Quantum Mechanical Models ....... 207
2.6. Overview and Concluding Remarks ..................... 227
Acknowledgments ..................................... 230
2.7. References and Bibliography ......................... 230
2.7.1. Numbered References ......................... 230
2.7.2. Bibliography ................................ 233
3. Data Analysis for Spectroscopic Ellipsometry .............. 237
3.1. Introduction ........................................ 237
3.2. Ellipsometry Parameters ............................. 239
3.2.1. Calculated Parameters: Jones Matrices ....... 240
3.2.2. Measured Parameters: Mueller Matrices ....... 241
3.2.3. Mueller-Jones Matrices ...................... 242
3.3. Calculation of Complex Reflection Coefficients ...... 246
3.3.1. Isotropic, Homogeneous Systems .............. 246
3.3.2. Anisotropic Systems ......................... 248
3.3.3. Inhomogeneous Layers ........................ 251
3.4. Models for Dielectric Functions ..................... 252
3.4.1. Tabulated Data Sets ......................... 253
3.4.2. Lorentz Oscillator Model .................... 254
3.4.3. Optical Functions of Amorphous Materials .... 255
3.4.4. Models for Crystalline Materials ............ 258
3.4.5. Effective Medium Theories ................... 260
3.5. Fitting Models to Data .............................. 262
3.5.1. Figures of Merit ............................ 263
3.5.2. Errors in Spectroscopic Ellipsometry ........ 265
3.5.3. Convergence Routines ........................ 268
3.5.4. An Example: (a-SixNy:H)...................... 271
3.6. Determination of Optical Functions from
Spectroscopic Ellipsometry Data ..................... 276
3.6.1. Optical Functions from Parameterization ..... 278
3.6.2. Newton-Raphson Algorithm .................... 280
3.6.3. Optical Functions of Bulk Isotropic
Semiconductors and Insulators ............... 282
3.6.4. Optical Functions of Anisotropic
Materials ................................... 285
3.6.5. Optical Functions of Thin Films ............. 286
3.7. Depolarization ...................................... 289
Acknowledgements .................................... 293
3.8. Further Reading and References ...................... 293
Optics and Ellipsometry ............................. 293
Data Reduction ...................................... 294
Numbered References ................................. 294
Part 2: Instrumentation ....................................... 297
4. Optical Components and the Simple PCSA (Polarizer,
Compensator, Sample, Analyzer) Ellipsometer ............... 299
4.1. General ............................................. 299
4.2. The Components ...................................... 301
4.2.1. Methods of Obtaining Polarized Light ........ 301
4.2.2. Double Refraction ........................... 302
4.2.3. Calcite Crystals ............................ 303
4.2.4. Polarizers and Analyzers .................... 305
4.2.5. Wollaston Prisms ............................ 307
4.2.6. Compensators, Quarter-Wave Plates, and
Retarders ................................... 308
4.2.7. Photoelastic Modulators ..................... 316
4.2.8. Monochromators .............................. 317
4.2.9. Goniometers ................................. 321
4.3. Ellipsometer Component Configurations ............... 322
4.3.1. Early Null Ellipsometer Configurations ...... 322
4.3.2. Photometric Ellipsometer Configurations ..... 323
4.3.3. Spectroscopic Ellipsometers ................. 324
4.3.4. Other Configurations ........................ 326
4.4. References .......................................... 327
5. Rotating Polarizer and Analyzer Ellipsometry .............. 329
5.1. Introduction ........................................ 329
5.2. Comparison of Ellipsometers ......................... 333
5.3. Instrumentation Issues .............................. 343
5.3.1. Optical Configuration ....................... 343
5.3.2. Optical Components and Spectral Range ....... 345
5.3.3. Alignment ................................... 351
5.3.4. Electronic Design and Components ............ 356
5.4. Data Reduction for the Rotating Polarizer and
Analyzer Ellipsometers .............................. 364
5.4.1. Ideal PXSAr Configuration ................... 364
5.4.2. Errors in the PXSAr Configuration ........... 371
5.4.3. PrXSA Configuration ......................... 378
5.5. Precision Considerations ............................ 386
5.6. Calibration Procedures .............................. 392
5.6.1. Ideal Rotating Polarizer and Analyzer
Ellipsometers ............................... 394
5.6.2. Detecting and Correcting Errors in
Calibration ................................. 407
5.6.3. Detecting and Correcting Compensator
Errors ...................................... 423
5.7. Summary: Recent and Future Directions ............... 425
5.8. References .......................................... 429
6. Polarization Modulation Ellipsometry ...................... 433
6.1. Introduction ........................................ 433
6.2. The Photoelastic Modulator (PEM) .................... 436
6.2.1. General Description and Historical
Perspective ................................. 436
6.2.2. Mathematical Description of a PEM ........... 440
6.2.3. Stokes Vector Descriptions of the PSG
and PSA ..................................... 442
6.3. Experimental Configurations of Polarization
Modulation Ellipsometers ............................ 446
6.3.1. Polarization Modulation Ellipsometry (PME)
with Analog Data Acquisition ................ 446
6.3.2. Phase Modulated Ellipsometry (PME) with
Digital Data Acquisition .................... 447
6.3.3. Two-Channel Spectroscopic Polarization
Modulation Ellipsometer (2-C SPME) .......... 449
6.3.4. Two-Modulator Generalized Ellipsometer
(2-MGE) ..................................... 450
6.4. Light Intensity Through a Polarization Modulation
Ellipsometer ........................................ 452
6.4.1. Mueller Matrices for Various Samples ........ 452
6.4.2. Intensity for a Standard PME ................ 455
6.4.3. Intensity for the 2-Modulator Generalized
Ellipsometer (2-MGE) ........................ 457
6.5. Waveform Analysis ................................... 461
6.5.1. Basis Function .............................. 463
6.5.2. Phase-Sensitive Detection ................... 465
6.5.3. Digital Waveform Analysis ................... 466
6.5.4. Two-Modulator Systems ....................... 467
6.6. Calibration Procedures .............................. 469
6.6.1. One-Modulator PMEs .......................... 470
6.6.2. Two-Modulator PMEs .......................... 472
6.7. Errors .............................................. 474
6.7.1. General Discussion .......................... 474
6.7.2. Systematic Errors of PMEs ................... 475
6.8. Further Reading and References ...................... 479
6.8.1. Further Reading ............................. 479
6.8.2. Numbered References ......................... 479
7. Multichannel Eilipsometry ................................. 481
7.1. Introduction ........................................ 481
7.2. Overview of Instrumentation ......................... 483
7.2.1. Self-Compensating Designs ................... 483
7.2.2. Rotating-Element Designs .................... 487
7.2.3. Phase-Modulation Designs .................... 491
7.2.4. Design Comparisons .......................... 493
7.2.5. Errors Unique to Multichannel Detection
Systems ..................................... 497
7.3. Rotating-Element Designs ............................ 502
7.3.1. Rotating Polarizer .......................... 502
7.3.2. Single Rotating Compensator ................. 523
7.3.3. Dual Rotating Compensator ................... 546
7.4. Concluding Remarks .................................. 562
Acknowledgements .................................... 564
7.5. References .......................................... 564
Part 3: Critical Reviews of Some Applications ................. 567
8. SiO2 Films ................................................ 569
8.1. Introduction 569
8.1.1. Preeminence of SiO2 in Microelectronics:
the Eilipsometry Connection ................. 569
8.1.2. Electronic Passivation ...................... 570
8.1.3. Properties of SiO2 Films .................... 571
8.2. Historical Perspective - Prior to 1970 .............. 578
8.3. Modern Studies - Since 1970 ......................... 585
8.3.1. Thick SiO2 Films ............................ 585
8.3.2. Thin SiO2 Films ............................. 599
8.3.3. Recent Results on Ultra Thin SiO2 Films
and the Si-SiO2 Interface ................... 619
8.4. Conclusions ......................................... 632
Acknowledgements .................................... 633
8.5. References .......................................... 633
9. Theory and Application of Generalized Ellipsometry ........ 637
9.1. Introduction ........................................ 637
9.2. The Generalized Ellipsometry Concept ................ 638
9.2.1. Comments on Notations in GE ................. 638
9.2.2. The Optical Jones Matrix .................... 640
9.2.3. The Generalized Ellipsometry Parameters ..... 643
9.2.4. Generalized Ellipsometry Acquisition
Techniques .................................. 647
9.3. Theory of Generalized Ellipsometry .................. 650
9.3.1. Birefringence in Stratified Media ........... 650
9.3.2. 4X4 Maxwell's Equations in Matrix Form ...... 652
9.3.3. Transmission and Reflection GE .............. 656
9.4. Special Generalized Ellipsometry Solutions .......... 657
9.4.1. Biaxial Films (Symmetrically Dielectric
Materials) .................................. 657
9.4.2. Bi-Biaxial or Magneto-Optical Films
(Non-Symmetrically Dielectric Materials) .... 661
9.4.3. Chiral Biaxial Films (Axially Twisted
Symmetrically Dielectric Materials) ......... 663
9.4.4. Isotropic Dielectric Films .................. 669
9.4.5. Further Solutions: [111] Superlattice
Ordering in III-V Compounds
(CuPt-Ordering) ............................. 671
9.5. Strategies in Generalized Ellipsometry .............. 675
9.5.1. Data Acquisition Strategies for
Anisotropic Samples ......................... 676
9.5.2. Strategies for Treatment of Sample
Backside Effects ............................ 679
9.5.3. Model Strategies ............................ 682
9.6. Generalized Ellipsometry Applications ............... 683
9.6.1. Anisotropic Bulk Materials .................. 684
9.6.2. Anisotropic Films ........................... 693
9.7. Conclusions ......................................... 710
Acknowledgements .................................... 710
9.8. Further Reading and References ...................... 711
9.8.1. General Reading ............................. 711
9.8.2. Numbered References ......................... 712
Part 4: Emerging Areas in EUipsometry ......................... 719
10. VUV Ellipsometry .......................................... 721
10.1. Introduction ........................................ 721
10.2. Historical Review of Short Wavelength
Ellipsometry ........................................ 722
10.2.1. BESSY Ellipsometer .......................... 722
10.2.2. EUV Ellipsometer ............................ 724
10.3. VUV EUipsometry Today ............................... 726
10.3.1. Current VUV Instrumentation ................. 726
10.4. Importance of VUV Ellipsometry ...................... 732
10.5. Survey of Applications .............................. 737
10.5.1. Lithography ................................. 740
10.5.2. Gate Dielectrics ............................ 748
10.5.3. High-energy Optical Constants ............... 749
10.6. Future of VUV Ellipsometry .......................... 757
10.7. Acknowledgments ..................................... 757
10.8. References .......................................... 757
11. Spectroscopic Infrared Ellipsometry ....................... 763
11.1. Experimental Tools .................................. 763
11.1.1. Two Kinds of Instruments .................... 763
11.1.2. Optical Equipment for the
Infrared-EIlipsometry ....................... 768
11.1.3. The Degree of Polarization .................. 771
11.1.4. Linearity of the Detection System ........... 775
11.1.5. Infrared Synchrotron Radiation .............. 775
11.2. Applications ........................................ 776
11.2.1. Optics of Absorbing Media ................... 776
11.2.2. Vibration Modes - the Concept of Weak and
Strong Oscillators .......................... 778
11.2.3. Inversion of Infrared Ellipsometric
Measurements ................................ 781
11.2.4. Anisotropy Features in the Infrared
Ellipsometric Spectra ....................... 786
11.3. References .......................................... 797
12. Ellipsometry in Life Sciences ............................. 799
Poem and Dedication ....................................... 799
12.1. Introduction ........................................ 800
12.2. Historical Background ............................... 802
12.3. The Interfaces Under Study .......................... 802
12.4. From Optics to Biology .............................. 804
12.4.1. The Unique Possibilities .................... 804
12.4.2. Verification of Ellipsometric Results ....... 805
12.5. Methodology for Data Evaluation - from ψ and Δ to
Biologically Related Parameters ..................... 806
12.5.1. A Thin Biolayer on a Flat Ideal Substrate ... 806
12.5.2. A Thick Biolayer on a Flat Ideal
Substrate ................................... 817
12.5.3. Adsorption of Biomolecules into Porous
Structures .................................. 817
12.5.4. Surface Roughness ........................... 819
12.5.5. Use of Dispersion Models .................... 820
12.5.6. Anisotropy .................................. 820
12.6. Methodology - Experimental .......................... 821
12.6.1. Instrumentation ............................. 821
12.6.2. Cell Designs ................................ 822
12.6.3. In situ Considerations for Biological
Interfaces .................................. 824
12.6.4. Some Model Surfaces ......................... 825
12.6.5. Studies on Real Biological Surfaces ......... 827
12.6.6. Complementary and Independent
Information ................................. 828
12.6.7. Experimental Design ......................... 828
12.7. Applications ........................................ 829
12.7.1. Introduction ................................ 829
12.7.2. Adsorption of Biomolecules to Model
Surfaces .................................... 830
12.7.3. Spectroscopy ................................ 839
12.7.4. Imaging ..................................... 841
12.7.5. Biological Surfaces ......................... 843
12.7.6. Biosensors Based on Ellipsometric Readout ... 844
12.7.7. Engineering Applications .................... 845
12.8. Outlook ............................................. 846
Acknowledgements .................................... 847
12.9. References .......................................... 847
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